a quadratic programming bibliography · 2001-02-27 · a quadratic programming bibliography n. n....

A Quadratic Programming Bibliography

NicholasI. M. Gould

ComputationalScienceandEngineeringDepartment

RutherfordAppletonLaboratory, Chilton

Oxfordshire,OX11 0QX, England,EU

Email : [email protected]

and

PhilippeL. Toint

Departmentof Mathematics,Universityof Namur

61, ruedeBruxelles,B-5000Namur, Belgium,EU.

Email : [email protected]

RAL NumericalAnalysisGroupInternalReport2000–1

Thisversion:February27,2001

Thefollowing is alist of all of thepublishedandunpublishedworksonquadraticprogramming

thatweareawareof. Somearegeneralreferencesto backgroundmaterial,while othersarecen-

tral to thedevelopmentof thequadraticprogrammingmethodsandto theapplicationswe in-

tendto coverin ourevolvingbookonthesubject.Wehavedeliberatelynotincludedany but the

mostrelevantof thehundreds,if not thousands,of citationsto sequential/successive/recursive

quadraticprogrammingmethodsfor nonlinearprogramming,nor to thoseon linearprogram-

ming or quadraticprogrammingwith quadraticconstraints.ThecompleteLATEX bibliography,

togetherwith up-to-dateadditions,is availableonlineat

ftp://ftp.numerical.rl.ac.uk/pub/qpbook/qpbook.bib

and

ftp://thales.math.fundp.ac.be/pub/qpbook/qpbook.bib .

We would bedelightedto receiveany correctionsor updatesto this list.

A Quadratic Programming Bibliography

N. N. Abdelmalek. Restorationof imageswith missinghigh-frequency componentsusingquadraticprogramming.AppliedOptics, 22(14),2182–2188,1983.

Abstract. A methodfor restoringan optical imagewhich is subjectedto low-passfrequency filtering is

presented.It is assumedthat the objectwhoseimageis restoredis of finite spatialextent.The problemis

treatedasanalgebraicimage-restorationproblemwhich is thensolvedasaquadraticprogrammingproblem

with boundedvariables.The regularizationtechniquefor the ill-posedsystemis to replacethe consistent

systemof thequadraticprogrammingproblemby anapproximatesystemof smallerrank.Therank which

givesa bestor near-bestsolutionis estimated.This methodis a novel one,andit comparesfavorablywith

otherknown methods.Computer-simulatedexamplesarepresented.Commentsandconclusionsaregiven.

N. N. AbdelmalekandT. Kasvand. Digital imagerestorationusingquadraticprogramming.AppliedOptics, 19(19),3407–3415,1980.

Abstract. The problemof digital imagerestorationis consideredby obtainingan approximatesolutionto

the Fredholmintegral equationof the first kind in two variables.The systemof linear equationsresulting

from thediscretizationof the integral equationis convertedto a consistentsystemof linearequations.The

problemis thensolved asa quadraticprogrammingproblemwith boundedvariableswherethe unknown

solutionis minimizedin theL2 norm.In thismethodminimumcomputerstorageis needed,andtherepeated

solutionsareobtainedin anefficient way. Also therankof theconsistentsystemwhich givesa bestor near

bestsolutionis estimated.Computersimulatedexamplesusingspatiallyseparablepointspreadfunctionsare

presented.Commentsandconclusionsaregiven.

R. A. Abramsand A. Ben Israel. A duality theoremfor complex quadraticprogramming.Journalof OptimizationTheoryandApplications, 4(4), 245–252,1969.

Abstract. A duality theoryfor complex quadraticprogrammingover polyhedralconesis developed,follow-

ing Dorn,by usinglinearduality theory.

J. W. Adams. Quadraticprogrammingapproachesto new optimal windows andantennaar-rays. ConferenceRecord. TwentyFourth AsilomarConferenceon Signals,SystemsandComputersMaplePress,SanJose, CA,USA, 1, 69–72,1990a.

Abstract. New window designproblemsareformulatedin termsof quadraticprogramming.Thenew win-

dowspermitthedesignerto controlthetradeoff betweenthepeaksidelobelevel andthetotalsidelobeenergy.

In addition,linearconstraintscanbeimposedon thedesignproblem.Theproposedmethodsareapplicable

to applicationsin thefieldsof signalprocessingandantennaarrays.

J.W. Adams.Quadraticprogrammingapproachesto new problemsin digital filter design.Con-ferenceRecord. TwentyFourthAsilomarConferenceonSignals,SystemsandComputersMaplePress,SanJose, CA,USA, 1, 307–310,1990b.

Abstract. New digital filter designproblemsareformulatedin termsof quadraticprogramming.The new

filters permit the designerto control the tradeoff betweenthe peakerrorsand the total squarederrors.In

2 A QUADRATIC PROGRAMMING BIBLIOGRAPHY

particular, themean-squarederrorcanbeminimizedsubjectto peakerrorsconstraints,asrequiredin many

practicaldesignproblems.In addition,equalityconstraintscanbeimposedfor specialapplications.

J. W. Adams,P. Kruethong,R. Hashemi,J. L. Sullivan, andD. R. Gleeson. New quadraticprogrammingalgorithmsfor designingFIR digital filters. ConferenceRecord of TheTwentySeventhAsilomarConferenceonSignals,SystemsandComputers.IEEEComput.SocPress,LosAlamitos,CA,USA, 2, 1206–1210,1993.

Abstract. TheParks-McClellanalgorithm(1973)is verypopularfor designingFIR digital filters. It is based

on a linear programmingalgorithmcalledthe Remezexchange.Our new algorithmis basedon quadratic

programming,which includeslinear programmingas a specialcase.The filters in this paperpermit the

designerto controlthetradeoff betweenthepeakerrorandthetotal squarederror. Thesefilters aredesigned

accordingto thepeak-constrainedleast-squares(PCLS)optimalitycriterion.

J. W. Adams,J. L. Sullivan, D. R. Gleeson,P. H. Chang,andR. Hashemi. Application ofquadraticprogrammingto FIR digital filter designproblems.ConferenceRecord of theTwentyEighthAsilomarConferenceon Signals,SystemsandComputers. IEEE Comput.SocPress,LosAlamitos,CA,USA, 1, 314–318,1994.

Abstract. Quadraticprogrammingproblemshavelongbeenof interestin thebusinesscommunity. Quadratic

programmingis often usedasthe basisfor ”programtrading” wherestocksareautomaticallyboughtand

soldby mutual fundsto optimizeprofits.Quadraticprogrammingalgorithmscanalsobe usedto optimize

digital filters, asdiscussedin this paper. We presentthe generalizedmultiple exchange(GME), simplified

generalizedmultipleexchange(SGME)andmodifiedgeneralizedmultipleexchange(MGME) algorithmsfor

designingconstrainedleast-squares(CLS)filters.TheCLSfiltersaregeneralizationsof thepopularminimax

andleast-squaresfilters.TheCLS filters areimportantnot only becauseof their generality, but alsobecause

they areneededfor many practicalapplications.

W. P. AdamsandP. M. Dearing. On the equivalencebetweenroof duality andLagrangianduality for unconstrained0–1quadraticprogrammingproblems.DiscreteAppliedMath-ematics, 48(1), 1–20,1994.

Abstract. Considerstechniquesfor computingupperboundson theoptimalobjective functionvalueto any

unconstrained0–1 quadraticprogrammingproblem(maximization).In particular, the authorsstudy three

methodsfor obtainingupperboundsaspresentedin arecentpaperby Hammer, Hansen,andSimeone(1984)

(1) generatingtwo classesof upper-boundinglinearfunctionsreferredto aspavedupperplanesandroofs,(2)

solving thecontinuousrelaxationof a mixed-integer linear problemby Rhys(1970),and(3) the quadratic

complementationof variableswhich resultsin a boundcalled the height.The authorsshow that all three

methodsdirectly result from standardpropertiesof a reformulationof the quadraticproblemasa mixed-

integer linear program,with methods(1) and (3) resultingfrom a Lagrangiandual of this reformulation.

Basedon this reformulation,they expanduponthepublishedresults.

W. P. Adamsand H. D. Sherali. A tight linearizationand an algorithm for 0–1 quadraticprogrammingproblems.ManagementScience, 32(10),1274–1290,1986.

Abstract. Thepaperis concernedwith thesolutionof linearlyconstrained0–1quadraticprogrammingprob-

lems.Problemsof thiskind arisein numerouseconomic,locationdecision,andstrategic planningsituations,

includingcapitalbudgeting,facility location,quadraticassignment,mediaselection,anddynamicsetcov-

ering.A new linearizationtechniqueis presentedfor this problemwhich is demonstratedto yield a tighter

continuousor linearprogrammingrelaxationthanis availablethroughothermethods.An implicit enumera-

tion algorithmwhichusesLagrangianrelaxation,Benders’cuttingplanes,andlocalexplorationsis designed

to exploit thestrengthof this linearization.Computationalexperienceis providedto demonstratetheuseful-

nessof theproposedlinearizationandalgorithm.

S.N. Afriat. Thequadraticform definiteon a manifold. Proceedingsof theCambridge Philo-

sophicalSociety, 47, 1–6,1951.

N. I. M. GOULD & PH.L. TOINT 3

A. Aggarwal andC. A. Floudas.A decompositionapproachfor globaloptimumsearchin QP,NLP andMINLP problems.Annalsof OperationsResearch, 25(1–4),119–145,1990.

Abstract. A new approachfor global optimumsearchis presentedwhich involvesa decompositionof the

variableset into two sets-complicatingandnoncomplicatingvariables.This resultsin a decompositionof

the constraintset leadingto two subproblems.The decompositionof the original probleminducesspecial

structurein the resultingsubproblemsanda seriesof thesesubproblemsarethensolved, usingthe gener-

alisedBenders’decompositiontechnique,to determinetheoptimalsolution.Mathematicalpropertiesof the

proposedapproacharepresented.Even thoughthe proposedapproachcannotguaranteethe determination

of theglobaloptimum,computationalexperienceon anumberof nonconvex QP, NLP andMINLP example

problemsindicatesthataglobaloptimumsolutioncanbeobtainedfrom variousstartingpoints.

F. G. Akhmadov. Computationalmethodfor solving the quadraticprogrammingproblemin Ln

2

0 T space. IzvestiyaAkademiiNauk Azerbaidzhanskoi SSR,Seriya Fiziko–

Tekhnicheskikhi MatematicheskikhNauk, 4(4), 102–106,1983.

Abstract. A methodfor solving the quadraticprogrammingproblemin the spaceof all vector-functions,

eachcomponentsquareof which is integrable,is given.It is supposedthat thematrix of thequadraticform

is positively defined.

F. A. Al-Khayyal. Linear, quadraticandbilinearprogrammingapproachesto the linearcom-

plementarityproblems.EuropeanJournalof OperationsResearch, 24, 216–227,1987.

F. A. Al-Khayyal. Jointly constrainedbilinearprogramsandrelatedproblems:An overview.

Computers in MathematicalApplications, 19, 53–62,1990.

F. A. Al-Khayyal andJ. E. Falk. Jointly constrainedbiconvex programming.Mathematicsof

OperationsResearch, 8, 273–286,1983.

C. AlessandriandA. Tralli. Frictionlesscontactwith BEM usingquadraticprogramming—

discussion.Journalof EngineeringMechanics-ASCE, 119(12),2538–2540,1993.

B. Alidaee, G. A. Kochenberger, andA. Ahmadian. 0–1 quadraticprogrammingapproachfor optimum solutionsof two schedulingproblems. International Journal of SystemsScience, 25(2), 401–408,1994.

Abstract. Two schedulingproblemsareconsidered:(1) schedulingn jobsnon-preemptively onasinglema-

chineto minimizetotal weightedearlinessandtardiness(WET); (2) schedulingn jobsnon-preemptively on

two parallelidenticalprocessorstominimizeweightedmeanflow time.In thesecondproblem,apre-ordering

of the jobs is assumedthatmustbe satisfiedfor any setof jobs scheduledon eachspecificmachine.Both

problemsareknown to be NP-complete.A 0–1 quadraticassignmentformulationof the problemsis pre-

sented.An equivalent0–1mixedinteger linearprogrammingapproachfor theproblemsareconsideredand

a numericalexampleis given.Theformulationspresentedenableoneto useoptimalandheuristicavailable

algorithmsof 0–1quadraticassignmentfor theproblemsconsideredhere.

K. S. Alsultan and K. G. Murty. Exterior point algorithmsfor nearestpoints and convex

quadraticprograms.MathematicalProgramming, 57(2), 145–161,1992.

A. Altman. QHOPDM—ahigher-orderprimal-dualmethodfor large-scaleconvex quadratic

programming.EuropeanJournalof OperationalResearch, 87(1), 200–202,1995.

A. Altman. Higher order primal-dual interior point methodfor separableconvex quadratic

optimization.Control andCybernetics, 25(4), 761–772,1996.


A. AltmanandJ.Gondzio.Regularizedsymmetricindefinitesystemsin interiorpointmethodsfor linearandquadraticoptimization. Logilab TechnicalReport1998.6,DepartmentofManagementSciences,Universityof Geneva,Geneva,Switzerland,1998.

Abstract. This paperpresentslinear algebratechniquesusedin the implementationof an interior point

methodfor solving linearprogramsandconvex quadraticprogramswith linearconstraints.New regulariza-

tion techniquesfor Newton systemsapplicableto bothsymmetricpositive definiteandsymmetricindefinite

systemsaredescribed.They transformthe latter to quasidefinitesystemsknown to bestronglyfactorizable

to a form of Cholesky-like factorization.Two differentregularizationtechniques,primal anddual,arevery

well suitedto the(infeasible)primal-dualinterior point algorithm.This particularalgorithm,with anexten-

sionof multiplecentralitycorrectors,is implementedin oursolverHOPDM.Computationalresultsaregiven

to illustratethe potentialadvantagesof the approachwhenappliedto the solutionof very large linear and

convex quadraticprograms.

H. Amato and G. Mensch. Rank restrictionson the quadraticform in indefinite quadraticprogramming.Unternehmensforschung, 15(3), 214–216,1971.

Abstract. A quadraticprogrammingproblem,whereq x aTx xT Qx is anindefiniteobjective function,

canbesolvedwith Swarup’s approachto optimizing cTx α dT x β only if therankof Q is two; if Q is

definite,therankof Q mustbeone.

D. E. AmosandM. L. Slater. Polynomialandsplineapproximationby quadraticprogramming.Communicationsof theACM, 12(7),379–381,1969.Seealso,CollectedAlgorithmsfromACM, 1985.

Abstract. The problemof approximationto a given function, or of fitting a given setof data,wherethe

approximatingfunctionis requiredto havecertainof its derivativesof specifiedsignover thewholerangeof

approximation,is studied.Two approachesarepresented,in eachof which quadraticprogrammingis used

to provide both theconstraintson thederivativesandtheselectionof thefunctionwhich yields thebestfit.

Thefirst is a modifiedBernsteinpolynomialscheme,andthesecondis asplinefit.

P. Anand. Decompositionprinciplefor indefinitequadraticprograme.TrabajosdeEstadistica

y deInvestigacion, 23, 61–71,1972.

S. C. Anand,F. E. Weisgerber, andH. S. Hwei. Direct solutionvs quadraticprogrammingtechniquein elastic-plasticfinite elementanalysis.ComputersandStructures, 7(2), 221–228,1977.

Abstract. Elastic-plasticplanestressfinite elementanalysisof adiskrolling onarigid trackis performedby

thedirectmethodaswell asthequadraticprogrammingtechnique.Trescaandvon Mises’ yield conditions

areusedin the former whereasanapproximatepiecewise linear Trescayield conditionis usedin the later

case.It is concludedfrom acomparisonof thecomputertimesneededin thetwo casesthatthedirectmethod

is far superiorto thequadraticprogrammingtechnique.

A. Anckonie. A quadraticprogramfor determiningefficient frontier portfolio compositions

using the SAS language. In ‘SUGI 10—Proceedingsof the TenthAnnual SAS Users

GroupInternationalConference’,Vol. 15,pp.55–60,1985.

W. Anfilof f. Gravity interpretationwith the aid of quadraticprogramming. Geophysics,

46(3), 340–341,1981.

P. L. DeAngelis,P. M. Pardalos,andG. Toraldo.Quadraticprogrammingwith boxconstraints.

In I. M. Bomze,ed., ‘Developmentsin Global optimization’, pp. 73–93,Kluwer Aca-

demicPublishers,Dordrecht,TheNetherlands,1997.


K. M. Anstreicher. On long steppathfollowing andSUMT for linearandquadraticprogram-ming. SIAMJournalon Optimization, 6(1), 33–46,1996.

Abstract. We considera long stepbarrieralgorithmfor the minimizationof a convex quadraticobjective

subjectto linear inequality constraints.The algorithm is a dual versionof a methoddevelopedby K. M.

Anstreicheret al. (1993),andrequiresO nL or O sqrtnL iterationsto solve a problemwith n constraints,

dependingon how the barrier parameteris reduced.As a corollary of our analysiswe demonstratethat

the classicalSUMT algorithm, exactly as implementedin 1968, solves linear andquadraticprogramsin

O nLlogL iterations,with properinitializationandchoiceof parameters.

K. M. Anstreicher. On theequivalenceof convex programmingboundsfor booleanquadraticprogramming.Working paper, Departmentof ManagementScience,Universityof Iowa,IowaCity, USA, 1997.

Abstract. Recentpapershave shown theequivalenceof several tractableboundsfor Booleanquadraticpro-

gramming.In this note we give simplified proofs for theseresults,andalso show that all of the bounds

consideredaresimultaneouslyattainedby onediagonalperturbationof thequadraticform.

K. M. AnstreicherandN. W. Brixius. Solving quadraticassignmentproblemsusingconvexquadraticprogrammingrelaxations. Working paper, Departmentof ManagementSci-ence,Universityof Iowa,IowaCity, USA, 2000.

Abstract. Wedescribeabranch-and-boundalgorithmfor thequadraticassignmentproblem(QAP)thatuses

a convex quadraticprogramming(QP) relaxationto obtain a boundat eachnode.The QP subproblems

areapproximatelysolved using the Frank-Wolfe algorithm, which in this caserequiresthe solution of a

linearassignmentproblemoneachiteration.Ourbranchingstrategy makesextensiveuseof dualinformation

associatedwith the QP subproblems.We obtainstate-of-the-artcomputationalresultson large benchmark

QAPs.

K. M. AnstreicherandN. W. Brixius. A new boundfor thequadraticassignmentproblembasedon convex quadraticprogramming.MathematicalProgramming, 89(3), 341–357,2001.

Abstract. We describea new convex quadraticprogrammingboundfor the quadraticassignmentproblem

(QAP).Theconstructionof theboundusesasemidefiniteprogrammingrepresentationof abasiceigenvalue

boundfor QAP. The new bounddominatesthe well-known projectedeigenvaluebound,andappearsto be

competitive with existing boundsin thetrade-off betweenboundqualityandcomputationaleffort.

K. M. Anstreicher, D. denHertog,C. Roos,andT. Terlaky. A long-stepbarriermethodforconvex quadraticprogramming.Algorithmica, 10(5), 365–382,1993.

Abstract. In this paperwe proposea long-steplogarithmicbarrier function methodfor convex quadratic

programmingwith linear equalityconstraints.After a reductionof the barrierparameter, a seriesof long

stepsalongprojectedNewton directionsaretakenuntil the iterateis in thevicinity of thecenterassociated

with the currentvalueof the barrierparameter. We prove that the total numberof iterationsis O nL or

O nL , dependingonhow thebarrierparameteris updated.

K. Aoki andT. Fujikawa. VAR planningandnonconvex quadraticprogramming.Transactionsof theInstituteof ElectricalEngineersof Japan, 100(3), 78–88,1980.

Abstract. So far, many methodsbasedon linear programming,nonlinearprogrammingand integer pro-

gramminghavebeenproposedfor varplanning.In thelinearprogrammingmethod,therelationshipbetween

voltage,active power and reactive power is representedby linear equations.The nonlinearprogramming

method,in which this relationshipis expressedexactly, is not suitedfor a large-scalepower system.Both

thesemethodsneglect discretevariablesrepresentingthe numberof the condenseror reactorunitsandthe

transformertap positions.The integer programmingmethod,which enableshandlingthesediscretevari-

ables,is of coursenot suitedfor a large-scalepower system.Theauthorsshow that thecondenserplanning

problemis formulatedinto aparametricconvex quadraticprogrammingandthereactorplanningproblemis

formulatedinto aparametricnonconvex quadraticprogramming.


K. Aoki and M. Kanezashi. A decompositionalgorithm for a dual angulartype quadraticprogramming.In A. Lew, ed.,‘Proceedingsof the6th Hawaii InternationalConferenceon SystemsSciences.WesternPeriodicals,North Hollywood,CA, USA’, pp. 358–360,1973.

Abstract. This paperdealswith a decompositionprocedurefor the problemwhoseobjective function is

linear for a couplingvariable.Moreover it is describedby takingadvantageof suchcharacteristicsthatone

caneasilyobtainthevariationof a couplingvariableby computinglinearforms.

K. Aoki andT. Satoh.Economicdispatchwith network securityconstraintsusingparametricquadraticprogramming. IEEE Transactionson Power Apparatusand Systems, PAS-101(12),4548–4556,1982.

Abstract. Thispaperpresentsanefficient methodto solveaneconomicloaddispatchproblemwith DC load

flow type network securityconstraints.The conventionallinear programmingandquadraticprogramming

methodscannotdealwith transmissionlossesasaquadraticform of generatoroutputs.In orderto overcome

this defect, the extensionof the quadraticprogrammingmethodis proposed,which is designatedas the

parametricquadraticprogrammingmethod.The upperboundingtechniqueandthe relaxationmethodare

coupledwith theproposedmethodfor thepurposeof computationalefficiency. Thetestresultsshow thatthe

proposedmethodis practicalfor real-timeapplications.

M. Arioli. The useof QR factorizationin sparsequadraticprogrammingandbackwarderrorissues.SIAMJournalon Matrix AnalysisandApplications, 21(3), 825–839,2000.

Abstract. We presenta roundoff erroranalysisof a null spacemethodfor solving quadraticprogramming

minimizationproblems.Thismethodcombinestheuseof adirectQRfactorizationof theconstraintswith an

iterative solver on thecorrespondingnull space.Numericalexperimentsarepresentedwhich give evidence

of thegoodperformancesof thealgorithmonsparsematrices.

B. ArmstrongandB. A. Wade. Nonlinearpid control with partial stateknowledge: designby quadraticprogramming.In ‘Proceedingsof the2000AmericanControlConference,Danvers,MA, USA’, Vol. 2, pp.774–778,2000.

Abstract. NonlinearPID (NPID)controlis implementedbyallowing thecontrollergainsto varyasafunction

of systemstate.NPID controllerswill in generaldependon knowledgeof thefull statevector. In this work,

NPID controllerswhichoperatewithout knowledgeof somestatevariablesaredemonstrated.A generalbut

conservative designmethodis presentedwith anexperimentaldemonstration.For a specialcase,complete

necessaryandsufficient conditionsareestablished.

D. A. Arsamastsev, P. I. Bartolomey, and S. K. Okulovski. New improved quadratic-

programmingmethodsfor super-large power-systemsanalysis. In ‘Proceedingsof the

EighthPowerSystemsComputationConference’,Vol. 178,pp.710–716,1984.

L. ArseneauandM. J. Best. Resolutionof degeneratecritical parametervaluesin parametric

quadraticprogramming.TechnicalReportCORR99-47,Departmentof Combinatorics

andOptimization,Universityof Waterloo,Ontario,Canada,1999.

J. Atkociunas.Quadraticprogrammingfor degenerateshakedown problemsof barstructures.

MechanicsResearch Communications, 23(2), 195–203,1996.

A. M. Aurela and J. J. Torsti. A quadraticprogrammingmethodfor stabilizedsolution ofunstablelinearsystems.AnnalesUniversitatisTurkuensis,SerAI (AstronomicaChemicaPhysicaMathematica), 123, 1968.


Abstract. An improved versionis presentedof thequadraticprogrammingmethodintroducedin 1967for

stabilizedsolutionof unstablesystemsof linear equations.The mostprobablesolutionandits confidence

limits arediscussed.In thepresentwork, themethodproperwasexaminedmoresystematicallyby usingthe

secondartificial exampleof Phillips (1962),in which thecorrectresultwasknown. Thedependenceof the

computingtimet onthenumberof variablesadjustedsimultaneously, l, wasstudied,takingatotalof m 15

variables,subjectto thenonnegativity constraint.Theoptimumwasachievedwith l 3 to 6 (final precision

ε 1 105, t 3 min in theIBM 1130).Thedifferentsolutionsexhibitedmarkedconsistency with eachother,

indicatingtheaccuracy andreliability of themethod

G. Auxenfants,L. Barthe,andP. Gibert. Architecturefor scientificsoftware.II. Analysisof aquadraticprogrammingalgorithm.RechercheAerospatiale, 4, 247–255,1982.

Abstract. Presentsa quadraticprogrammingalgorithmwith linearconstraints,working in thecaseof large-

scaleoptimizationproblems.The numberof variablesis reducedby a partial dualizationof constraintre-

lations.It enablesoneto determinewhetheror not theadmissiblesetis empty. Theprogramminghasbeen

implementedon a CYBER 170/750usinga methodof architecturebasedon (1) datacentralizationand(2)

managementof informationexchangebetweenprocessorsby databasemanagementsystem.This algorithm

representsoneof theelementsof amoreoptimizationcode.

T. Aykin. Onaquadraticinteger-programfor thelocationof interactinghubfacilities.European

Journalof OperationalResearch, 46(3), 409–411,1990.

A. BachemandB. Korte.An algorithmfor quadraticoptimizationovertransportationpolytopes.

Zeitschrift fur AngewandteMathematikundMechanik, 58, 459–461,1978.

R. BacherandH. P. vanMeeteren.Securitydispatchbasedon couplingof linearandquadraticprogrammingtechniques. Power Systems,Modelling and Control Applications.IFACSymposiumPergamon,Oxford, England, pp.211–217,1989.

Abstract. Securitydispatchcan be definedas the real-timeclosedloop cost-optimalallocationof active

generatoroutputwhile consideringbranchflow limits of theintactnetwork andlower anduppergeneration

limits. MostOPFalgorithmsfail to guaranteeaccuracy. reliability andspeedatthesametimeandcanthusnot

beusedin real-timeclosedloop application.Accuracy, reliability andspeedcanbe obtainedby executing

a LP basedOPF and a QP basedconstrainedeconomicdispatchat different executionfrequencies.The

QP basedalgorithmusesthe critical constraintsetasdeterminedby the LP basedalgorithm.Constrained

economicdispatchcansubstitutetheclassicaleconomicdispatchandwill provide asecuredispatch.

W. E. Baethgen,D. B. Taylor, andM. M. Alley. Quadratic-programmingmethodfor deter-

mining optimum nitrogenratesfor winter-wheatduring tillering. AgronomyJournal,

81(4), 557–559,1989.

A. BagchiandB. Kalantari. A methodfor computingapproximatesolutionof thetrust region

problemwith applicationto projectivemethodsfor quadraticprogramming.Workingpa-

per, Departmentof ComputerScience,RutgersUniversity, New Brunswick,New Jersey,

USA, 1988.

J. R. Baker. Determinationof an optimal forecastmodel for ambulancedemandusinggoal

andquadraticprogramming.In ‘Proceedings—SoutheasternChapterof the Instituteof

ManagementSciencesTwentiethAnnualMeeting’,Vol. 6, pp.154–157,1984.

E. Balas. Duality in discreteprogramming: II. The quadraticcase. ManagementScience,

16, 14–32,1969.


E. Balas.Nonconvex quadraticprogrammingvia generalizedpolars.SIAMJournalonAppliedMathematics, 28(2), 335–349,1975.

Abstract. A new approachis proposedto linearly constrainednonconvex quadraticprogramming.Theap-

proachis basedon generalizedpolar sets,andis akin to the convex analysisapproachto integer program-

ming.Theauthorconstructsageneralizedpolarof theKuhn-Tucker polyhedronassociatedwith a quadratic

program.This generalizedpolaris aconvex polyhedralconewhoseinterior containsnocomplementaryfea-

sible solutionbetterthanthe bestknown one.An algorithmis thenproposed,which doesnot usecutting

planes,but constructsapolytopecontainingthefeasiblesetandcontainedin thepolarof thelatter. Thebest

complementarysolutionfoundin theprocessis optimal,or noneexists.

C. C. Baniotopoulos,K. M. Abdalla, and P. D. Panagiotopoulos.A variational inequalityandquadraticprogrammingapproachto theseparationproblemof steelboltedbrackets.ComputersandStructures, 53(4), 983–991,1994.

Abstract. A variationalinequalityandquadraticprogrammingapproachis proposedfor theinvestigationof

the separationproblemof steelboltedbrackets.By applyingthe classicunilateralcontactlaw to describe

theseparationprocessalongthecontactsurfacesbetweenthebracket andthecolumnflange,thecontinuous

problemis formulatedasa variationalinequalityor asa quadraticprogrammingproblem.By applyingan

appropriatefinite elementdiscretizationscheme,thediscreteproblemis formulatedasaquadraticoptimiza-

tion problemwith inequalityconstraintswhich, in turn, canbeeffectively treatednumericallyby meansof

an appropriatequadraticoptimizationalgorithm.The applicability and the effectivenessof the methodis

illustratedby meansof anumericalapplication.

C. C. Baniotopoulosand K. M. Abdalla. Steelcolumn-to-columnconnectionsundercom-binedload—aquadratic-programmingapproach.Computers andStructures, 46(1), 13–20,1993.

Abstract. The aim of this paperis the investigationof the mechanicalbehaviour of boltedsteelcolumn-

to-columnconnectionsundermomentand axial loadsby meansof a methodthat takes into accountthe

possibility of the appearanceof detachmentphenomenabetweenthe spliceplates.As is well known, re-

gionsof detachment(callednonactive contactregionsbelow), dueto the appearanceof the prying-action

phenomenon,do appearon the adjacentfronts of suchsteelspliceplates,greatlyaffecting the mechanical

responseof steelconnectionsof this type.The significanceof theproblemunderinvestigationarisesfrom

thefactthatcolumn-to-columnsplicesareextensively appliedin any possiblecombinationto thedesignand

constructionof steelstructures.It is thereforeobvious that,sincelocal failurephenomenaon suchconnec-

tionsdueto undesirable-andnotapriori defined-detachmentbetweenthespliceplates(asconsequenceof the

developmentof theprying- actionphenomenon)maycausea total destructionof thewholesteelstructure.

For this reason,it is importantfor suchabehaviour to beaccuratelypredictedandthepreviously mentioned

nonactive contactregionson thespliceplatesto bedefined.In this sense,suchan investigationleadsto an

ameliorationof thedesignprinciplesfor boltedsteelcolumn-to-columnsplicesandto a refinementof the

respective steelconstructionstandards.

B. BankandR. Hansel.Stability of mixed-integerquadratic-programmingproblems.Mathe-

maticalProgrammingStudies, 21, 1–17,1982.

F. Barahona,M. Junger, andG. Reinelt. Experimentsin quadratic0–1programming.Mathe-

maticalProgramming, 44(2), 127–137,1989.

E.W. BarankinandR.Dorfman.Towardquadraticprogramming.Reportto thelogisticsbranch,

Officeof Naval Research,1955.

E. W. BarankinandR. Dorfman. A methodfor quadraticprogramming.Econometrica, 24,

1956.


E. W. BarankinandR. Dorfman. On quadraticprogramming.University of California Publi-

cationsin Statistics, 2(13),285–318,1958.

H. J.C. Barbosa,F. M. P. Raupp,andC. C. H. Borges.Numericalexperimentswith algorithmsfor boundconstrainedquadraticprogrammingin mechanics.ComputersandStructures,64(1–4),579–594,1997.

Abstract. In this work, the computationalperformanceof somealgorithmsfor solving boundconstrained

quadraticprogrammingproblemsis comparedby meansof numericalexperiments.The model problems

usedto testthebehaviour of thealgorithmsconsideredweretheobstacleproblemfor a membraneandthe

contactproblemin infinitesimalelasticity. Bothproblemsinvolveddifferentloadconditionsandparameters.

Thefinite elementmethodwasusedfor thespatialdiscretizationprocess.

J.L. Barlow andG. Toraldo. Theeffect of diagonalscalingon projectedgradientmethodsfor

boundconstrainedquadraticprogrammingproblems. OptimizationMethodsand Soft-

ware, 5(3), 235–245,1995.

R. O. Barr. An efficient computationalprocedurefor a generalizedquadraticprogramming

problem.SIAMJournalon Control, 7(3), 415–429,1999.

R. H. Bartels,G. H. Golub,andM. A. Saunders.Numericaltechniquesin mathematicalpro-

gramming. In J. B. Rosen,O. L. MangasarianandK. Ritter, eds,‘NonlinearProgram-

ming’, pp.123–176.AcademicPress,London,England,1970.

G. Basheinand M. Enns. Computationof optimal controlsby a methodcombiningquasi-linearizationandquadraticprogramming.InternationalJournal of Control, 16(1), 177–187,1972.

Abstract. Quadraticprogramming(QP)haspreviously beenappliedto thecomputationof theoptimalcon-

trols for linear systemswith quadraticcost criteria. This paperextendsthe applicationof QP to nonlin-

earproblemsthroughquasi-linearizationand the solutionof a sequenceof linear-quadraticsub-problems

whosesolutionsconverge to the solutionof the original non-linearproblem.The methodis calledquasi-

linearization-quadraticprogrammingor Q-QP.

E. M. L. Beale. On quadraticprogramming.NavalResearch LogisticsQuarterly, 6(3), 227–

243,1959.

E.M. L. Beale.Theuseof quadraticprogrammingin stochasticlinearprogramming.Technical

ReportP-2404-1,TheRAND Corporation,SantaMonica,CA, USA, 1961.

E. M. L. Beale.Noteon ’a comparisonof two methodsin quadraticprogrammng.Operations

Research, 14, 442–443,1966.

E.M. L. Beale.An introductionto Beale’smethodof quadraticprogramming.In J.Abadie,ed.,

‘Nonlinear programming’,pp. 143–153,North Holland, Amsterdam,the Netherlands,

1967.

E. M. L. BealeandR. Benveniste. Quadraticprogramming. In ‘Design andImplementationof OptimizationSoftware’, pp. 249–258.Sijthoff andNoordhoff, AlphenaandenRijn,Netherlands,1978.


Abstract. Following a generalintroductionto the theory of quadraticprogramming,the paperdescribes

computationalaspectsof anew algorithmfor convex quadraticprogramming.An essentialfeatureis thatthe

only informationneededaboutthe objective function is the gradientdirectionat successive trial solutions

(andthevalueof theobjective functionat thefinal solution).Theconstraintsarehandledasin theReduced

GradientMethod.The methodis essentiallya generalizationof the methodof ConjugateGradients.But

pureConjugateGradients,althoughfinite, requirea completerestartwhenever thesetof active constraints

changes.If storagespaceis available,thealgorithmstoresadditionaldirectionsin awaythatavoidstheneed

for acompleterestart.

J.E. Beasley. Heuristicalgorithmsfor theunconstrainedbinaryquadraticprogrammingprob-lem. Technicalreport, Departmentof Mathematics,Imperial College of ScienceandTechnology, London,England,1998.

Abstract. In this paperwe considerthe unconstrainedbinary quadraticprogrammingproblem.This is the

problemof maximisinga quadraticobjective by suitablechoiceof binary (zero-one)variables.We present

two heuristicalgorithmsbasedupontabu searchandsimulatedannealingfor this problem.Computational

resultsarepresentedfor a numberof publically availabledatasetsinvolving up to 2500variables.An inter-

estingfeatureof our resultsis thatwhilst for mostproblemstabu searchdominatessimulatedannealingfor

theverylargestproblemsweconsidertheconverseis true.Thispapertypifiesa”multiple solutiontechnique,

singlepaper”approach,i.e.anapproachthatwithin thesamepaperpresentsresultsfor anumberof different

heuristicsappliedto thesameproblem.Issuesrelatingto algorithmicdesignfor suchpapersarediscussed.

C. R. Bector. Indefinitequadraticprogrammingwith standarderrorsin objective. Cahiers du

Centre d’EtudesdeRechercheOperationalle, 10, 247–253,1968.

C. R. BectorandM. Dahl. Simplex type finite iterative techniqueandduality for a special

type of pseudo-concave quadraticprogram,. Cahiers du Centre d’Etudesde Recherche

Operationalle, 16, 207–222,1974.

L. BehjatandA. Vannelli.VLSI concentricpartitioningusinginteriorpointquadraticprogram-ming. In ‘ISCAS’99.Proceedingsof the1999IEEEInternationalSymposiumonCircuitsandSystemsVLSI. IEEE,Piscataway, NJ,USA’, Vol. 6, pp.93–96,1999.

Abstract. This paperpresentsa novel approachfor solvingthestandardcell placementproblem.A relaxed

quadraticformulationof theproblemis solvediteratively incorporatingtechniquesto increasethespreading

of cells,includingintroducingattractorsanddynamicfirst momentconstraints.At eachiteration,a percent-

ageof thecells thatarecloseto theboundaryof thechip arefixed.This procedureis donerecursively until

at leasteightypercentof thecellsarefixed.Numericalsimulationof theproposedapproachis presentedfor

testsystems.

L. Y. Belousov. Quadraticprogrammingin problemsof optimal planningof trajectorymea-surements.CosmicResearch, 9(6), 750–759,1971.

Abstract. Theproblemof optimalplanningof trajectorymeasurementsof two differentmeasuredparameters

is investigatedfor thecaseof alimiteddispersion,anarbitrarycorrelationcouplingof themeasurementerrors

of eachparameterseparately, andabsenceof correlationbetweenmeasurementsof differentparameters.It

is shown that the problemposedcanbe reducedto solving a problemin quadraticprogrammingbasedon

thelinear-programmingmethodgeneralizedfor thecontinuouscase.In conclusion,theproblemis statedby

inductionfor anarbitrarynumberof independentmeasuredparameters.

T. Belytschko. Discussionof elastic-plasticanalysisby quadraticprogramming. American

Societyof Civil Engineering, Journal of theEngineeringMechanicsDivision, 100, 130–

131,1974.


A. Bemporad,M. Morari, V. Dua, andE. N. Pistikopoulos. The explicit solutionof modelpredictive control via multiparametricquadraticprogramming. In ‘Proceedingsof the2000AmericanControlConference,Danvers,MA, USA’, Vol. 2, pp.872–876,2000.

Abstract. The control basedon online optimization,popularlyknown asmodelpredictive control (MPC),

haslong beenrecognizedasthe winning alternative for constrainedsystems.The main limitation of MPC

is, however, its onlinecomputationalcomplexity. For discrete-timelinear time-invariantsystemswith con-

straintson inputsandstates,we developanalgorithmto determineexplicitly thestatefeedbackcontrol law

associatedwith MPC,andshow thatit is piecewise linearandcontinuous.Thecontrollerinheritsall thesta-

bility andperformancepropertiesof MPC,but theonlinecomputationis reducedto a simplelinearfunction

evaluationinsteadof theexpensive quadraticprogram.Thenew techniqueis expectedto enlarge thescope

of applicabilityof MPC to small-size/fast-samplingapplicationswhichcannotbecoveredsatisfactorily with

anti-windupschemes.

M. BenDaya.Line searchtechniquesfor thelogarithmicbarrierfunctionin quadraticprogram-ming. Journalof theOperationalResearch Society, 46(3), 332–338,1995.

Abstract. In thispaper, weproposealine-searchprocedurefor thelogarithmicbarrierfunctionin thecontext

of aninteriorpointalgorithmfor convex quadraticprogramming.Preliminarytestingshowsthattheproposed

procedureis superiorto someother line-searchmethodsdevelopedspecificallyfor the logarithmicbarrier

functionin theliterature.

M. Ben DayaandK. S. Al Sultan. A new penaltyfunction algorithmfor convex quadraticprogramming.EuropeanJournalof OperationalResearch, 101(1), 155–163,1997.

Abstract. Wedevelopanexteriorpointalgorithmfor convex quadraticprogrammingusingapenaltyfunction

approach.Eachiteration in the algorithmconsistsof a singleNewton stepfollowed by a reductionin the

valueof thepenaltyparameter. Thepointsgeneratedby thealgorithmfollow anexterior paththatwedefine.

Convergenceof thealgorithmis established.Theproposedalgorithmwasmotivatedby thework of Al-Sultan

andMurty (1991)on nearestpoint problems,a specialquadraticprogram.A preliminaryimplementationof

thealgorithmproducedencouragingresults.In particular, thealgorithmrequiresasmallandalmostconstant

numberof iterationsto solve thesmallto mediumsizeproblemstested.

M. BenDayaandC. M. Shetty. Polynomialbarrierfunctionalgorithmsfor convex quadratic

programming.ArabianJournal for ScienceandEngineering, 15(4B), 656–670,1990.

J.M. Bennett.Quadraticprogrammingandpiecewiselinearnetworkswith structuralengineer-

ing applications,. In A. Prekopa,ed., ‘Survey of MathematicalProgramming,Vol. 3’,

pp.95–105,North Holland,Amsterdam,theNetherlands,1979.

P. Benson,R. L. Smith, I. E. Schochetman,andJ. C. Bean. Optimal solutionapproximationfor infinite positive-definitequadraticprogramming.Journalof OptimizationTheoryandApplications, 85(2), 235–248,1995.

Abstract. We considera generaldoubly-infinite, positive-definite,quadraticprogrammingproblem.We

show that the sequenceof uniqueoptimal solutionsto the naturalfinite-dimensionalsubproblemsstrongly

converges to the uniqueoptimal solution.This offers the opportunityto arbitrarily well approximatethe

infinite-dimensionaloptimalsolutionby numericallysolving a sufficiently large finite-dimensionalversion

of theproblem.We thenapplyour resultsto a generaltime-varying, infinite- horizon,positive-definite,LQ

controlproblem.

R. Benveniste.A quadraticprogrammingalgorithmusingconjugatesearchdirections.Mathe-maticalProgramming, 16(1), 63–80,1979.


Abstract. A quadraticprogrammingalgorithmis presented,resemblingBeale’s (1955)quadraticprogram-

ming algorithmandWolfe’s reducedgradientmethod.It usesconjugatesearchdirections.Thealgorithmis

conceivedasbeingparticularlyappropriatefor problemswith a largeHessianmatrix.An outlineof thesolu-

tion to thequadraticcapacity-constrained transportationproblemusingtheabove methodis alsopresented.

R.Benveniste.Onewayto solvetheparametricquadraticprogrammingproblem.MathematicalProgramming, 21(2), 224–228,1981.

Abstract. A methodis presentedfor thesolutionof theparametricquadraticprogrammingproblemby the

useof conjugatedirections.

C. Bergthaller. A quadraticequivalentof the minimum risk problem. RevueRoumainedo

MathematiquesPureet Appliquees, 15, 17–23,1970.

C. Bergthaller. Parametricquadraticprogramming.In ‘4th conferenceon probability theory.Abstracts.AcadSocialistRepublicof Rumania,Bucharest,Romania’,pp.23–24,1971a.

Abstract. Thispaperdealswith theparametricprogrammingproblemmincTx 1 2xT Dx A λ x b, x 0,

where: c x Rn, b Rm, D is a symmetricnxn positive definitematrix, A λ Ao λAl , Ao andAl are

fixedm n matrices,suchthattherankof Al is l andλ 0 is realparameter. Someparticularcasesare:1)

Oneelementof thematrix A is a linear functionof lambdaandall othersareconstant.2) A columnof mod

A is a linear (vectorial)function of λ aj aoj λal

j andtheothersareconstant.3) A row of A is a linear

(vectorial)functionof λ α j αoj λαl

i andtheothersareconstant.

C. Bergthaller. Quasi-convex quadraticprogramming. ComptesRendusHebdomadairesdesSeancesde l’AcademiedesSciences,SerieA (ScienceMathematiques), 273(15), 685–686,1971b.

Abstract. A simplicial algorithm is given for the programminqTx 1 2xT Qx Ax b x 0 whereq is

an n-dimensionalvector andQ is a symmetricalmatrix suchthat the objective function f x identical to

qTx 1 2xT Qx is quasi-convex for x 0 without beingconvex.

A. B. Berkelaar. Sensitivity analysisin (degenerate)quadraticprogramming. EconometricInstituteReport30, EconometricInstitute,ErasmusUniversity, Rotterdam,TheNether-lands,1997.

Abstract. In this paperwe dealwith sensitivity analysisin convex quadraticprogramming,without making

assumptionson nondegeneracy, strict convexity of the objective function, and the existenceof a strictly

complementarysolution.We show that theoptimalvalueasa functionof a right–handsideelement(or an

elementof the linearpartof theobjective) is piecewisequadratic,wherethepiecescanbecharacterizedby

maximalcomplementarysolutionsandtripartitions.Further, we investigatedifferentiabilityof this function.

A new algorithmto computetheoptimalvaluefunction is proposed.Finally, we discusstheadvantagesof

thisapproachwhenappliedto mean–varianceportfolio models.

A. B. Berkelaar, B. Jansen,C. Roos,andT. Terlaky. Sensitivity analysisin quadraticpro-gramming. Report96-11,EconometricInstitute,ErasmusUniversity, Rotterdam,TheNetherlands,1996.

Abstract. In this paperwe dealwith sensitivity analysisin convex quadraticprogramming,without making

assumptionson nondegeneracy, strict convexity of the objective function, and the existenceof a strictly

complementarysolution.We show that theoptimalvalueasa functionof a right–handsideelement(or an

elementof the linearpartof theobjective) is piecewisequadratic,wherethepiecescanbecharacterizedby

maximalcomplementarysolutionsandtripartitions.Further, we investigatedifferentiabilityof this function.

A new algorithmto computetheoptimalvaluefunction is proposed.Finally, we discusstheadvantagesof

thisapproachwhenappliedto mean–varianceportfolio models.


A. B. Berkelaar, B. Jansen,K. Roos,andT. Terlaky. Basis-andpartition identificationforquadraticprogrammingandlinearcomplementarityproblems.MathematicalProgram-ming, 86(2), 261–282,1999.

Abstract. Optimal solutionsof interior point algorithmsfor linear andquadraticprogrammingand linear

complementarityproblemsprovide maximally complementarysolutions.Maximally complementarysolu-

tionscanbecharacterizedby optimalpartitions.Ontheotherhand,thesolutionsprovidedby simplex-based

pivot algorithmsaregivenin termsof complementarybases.A basisidentificationalgorithmis analgorithm

whichgeneratesacomplementarybasis,startingfrom any complementarysolution.A partitionidentification

algorithmis analgorithmwhichgeneratesa maximallycomplementarysolution(andits correspondingpar-

tition), startingfrom any complementarysolution.In linearprogrammingsuchalgorithmswererespectively

proposedby Megiddo in 1991andBalinski andTucker in 1969.In this paperwe will presentidentifica-

tion algorithmsfor quadraticprogrammingandlinear complementarityproblemswith sufficient matrices.

Thepresentedalgorithmsarebasedon theprincipalpivot transformandtheorthogonalitypropertyof basis

tableaus.

A. B. Berkelaar, K. Roos,and T. Terlaky. The optimal partition and optimal set approachto linear and quadraticprogramming. EconometricInstitute Report51, EconometricInstitute,ErasmusUniversity, Rotterdam,TheNetherlands,1997.

Abstract. In this chapterwe describetheoptimalsetapproachfor sensitivity analysisfor LP. We show that

optimalpartitionsandoptimalsetsremainconstantbetweentwo consecutive transition-pointsof theoptimal

valuefunction.Theadvantageof usingthisapproachinsteadof theclassicalapproach(usingoptimalbases)

is shown. Moreover, we presentanalgorithmto computethepartitions,optimalsetsandtheoptimalvalue

function. This is a new algorithm and usesprimal and dual optimal solutions.We also extend someof

the resultsto parametricquadraticprogramming,anddiscussdifferencesandresemblanceswith the linear

programmingcase.

H. Bernau. Upper boundtechniquesfor quadraticprogramming. AlkalmazottMatematikaiLapok, 3(1–2),161–170,1977. Seealso,A. Prekopa,ed.Survey of MathematicalPro-gramming, Vol.1, North-Holland,Amsterdam,pp.347–356,1979.

Abstract. An extensionof themethodsof Wolfe, JagannathanandBealeis presentedfor quadraticprogram-

ming problemswith upperboundsfor thevariables.It is shown that the upperboundstechniquefor linear

programmingproblemscanbeveryeasilyincorporatedin thesemethods.

H. Bernau. Quadraticprogrammingproblemsandrelatedlinear complementarityproblems.Journalof OptimizationTheoryandApplications, 65(2), 209–222,1990.

Abstract. Investigatesthegeneralquadraticprogrammingproblem,i.e. theproblemof finding theminimum

of aquadraticfunctionsubjectto linearconstraints.In thecasewhere,over thesetof feasiblepoints,theob-

jective functionis boundedfrom below, thisproblemcanbesolvedby theminimizationof a linearfunction,

subjectto thesolutionsetof a linearcomplementarityproblem,representingtheKuhn-Tucker conditionsof

the quadraticproblem.To detectin the quadraticproblemthe unboundednessfrom below of the objective

function,necessaryandsufficientconditionsarederived.It is shown that,whentheseconditionsareapplied,

thegeneralquadraticprogrammingproblembecomesequivalentto theinvestigationof anappropriatelyfor-

mulatedlinearcomplementaryproblem.

O.Bertoldi,M. V. Cazzol,A. Garzillo,andM. Innorta.A dualquadraticprogrammingalgorithmorientedto the probabilisticanalysisof large interconnectednetworks. In ‘PSCC.Pro-ceedingsof theTwelfth Power SystemsComputationConference.Power Syst.Comput.Conference,Zurich,Switzerland’,Vol. 2, pp.1249–1255,1996.

Abstract. A fastandrobust innovative computingprocedurehasbeendevelopedaimedat allowing theuse

of optimalpower flow techniquesin theframework of theprobabilisticadequacy assessmentof large inter-

connectedpower systems.Thepaperdescribesthemethodologicalapproachandthe relevant implemented


algorithm.Several numericalresultsaresuppliedwhich demonstratethe high computingefficiency of the

proceduresothat it is suitablein theprobabilisticsimulationdomain.

M. J.Best. Equivalenceof somequadraticprogrammingalgorithms.MathematicalProgram-ming, 30(1), 71–87,1984.

Abstract. The authorformulatesa generalalgorithmfor the solutionof a convex (but not strictly convex)

quadraticprogrammingproblem.Conditionsaregivenunderwhich theiteratesof thealgorithmareuniquely

determined.The quadraticprogrammingalgorithmsof Fletcher(1971),Gill andMurray (1978),Bestand

Ritter(1976),andvandePanneandWhinston/Dantzig(1969)areshown to bespecialcasesandconsequently

areequivalentin thesensethatthey constructidenticalsequencesof points.Thevariousmethodsareshown

to differ only in the mannerin which they solve the linear equationsexpressingthe Kuhn-Tucker system

for theassociatedequalityconstrainedsubproblems.Equivalenceresultshave beenestablishedby Goldfarb

(1972)andDjang (1979) for the positive definite Hessiancase.The analysisextendstheseresultsto the

positive semi-definitecase.

M. J.Best.An algorithmfor parametricquadraticprogramming.In H. Fischer, B. Rieduller and

S. Schaffler, eds,‘Applied MathematicsandParallel Computing—Festschriftfor Klaus

Ritter’, pp.57–76.Physica-Verlag,Heidelburg, 1996.

M. J. Best and R. J. Caron. A methodto increasethe computationalefficiency of certainquadraticprogrammingalgorithms.MathematicalProgramming, 25(3), 354–358,1983.

Abstract. Presentsamethodfor computingtheKuhn-Tuckermultipliersassociatedwith equalityconstraints

in quadraticprogrammingproblems.Whenappliedto acertainclassof algorithmsasignificantreductionin

computationtime andin storageis achieved.

M. J.BestandR. J.Caron.A parameterizedHessianquadraticprogrammingproblem.Annalsof OperationsResearch, 5(1–4),373–394,1986.

Abstract. Presentsa generalactive setalgorithmfor thesolutionof a convex quadraticprogrammingprob-

lemhaving aparametrizedHessianmatrix.TheparametricHessianmatrix is apositive semidefiniteHessian

matrix plusa real parametermultiplying a symmetricmatrix of rank oneor two. The algorithmsolvesthe

problemfor all parametervaluesin theopeninterval uponwhich theparametricHessianis positive semidef-

inite. The algorithm is generalin that any of several existing quadraticprogrammingalgorithmscan be

extendedin astraightforwardmannerfor thesolutionof theparametricHessianproblem.

M. J. BestandN. Chakravarti. Stability of linearly constrainedconvex quadraticprograms.

Journalof OptimizationTheoryandApplications, 64(1), 43–53,1990.

M. J. BestandN. Chakravarti. An On2 active setmethodfor solving a certainparametric

quadraticprogram. Journal of OptimizationTheoryand Applications, 72(2), 213–224,1992.

Abstract. The paper presents an O n2 method for solving the parametric quadratic program

min 1 2 xT Dx aT x λ 2 ∑nj 1 γ j xj c 2, having lower andupperboundson thevariables,for all non-

negative valuesof theparameterlambda. Here,D is apositive diagonalmatrix,aanarbitraryn-vector, each

γ j , j 1 n, andc arearbitraryscalars.An applicationto economicsis alsopresented.

M. J.BestandK. Ritter. An effectivealgorithmfor quadraticminimizationproblems.Technical

report1691,Universityof Wisconsin,Madison,Wisconsin,USA, 1976.

M. J. BestandK. Ritter. A quadraticprogrammingalgorithm. ZOR,MethodsandModelsofOperationsResearch, 32(5), 271–297,1988.


Abstract. By usingconjugatedirectionsa methodfor solving convex quadraticprogrammingproblemsis

developed.Thealgorithmgeneratesa sequenceof feasiblesolutionsandterminatesaftera finite numberof

iterations.Extensionsof thealgorithmfor nonconvex andlargestructuredquadraticprogrammingproblems

arediscussed.

D. Bhatia. Duality for quadraticprogrammingin complex space.Zeitschrift fur AngewandteMathematikundMechanik, 54(1), 55–57,1974.

Abstract. The duality theoremsof Rami (1972) for symmetricdual quadraticprogramsare extendedin

complex spaceover arbitrarypolyhedralcones.

S. Bhowmik, S. K. Goswami, andP. K. Bhattacherjee.Distribution systemplanningthroughcombinedheuristicandquadraticprogrammingapproach.Electric MachinesandPowerSystems, 28(1), 87–103,2000.

Abstract. Thepresentpaperreportsanew techniquefor theplanningof a radialdistribution system.Distri-

butionsystemplanninghasbeenformulatedasaproblemof quadraticmixedintegerprogramming(QMIP).

A two-stageiterative solution techniquehasbeenproposedwherethe first stagedeterminesthe optimum

substationsitesand the secondstagedeterminesthe optimum network configurations.To reducethe di-

mensionalityproblem,the integer constraintsarefirst related,thusconverting the quadraticmixed integer

programmingprobleminto a quadraticprogramming(QP)problem.After thesolutionof theQPproblem,

integerconstraintsareimposedusingheuristictechniques.

D. Bienstock.Computationalstudyof a family of mixed-integerquadraticprogrammingprob-lems. MathematicalProgramming, 74(2), 121–140,1996. Seealso, Integer Program-ming andCombinatorialOptimization.4th InternationalIPOCConference,Proceedings(Balas,E. andClausen,J.,eds.),Springer-Verlag,Berlin, Germany, pages90–94,1995.

Abstract. Wepresentcomputationalexperiencewith abranch-and-cutalgorithmto solvequadraticprogram-

ming problemswherethereis anupperboundon thenumberof positive variables.Suchproblemsarisein

financialapplications.Thealgorithmsolvesthelargestreal-life problemsin a few minutesof run-time.

A. Billionnet andA. Sutter. Persistency in quadratic0–1optimization.MathematicalProgram-

ming, 54(1), 115–119,1992.

A. Bjorck. Constrainedleast-squaresproblems. In ‘Numerical Methodsfor LeastSquares

Problems’,chapter5, pp.187–213.SIAM, Philadelphia,USA, 1996.

E. Blum and W. Oettli. Direct proof of the existencetheoremfor quadraticprogramming.OperationsResearch, 20(1), 165–167,1972.

Abstract. A directanalyticalproof is givenfor thefollowing theorem:If theinfimumof aquadraticfunction

onanonempty(possiblyunbounded)polyhedralsetRcontainedin IRn is finite, thentheinfimumis assumed

somewhereonR, thusbeingaminimum.

P. T. Boggs,P. D. Domich,andJ.E. Rogers.An interior point methodfor generallarge-scalequadraticprogrammingproblems.Annalsof OperationsResearch, 62, 419–437,1996a.

Abstract. Presentsan interior point algorithmfor solvingbothconvex andnonconvex quadraticprograms.

The method,which is an extensionof the authors‘ interior point work on linear programmingproblems,

efficiently solvesa wide classof large-scaleproblemsandforms the basisfor a sequentialquadraticpro-

gramming(SQP)solver for generallarge scalenonlinearprograms.The key to the algorithm is a three-

dimensionalcostimprovementsubproblem,which is solvedat every iteration.Theauthorshave developed

anapproximaterecenteringprocedureandanovel, adaptive big-M PhaseI procedurethatareessentialto the

successof thealgorithm.Theauthorsdescribethebasicmethodalongwith therecenteringandbig-M Phase

I procedures.Detailsof theimplementationandcomputationalresultsarealsopresented.


P. T. Boggs,P. D. Domich,J. E. Rogers,andC. Witzgall. An interior-point methodfor linear

andquadraticprogrammingproblems.COAL Newsletter, 19, 32–40,1991.

P. T. Boggs,P. D. Domich,J.E. Rogers,andC. Witzgall. An interior point methodfor generallargescalequadraticprogrammingproblems.Annalsof OperationsResearch, 62, 419–437,1996b.

Abstract. In this paperwe presentan interior point algorithm for solving both convex and nonconvex

quadraticprograms.The method,which is an extensionof our interior point work on linear program-

ming problems,efficiently solvesa wide classof largescaleproblemsandformsthebasisfor a sequential

quadraticprogramming(SQP)solver for generallarge scalenonlinearprograms.The key to the algorithm

is a 3-dimensionalcost-improvementsubproblem,which is solvedat every iteration.Wehave developedan

approximaterecenteringprocedureanda novel, adaptive big-M PhaseI procedurethatareessentialto the

success.We describethebasicmethodalongwith therecenteringandbig-M PhaseI procedures.Detailsof

theimplementationandcomputationalresultsarealsopresented.

N. L. Boland.A dual-active-setalgorithmfor positivesemidefinitequadraticprogramming.In

‘Optimization: TechniquesandApplications’,pp.80–89,1992.

N. L. Boland. A dual-active-setalgorithmfor positive semi-definitequadraticprogramming.MathematicalProgramming, 78(1), 1–27,1997.

Abstract. Becauseof themany importantapplicationsof quadraticprogramming,fastandefficient methods

for solving quadraticprogrammingproblemsare valued.Goldfarb and Idnani (1983) describeone such

method,Well known to beefficientandnumericallystable,theGoldfarbandIdnanimethodsuffersonly from

therestrictionthatin its original form it cannotbeappliedto problemswhicharepositivesemi-definiterather

thanpositive definite.In this paper, we presenta generalizationof the Goldfarb andIdnani methodto the

positive semi-definitecaseandprove finite terminationof thegeneralizedalgorithm.In our generalization,

wepreserve thespirit of theGoldfarbandIdnanimethod,andextendtheirnumericallystableimplementation

in anaturalway.

I. M. BomzeandG. Danninger. A global optimizationalgorithmfor concave quadraticpro-

grammingproblems.SIAMJournalonOptimization, 3(4), 826–842,1993.

J. F. Bonnansand M. Bouhtov. The trust region affine interior point algorithm for convexandnonconvex quadraticprogramming.RAIRO-RechercheOperationnelle—OperationsResearch, 29(2), 195–217,1995.

Abstract. Westudyfrom a theoreticalandnumericalpoint of view aninterior point algorithmfor quadratic

QPusingatrustregion idea,formulatedby YeandTse(1989).Weshow that,underanondegeneracy hypoth-

esis,thealgorithmconvergesglobally in theconvex case.For anonconvex problem,underamild additional

hypothesis,thesequenceof pointsconvergesto a stationarypoint.Weobtainalsoanasymptoticlinearcon-

vergenceratefor thecostthatdependsonly on thedimensionof theproblem.Whenweshow that,provided

somemodificationsareaddedto thebasicalgorithm,themethodhasagoodnumericalbehaviour.

J. C. G. Boot. Noteson quadraticprogramming:The Kuhn-Tucker andThiel-Van de Panne

conditions,degeneracy andequalityconstraints.ManagementScience, 8, 85–98,1961.

J. C. G. Boot. Binding constraintproceduresof quadraticprogramming. Econometrica,

31, 464–498,1963a.

J.C. G. Boot. On sensitivity analysisin convex quadraticprogrammingproblems.Operations

Research, 11, 771–786,1963b.


J.C. G. Boot. Quadratic Programming. Algorithms—Anomalies—Algorithms. North Holland,

Amsterdam,theNetherlands,1964.

J.C. G. Boot andH. Theil. A procedurefor integermaximizationof a definitequadraticform.

In G. KrewerasandG. Morlat, eds,‘Proceedingsof the3rd InternationalConferenceon

OperationsResearch’.EnglishUniversitiesPress,1964.

V. L. Borre andS. G. Kapoor. A multi-stagequadratic-programmingoptimizationtechnique

for optimal managementof large hydro-power operations. EngineeringOptimization,

8(2), 103–118,1985.

J.M. Borwein. Necessaryandsufficient conditionsfor quadraticminimality. NumericalFunc-

tional AnalysisandOptimization, 5, 127–140,1982.

R. J.Bosch,Y. Ye,andG. G. Woodworth. A convergentalgorithmfor quantileregressionwithsmoothingsplines.ComputationalStatisticsandDataAnalysis, 19, 613–630,1995.

Abstract. An importantpracticalproblemis thatof determininganonparametricestimateof theconditional

quantileof y givenx. If webalancefidelity to thedatawith asmoothnessrequirement,theresultingquantile

functionis a cubicsmoothingspline.Wereformulatethis estimationprocedureasa quadraticprogramming

problem,with associatedoptimality conditions.A recentlydevelopedinterior point algorithmwith proven

convergenceis extendedto solve the quadraticprogram.This solutioncharacterizesthe desirednonpara-

metric conditionalquantilefunction. Thesemethodsare illustratedin a study of audiologicperformance

following cochlearimplants.

A. Bouzaher. SymmetricQP andlinearprogrammingunderprimal-dualuncertainty. Opera-tionsResearch Letters, 6(5), 221–225,1987.

Abstract. A saddle-pointformulationof linearprogrammingproblemswith randomobjective functionand

RHS coefficients is proposed.Under a certaintyequivalent criterion, a pair of primal-dualdeterministic

equivalentsis derived. Theseproblemsaresymmetricdual quadraticprograms,andcanbe interpretedas

generalizationsof theclassicalmean-variancemodel.

J. Bouzitat. On Wolfe’s methodand Dantzig’s methodfor convex quadraticprogramming.RAIRO RechercheOperationelle, 13(2), 151–184,1979.

Abstract. Both Wolfe’s and Dantzig’s methodssolve linear-constrainedconvex quadraticprogramming

problemsby simplex-like algorithms.They usetheKuhn-Tucker conditions,which arenecessaryandsuf-

ficient for suchproblems.The authorpresentsthosetwo methods,with completetheoreticalproofs the

greaterpart of which, to the author’s knowledge,is new. Wolfe’s methodreceivesherea complementand

is thenfound to bemoreefficient thanit previously appeared,on accountof ’blocking’ phenomenawhich

areprovednot to stoptheconvergentprocessof thecompletedalgorithm.The’shearform’ of themethodis

consequentlyapplicableto solve any nonparametricproblem,andthe’ long form’ maybereservedfor para-

metricproblemsonly. Thepresentproofof Dantzig’s algorithmconvergenceis notbasedonthedirectstudy

of computationschemata,but usestheconvexity of thequadraticfunctionto beminimized,which leadsto

quitesimpleproof.Thegeneralpresentationof Wolfe’s andDantzig’s methodsis illustratedby a numerical

problemwhich is solvedby bothof them,soasto permitacomparison.

R. J. Braitsch. A computercomparisonof four quadraticprogrammingalgorithms. Manage-mentScience, 18(11),632–643,1972.

Abstract. This papercomparesthecomputationalperformanceof four quadraticprogrammingalgorithms.

Problemsaregeneratedandsolvedon thecomputerwith iterationcountservingastheprincipalmethodof

comparison.The effect of certainproblemparameterson rateof convergenceis consideredandcomputer

timeandstoragerequirementsof thefour algorithmsarediscussed.


N. J.Breytenbach.A structuredquadraticprogram.In J.A. Snyman,ed.,‘Proceedingsof the

TenthSouthAfrican Symposiumon NumericalMathematics’,p. 128,1984.

J. F. Brotchie. A generalizeddesign approachto solution of the nonconvex quadratic-

programmingproblem.AppliedMathematicalModelling, 11(4), 291–295,1987.

B. M. Brown andC. R.Goodall.Applicationsof quadraticprogrammingin statistics.Technical

Report93-09,Departmentof Statistsics,ThePennsylvaniaStateUniversity, USA, 1993.

K. S. Brown andK. A. Reiman. An ALGOL programfor quadraticprogrammingusingthemethodof Wolfe andFrank. Bulletin of the OperationsResearch Societyof America,23, B374,1975.

Abstract. Tomeettheneedsof makingavailablecomputerprograms,thispaperpresentsanALGOL program

for applyingthe methodof Wolfe andFrankto properlystructuredquadraticprogrammingproblems.The

procedureis thendemonstratedby solvinga representative sampleof structuredproblems.Wherefeasible,

problemsareformulatedin suchaway thatrelevanceto realproblemsis readilynoted.

C. G. BroydenandN. F. Attia. Penaltyfunctions,Newton’s method,andquadraticprogram-

ming. Journalof OptimizationTheoryAndApplications, 58(3), 377–385,1988.

J.R.BunchandL. Kaufman.A computationalmethodfor theindefinitequadraticprogrammingproblem.LinearAlgebra andIts Applications, 34, 341–370,1980.

Abstract. Presentsanalgorithmfor thequadraticprogrammingproblemof determininga localminimumof

f x 1 2xT Qx cT x suchthatATx b whereQ is asymmetricmatrixwhichmaynotbepositive definite.

C. A. Burdet. Generalquadraticprogramming. Technicalreport, Carnegie Mellon Univ ,Pittsburgh,PA, USA, 1971.

Abstract. An algorithmis presentedfor thegeneral(not necessarilyconvex or concave) quadraticprogram-

mingproblemoveralinearlyconstrainedset.Thealgorithmis finitely convergentandmakesuseof aconvex

quadraticprogrammingmethodasa subroutine(like thequadraticsimplex for instance).Thebasictool for

thismethodis a facialdecompositionfor polyhedralsets.

S.J.Byrne.Solutionof quadraticprogrammingproblems.New ZealandOperationalResearch,12(2), 73–89,1984.

Abstract. Quadraticprogrammingproblemsarisein anumberof situations.Typicalexamples,e.g.portfolio

selection,economicmodelling,regressionanalysis,andsolutionof non-linearprogrammingproblems,are

briefly described.Themethodsthathavebeenproposedfor solvingthisproblemarereviewed.Theapproach

usinga linear complementarityprogramis selected,andan efficient, numericallywell-behaved procedure

is developed,basedon Lemke’s algorithmandusingorthogonalfactorizations.A principalpivot versionis

alsopresented,which parallelsDantzigandCottle’s solutionprocedure,but is applicableto thesamevery

wideclassof matricesprocessedby Lemke’salgorithm.A restartfacility is developedfor thisversion,which

acceleratesthesolutionof asequenceof relatedquadraticprograms,by proceedingfrom theoptimumof the

lastproblem.Themethodshavebeencodedin FortranIV andperformwell. Theprocessis easilyextendedto

handlebothupperandlowerboundsonconstraintsandhasprovedacceptablefor usein ageneralnon-linear

programmingalgorithm.

R. CaballeroandA. Santos. A new dual methodfor solving strictly convex quadraticpro-

grams. Technicalreport,Departmentof Applied Economics(Mathematics),University

of Malaga,Spain,1998.


A. V. CabotandR. L. Francis.Solvingnonconvex quadraticminimizationproblemsby ranking

theextremepoints.OperationsResearch, 18, 82–86,1970.

L. M. Cabral. An efficient algorithmfor the quadraticprogrammingproblemwith inequalityconstraints.In ‘Proceedingsof the1982AmericanControlConference.IEEE,New York,NY, USA’, Vol. 3, pp.1016–1017,1982.

Abstract. The solution to the generallinear problemAx y with side conditionscan be interpretedas

a quadraticoptimizationproblem.Sucha requirementarisesin the solution of illposed problemswhere

the methodof regularizationis exploited.A compactquadraticoptimizationprocedureis presentedwhich

obviatesthe requirementto invert large matrices,encounteredin typical applications,and therebyreduce

computationtime anderrorsdueto numericalround-off.

P. H. Calamaiand L. N. Vicente. ALGORITHM 728: Fortran subroutinesfor generating

quadraticbilevel programmingtestproblems.ACM Transactionson MathematicalSoft-

ware, 20(1), 120–123,1994a.

P. H. CalamaiandL. N. Vicente. Generatingquadraticbilevel programmingtestproblems.

ACM Transactionson MathematicalSoftware, 20(1), 103–119,1994b.

P. H. Calamai,J. J. Judice,andL. N. Vicente. Generationof disjointly constrainedbilinear

programmingtestproblems.ComputationalOptimizationandApplications, 1, 299–306,

1992.

P. H. Calamai,L. N. Vicente,and J. J. Judice. A new techniquefor generatingquadratic-programmingtestproblems.MathematicalProgramming, 61(2), 215–231,1993.

Abstract. Thispaperdescribesanew techniquefor generatingconvex, strictly concave andindefinite(bilin-

earor not) quadraticprogrammingproblems.Theseproblemshave a numberof propertiesthatmake them

usefulfor testpurposes.For example,strictly concave quadraticproblemswith theirglobalmaximumin the

interiorof thefeasibledomainandwith anexponentialnumberof localminimawith distinctfunctionvalues

andindefiniteandjointly constrainedbilinearproblemswith nonextremeglobalminima,canbegenerated.

Unlikemostexistingmethodsourconstructiontechniquedoesnotrequirethesolutionof any subproblemsor

systemsof equations.In addition,theauthorsknow of no othertechniquefor generatingjointly constrained

bilinearprogrammingproblems.

A. J. Calise. Statisticaldesignof sampled-datacontrol systemsvia quadraticprogramming.Technicalreport,Univ Pennsylvania,PA, USA, 1968.

Abstract. Quadraticprogrammingis appliedto the statisticaldesignof Linear sampled-datacontrol sys-

tems.Given a fixed plant, a compensatoris chosenwhich minimizesthe mean-squarevalue of an error

sequencesubjectto a setof constraints.The systeminputsaredescribedin termsof sampledvaluesfrom

autocorrelationandcross-correlationfunctions,eliminatingtheneedfor theanalyticalexpressionsrequired

in theWiener-Hopf equation.Constraintswhichcharacterizetheclosed-loopresponseto deterministicinputs

andconstraintswhich limit thecomplexity of thecompensatingnet-work canbesimultaneouslyemployed.

Thesolutiongeneratestheparametersof theoptimumcompensator. Thecasewheretheerrorsignalis not

sampledandwherethemean-squarevalueof theerroris theindex of theperformanceis alsoconsidered.

E. K. Can. A quadratic-programmingsolution to cost-timetrade-off for CPM. In ‘Applied

SimulationandModelling—ASM ’85’, Vol. 3, pp.253–256,1985.

W. CandlerandR. J.Townsley. Themaximizationof a quadraticfunctionof variablessubject

to linearinequalities.ManagementScience, 10, 515–523,1964.


E. Canestrelli,S.Giove,andR. Fuller. Stability in possibilisticquadraticprogramming.FuzzySetsandSystems, 82(1), 51–56,1996.

Abstract. Weshow thatpossibilisticquadraticprogramswith crispdecisionvariablesandcontinuousfuzzy

numbercoefficientsarewell-posed,i.e. small changesin the membershipfunction of the coefficientsmay

causeonly a smalldeviation in thepossibilitydistribution of theobjective function.

J.CanisiusandJ.L. vanHemmen.A polynomialtimealgorithmin generalquadraticprogram-ming andground-statepropertiesof spin glasses.EurophysicsLetters, 1(7), 319–326,1986.

Abstract. An algorithmis presentedwhich findsa surprisinglygoodapproximationto theglobalminimum

of a not necessarilyconvex, quadraticfunction in N variables,restrictedto a N-dimensionalcube.For in-

stance,all Ising Hamiltonianswith pair interactionsbelongto this class.In general,the time complexity of

thealgorithmis O N4 . For nearest-neighbourinteractions,it reducesto O N3 . Thealgorithmis usedto find

thegroundstateof variousIsingspinglassmodelsandto studythezero-temperaturebehavior of themagne-

tizationasafunctionof theexternalfield h in bothandthreedimensions.It is foundthatthetwo-dimensional

+or-J modelhasa nonzeromagnetizationash 0.

M. D. CanonandJ.H. Eaton.A new algorithmfor aclassof quadraticprogrammingproblems

with applicationto control. SIAMJournalon Control, 4, 34–45,1996.

J.M. Cao.Necessaryandsufficientconditionfor localminimaof aclassof nonconvex quadratic

programs.MathematicalProgramming, 69(3), 403–411,1995.

M. Capurso. A quadraticprogrammingapproachto the impulsive loading analysisof rigidplasticstructures.Meccanica, 7(1), 45–57,1972.

Abstract. This paperdiscussesthedynamicproblemof rigid plasticstructuressubjectedto impulsive load-

ing. A coupleof ’dual’ extremumtheoremsreducesthe problemto the optimizationof convex quadratic

functionssubjectto linearequalitiesandequations:thefirst theoremtakesasvariablesstressandaccelera-

tions, thesecondaccelerationsandplasticmultiplier rates.The problemis discussedin matrix notationon

the basisof finite elementdiscretizationof the structureandpiecewise linear approximationof the yield

surfaces,usingsomequadraticprogrammingconcepts.The procedureis illustratedby a simplenumerical

example.

R. J.Caron.Parametricquadraticprogramming.WindsorMathematicsReport86-01,Depart-

mentof Mathematics,Universityof Windsor, Ontario,Canada,1986.

T. J. CarpenterandD. F. Shanno.An interior point methodfor quadraticprogramsbasedonconjugateprojectedgradients.ComputationalOptimizationandApplications, 2(1), 65–28,1993.

Abstract. We proposean interior point methodfor large-scaleconvex quadraticprogrammingwhereno

assumptionsaremadeaboutthe sparsitystructureof the quadraticcoefficient matrix Q. The interior point

methoddescribedis a doubly iterative algorithmthat invokes a conjugateprojectedgradientprocedureto

obtain the searchdirection.The effect is that Q appearsin a conjugatedirection routine ratherthan in a

matrix factorization.By doing this, thematricesto be factoredhave thesamenonzerostructureasthosein

linear programming.Further, onevariantof this methodis theoreticallyconvergent with only onematrix

factorizationthroughouttheprocedure.

T. J.Carpenter, I. J. Lustig, J. M. Mulvey, andD. F. Shanno.Higher-orderpredictor-corrector

interior point methodswith applicationto quadraticobjectives. SIAMJournal on Opti-

mization, 3(4), 696–725,1993a.


T. J.Carpenter, I. J.Lustig,J.M. Mulvey, andD. F. Shanno.Separablequadraticprogrammingvia a primal-dual interior point methodand its usein a sequentialprocedure. ORSAJournalon Computing, 5(2), 182–191,1993b.

Abstract. Extendsa primal-dualinterior point procedurefor linearprogramsto thecaseof convex separa-

ble quadraticobjectives.Includedareefficient proceduresfor: attainingprimal anddualfeasibility, variable

upperbounding,andfreevariables.A sequentialprocedurethatinvokesthequadraticsolver is proposedand

implementedfor solving linearly constrainedconvex separablenonlinearprograms.Computationalresults

areprovided for several large testcasesfrom stochasticprogramming.Theproposedmethodscomparefa-

vorablywith MINOS, especiallyfor the largerexamples.Thenonlinearprogramsrangein sizeup to 8700

constraintsand22000variables.

J. L. Carpentier, G. Cotto, andP. L. Niederlander. New conceptsfor automaticgenerationcontrolin electricpowersystemsusingparametricquadraticprogramming.In A. Alonso-Concheiro,ed.,‘Real Time Digital ControlApplications.Proceedingsof the IFAC/IFIPSymposium.Pergamon,Oxford,England’,pp.595–600,1984.

Abstract. New conceptsfor automaticgenerationcontrol in electricpower systemsarepresented,where

the two componentsof automaticgenerationcontrol, load frequency control and economicdispatchare

performedat the samerate, i.e. a few seconds,andwhereeconomicdispatchtakes network securityinto

account.This givesnetwork securityandgoodtransients,avoiding contradictoryactionsof load frequency

controlandeconomicdispatchon thegeneratingunits.Thecornerstoneof thesolutionis theuseof a new

faston-lineoptimalpower flow, usinganew parametricquadraticprogrammingmethod,which is presented

in details.

P. Carraresi,F. Farinaccio,andF. Malucelli. Testingoptimality for quadratic0-1 problems.TechnicalReportTR-95-11,Dipartimentodi Informatica,Universitadi Pisa,Italy, 1995.

Abstract. The issuetackledis testingwhethera given solutionof a quadratic0-1 problemis optimal.The

paperpresentsanalgorithmbasedonthenecessaryandsufficientoptimalityconditionintroducedby Hirriart-

Urruty for generalconvex problems.A measureof the quality of the solutionis provided. Computational

resultsshow theapplicabilityof themethod.Themethodis extendedto constrainedquadratic0-1 problems

suchasquadraticassignmentandquadraticknapsack.

E. Casasand C. Pola. An algorithm for indefinite quadraticprogrammingbasedon a par-tial Cholesky factorization. RAIRO-Recherche Operationnelle-Operations Research,27(4), 401–426,1993.

Abstract. A new algorithm is describedfor quadraticprogrammingthat is basedon a partial Cholesky

factorizationthatusesa diagonalpivoting strategy andallows computationof thenull of negative curvature

directions.The algorithm is numericallystableand hasshown efficiency in solving positive-definiteand

indefiniteproblems.It is speciallyinterestingin indefinitecasesbecausetheinitial pointdoesnotneedto be

a vertex of the feasibleset.The authorsthusavoid introducingartificial constraintsin the problem,which

turnsout to bevery efficient in parametricprogramming.At thesametime, techniquesfor updatingmatrix

factorizationsareused.

Y. ChabrillacandJ.-P. Crouzeix.Definitenessandsemidefinitenessof quadraticformsrevisited.

LinearAlgebra andits Applications, 63, 283–292,1984.

T. F. Chan,J.A. Olkin, andD. W. Cooley. Solvingquadraticallyconstrainedleastsquaresusing

blackboxsolvers.BIT, 32, 481–495,1992.

S. W. Chang. A methodfor quadraticprogramming. Naval Research Logistics Quarterly,33(3), 479–487,1986.


Abstract. A solutionto thequadraticprogrammingis presentedwith theconstraintof theform Ax b using

thelinearcomplementaryproblemapproach.

Y. Y. ChangandR. W. Cottle. Least-index resolutionof degeneracy in quadraticprogramming.MathematicalProgramming, 18(2), 127–137,1980.

Abstract. Combinesleast-index pivot selectionruleswith Keller’s algorithmfor quadraticprogrammingto

obtainafinite methodfor processingdegenerateproblems.

A. CharnesandJ. Semple.Practicalerrorboundsfor a classof quadraticprogrammingprob-lems. Informatica, 2(3), 352–366,1991.

Abstract. The absoluteerror betweenan approximatefeasiblesolution,generatedvia a dual formulation,

andthetrueoptimalsolutionis measured.Theseerrorboundsinvolve considerablylesscomputationalwork

thanexisting estimates.

B. ChenandP. T. Harker. A noninteriorcontinuationmethodfor quadraticandlinearprogram-

ming. SIAMJournalon Optimization, 3(3), 503–515,1993.

M. ChenandJ.A. Filar. Hamiltoniancycles,quadraticprogramming,andrankingof extreme

points. In C. FloudasandP. Pardalos,eds,‘Global Optimization’, pp. 32–9.Princeton

UniversityPress,USA, 1992.

Y.-H. Chenand S. C. Fang. Neurocomputingwith time delay analysisfor solving convex

quadraticprogrammingproblems.IEEETransactionsonNeural Networks, p. (to appear),

1999.

Y. H. Chenand S. C. Fang. Neurocomputingwith time delay analysisfor solving convexquadraticprogrammingproblems.IEEE Transactionson Neural Networks, 11(1), 230–240,2000.

Abstract. This paperpresentsa neural-network computationalschemewith time-delayconsiderationfor

solvingconvex quadraticprogrammingproblems.Basedonsomeknown results,adelaymargin is explicitly

determinedfor the stability of the neuraldynamics,underwhich the statesof the neuralnetwork doesnot

oscillate.The configurationof the proposedneuralnetwork is provided. Operationalcharacteristicsof the

neuralnetwork aredemonstratedvia numericalexamples.

Z. ChenandN. Y. Deng.Somealgorithmsfor theconvex quadraticprogrammingproblemvia

theABS approach.OptimizationMethodsandSoftware, 8(2), 157–170,1997.

F. T. Cheng, T. H. Chen, and Y. Y. Sun. Efficient algorithm for resolving manipulator

redundancy—the compactQP method. In ‘1992 IEEE InternationalConferenceon

RoboticsandAutomation: Proceedings’,Vol. 1–3,pp.508–513,1992a.

F. T. Cheng,T. H. Chen,Y. S. Wang,andY. Y. Sun. Efficient algorithmfor resolvingmanip-ulator redundancy-thecompactQP method. In ‘Proceedings.1992IEEE InternationalConferenceonRoboticsAnd Automation.IEEEComput.Soc.Press,LosAlamitos,CA,USA’, Vol. 1, pp.508–513,1992b.

Abstract. Dueto hardwarelimitations,physicalconstraints,suchasjoint rateboundsandjoint anglelimits,

alwaysexist. In thepresentwork, theseconstraintsareincludedin thegeneralformulationof theredundant

inversekinematicproblem.To take into accountthesephysicalconstraints,the computationallyefficient

compactQP (quadraticprogramming)methodis derived to resolve the kinematicredundancy problem.In

addition,thecompact-inverseQPmethodis developedto remedythesingularityproblem.ThecompactQP


(compactandinverseQP)methodmakesuseof thecompactformulationto obtainthegeneralsolutionsand

to eliminatethe equalityconstraints.As such,the variablesaredecomposedinto basicandfree variables,

andthe basicvariablesareexpressedby the free variables.Thus, the problemsize is reducedandit only

requiresanoptimizationalgorithm,suchasQP, for thefreevariablessubjectto pureinequalityconstraints.

Thisapproachwill expeditetheoptimizationprocessandmake real-timeimplementationpossible.

F. T. Cheng,T. H. Chen,Y. S.Wang,andY. Y. Sun.Obstacleavoidancefor redundantmanipu-latorsusingthecompactQPmethod.In ‘ProceedingsIEEE InternationalConferenceonRoboticsandAutomation.IEEE Comput.Soc.Press,Los Alamitos,CA, USA’, Vol. 3,pp.262–269,1993.

Abstract. ThecompactQP(quadraticprogramming)methodis proposedto resolve theobstacleavoidance

problemfor a redundantmanipulator. The drift-free criterion is consideredwhena redundantmanipulator

performsa repeatedmotion.Dueto thecomputationalefficiency andversatilityof thecompactQPmethod,

real-timeimplementationis ableto beachieved,andphysicallimitationssuchasjoint rateboundsandjoint

anglelimits canbeeasilytaken into account.An exampleis given to demonstratethat this methodis able

to avoid the throatof a cavity, andto remedythedrift problemwhile a primarygoalof themanipulatorsis

carriedout.Simulationresultsshow thatmultiplegoalscaneasilybefulfilled by thismethod.

F. T. Cheng,R. J. Sheu,T. H. Chen,Y. S. Wang, and F. C. Kung. The improved com-pactQP methodfor resolvingmanipulatorredundancy. In ‘IROS ’94. ProceedingsoftheIEEE/RSJ/GIInternationalConferenceonIntelligentRobotsandSystems.AdvancedRoboticSystemsandtheRealWorld. IEEE,New York,NY, USA’, Vol. 2,pp.1368–1375,1994.

Abstract. The compactQP methodis an effective and efficient algorithm for resolvingthe manipulator

redundancy underinequalityconstraints.In this paper, a morecomputationallyefficient schemewhich will

improve the efficiency of the compactQP method-theimproved compactQP method-is developed.With

the techniqueof workspacedecomposition,the redundantinversekinematicsproblemcanbe decomposed

into two subproblems.Thus,thesizeof theredundancy problemcanbereduced.For ann degree-of-freedom

spatialredundantmanipulator, insteadof a 6n matrix, only a 3 n 3 matrix is neededto be manipulated

by Gaussianeliminationwith partialpivoting for selectingthe freevariables.Thesimulationresultson the

CESARmanipulatorindicatethatthespeedupof thecompactQPmethodascomparedwith theoriginalQP

methodis about4.3.Furthermore,thespeedupof theimprovedcompactQPmethodis about5.6.Therefore,

it is believed that the improvedcompactQPmethodis oneof themostefficient andeffective optimization

algorithmfor resolvingthemanipulatorredundancy underinequalityconstraints.

C. C. N. ChuandD. F. Wong.A quadraticprogrammingapproachto simultaneousbuffer inser-tion/sizingandwire sizing. IEEE Transactionson ComputerAidedDesignof IntegratedCircuitsandSystems, 18(6), 787–798,1999.

Abstract. In this paper, we presenta completelynew approachto the problemof delayminimizationby

simultaneousbuffer insertionandwire sizing for a wire. We show that the problemcanbe formulatedas

a convex quadraticprogram,which is known to be solvable in polynomial time. Nevertheless,we explore

somespecialpropertiesof our problemandderive anoptimalandvery efficient algorithm,modifiedactive

setmethod(MASM), to solve the resultingprogram.Given m buffers anda setof m discretechoicesof

wire width, the runningtime of our algorithmis O mn2 andis independentof thewire lengthin practice.

For example,aninstanceof 100buffersand100choicesof wire width canbesolvedin 0.92s. In addition,

we extendMASM to considersimultaneousbuffer insertion,buffer sizing, andwire sizing. The resulting

algorithmMASM-BS is againoptimal andvery efficient. For example,with six choicesof buffer sizeand

10 choicesof wire width, the optimalsolutionfor a 15000µ m long wire canbe found in 0.05s. Besides,

our formulationis soversatilethat it is easyto considerotherobjectiveslike wire areaor power dissipation,

or to addconstraintsto thesolution.Also, wire capacitancelookuptables,or very generalwire capacitance

modelswhichcancaptureareacapacitance,fringing capacitance,couplingcapacitance,etc.canbeused.


C. S. ChungandD. Gale. A complementarityalgorithmfor optimal stationaryprogramsin

growth modelswith quadraticutility. TechnicalReportORC81-10,OperationsResearch

Center, Universityof California,Berkeley, CA, USA, 1981.

S. J.ChungandK. G. Murty. Polynomiallyboundedellipsoidalgorithmfor convex quadratic

programming.In O. L. Mangasarian,R. R. Meyer andS. M. Robinson,eds,‘Nonlinear

Programming,4’, pp.439–485,AcademicPress,LondonandNew York, 1981a.

S.J.ChungandK. G. Murty. Polynomiallyboundedellipsoidalgorithmsfor convex quadraticprogramming.Methodsof OperationsResearch, 40, 63–66,1981b.

Abstract. LetB, b berespectively agivensquarenonsingularintegermatrixof ordern, andanintegercolumn

vectorin IRn. It is requiredto find thenearestpoint to b in theConePos B x : x Bz z 0 . This leads

to thelinearcomplementarityproblem:find w w1 wn , z z1 zn satisfyingW BTB z BTb,

w 0 z 0, wTz 0.

T. T. Chung. Analysisof platebendingby the quadraticprogrammingapproach.Technicalreport,WashingtonUnivesity, St.Louis,MO, USA, 1974.

Abstract. Finite elementanalysisof platebendingis interpretedasa quadraticprogrammingproblem.The

total potentialenergy, expressedin termsof thecoefficientsof theapproximatingpolynomialsis theobjec-

tive function theminimumof which is soughtsubjectto linearequalityconstraints.Theconstraintsrequire

satisfactionof all kinematicboundaryconditionsandinter-elementcontinuityconditions.Convergencechar-

acteristicsof this approachwith respectto increasingordersof polynomialapproximation,aswell aswith

respectto progressively reducedelementsizes,arediscussedand illustratedwith a numberof examples.

Advantagesof theproposedapproacharediscussed,andtopicsrequiringfurtherinvestigationsareoutlined.

L. Churilov, D. Ralph,andM. Sniedovich. A noteon compositeconcave quadraticprogram-ming. OperationsResearch Letters, 23(3–5),163–169,1998.

Abstract. Wepresentapivotal-basedalgorithmfor theglobalminimizationof compositeconcave quadratic

functionssubjectto linearconstraints.It is shown thatcertainsubclassesof this family yield easy-to-solve

line searchsubproblems.Sincethe proposedalgorithmis equivalent in efficiency to a standardparametric

complementarypivoting procedure,the implication is thatconventionalparametricquadraticprogramming

algorithmscannow be usedastools for the solutionof muchwider classof complex global optimization

problems.

T. F. Colemanand L. A. Hulbert. A direct active set algorithm for large sparsequadratic

programswith simplebounds. MathematicalProgramming, SeriesB, 45(3), 373–406,

1989.

T. F. ColemanandL. A. Hulbert.A globallyandsuperlinearlyconvergentalgorithmfor convex

quadraticprogramswith simplebounds.SIAMJournal on Optimization, 3(2), 298–321,

1993.

T. F. ColemanandY. Li. A reflective Newton methodfor minimizing a quadraticfunctionsubjectto boundson someof thevariables.SIAMJournal on Optimization, 6(4), 1040–1058,1996.

Abstract. We proposea new algorithm,a reflective Newton method,for the minimizationof a quadratic

functionof many variablessubjectto upperandlower boundsonsomeof thevariables.Themethodapplies

to a general(indefinite)quadraticfunction for which a local minimum subjectto boundsis requiredand

is particularlysuitablefor the large-scaleproblem.Our new methodexhibits strongconvergenceproperties

andglobalandsecond-orderconvergenceandappearsto havesignificantpracticalpotential.Strictly feasible


pointsaregenerated.We provide experimentalresultson moderatelylarge andsparseproblemsbasedon

bothsparseCholesky andpreconditionedconjugategradientlinearsolvers.

T. F. ColemanandJ.G. Liu. An interior Newton methodfor quadraticprogramming.Mathe-maticalProgramming, 85(3), 491–523,1999.

Abstract. We proposea new (interior) approachfor the generalquadraticprogrammingproblem.We es-

tablishthat thenew methodhasstrongconvergenceproperties:thegeneratedsequenceconvergesglobally

to a point satisfyingthesecond-ordernecessaryoptimality conditions,andtherateof convergenceis 2-step

quadraticif thelimit point is astronglocalminimizer. Publishedalternative interiorapproachesdonotshare

suchstrongconvergencepropertiesfor the nonconvex case.We also report on the resultsof preliminary

numericalexperiments:theresultsindicatethattheproposedmethodhasconsiderablepracticalpotential.

T. F. ColemanandJ.G. Liu. An exteriorNewtonmethodfor strictly convex quadraticprogram-ming. ComputationalOptimizationandApplications, 15(1), 5–32,2000.

Abstract. WeproposeanexteriorNewtonmethodfor strictly convex quadraticprogramming(QP)problems.

Thismethodis basedonadualformulation:asequenceof pointsisgeneratedwhichmonotonicallydecreases

thedualobjective function.Weshow thatthegeneratedsequenceconvergesgloballyandquadraticallyto the

solution(if theQPis feasibleandcertainnondegeneracy assumptionsaresatisfied).Measuresfor detecting

infeasibilityareprovided.Themajorcomputationin eachiterationis to solveaKKT-like system.Therefore,

givenaneffective symmetricsparselinearsolver, theproposedmethodis suitablefor largesparseproblems.

Preliminarynumericalresultsarereported.

D. C. Collins. Terminalstatedynamicprogramming:quadraticcosts,lineardifferentialequa-tions. Journalof MathematicalAnalysisandApplications, 31(2), 235–253,1970.

Abstract. A fairly generalclassof controlproblemscanbeposedin termsof minimizing a costfunctional

involving thestateof thesystemto becontrolledandthecontrolexertedover a fixed interval of time. The

stateandcontrolvariablesarerelatedby a stateequation,oftenwith additionalconstraintsof variousforms

uponthecontrolor state.That is, it is desiredto find miny tεY p x T !"# To q x t $ dt , wherex andy are

theN-dimensionalstatevectorandtheM-dimensionalcontrolvector, respectively, while p andq arescalar-

valuedfunctionsof their arguments.The stateequationis dx dt h x y , x 0 c, with y constrainedto

someclassof admissiblecontrols,Y. This paperdiscussesa specializationof theabove problem,a terminal

controlproblem,in which thecostof stateis measuredonly at theterminaltime,T.

A. R.ConnandJ.W. Sinclair. Quadraticprogrammingvia anon-differentiablepenaltyfunction.

TechnicalReportCORR75/15,Facultyof Mathematics,Universityof Waterloo,Ontario,

Canada,1975.

A. R. Conn, N. I. M. Gould, and Ph. L. Toint. A primal-dualalgorithm for minimizing anonconvex functionsubjectto boundandlinearequalityconstraints.In G. Di Pillo andF. Giannessi,eds,‘NonlinearOptimizationandRelatedTopics’,pp.15–50,Kluwer Aca-demicPublishers,Dordrecht,TheNetherlands,1999.

Abstract. A new primal-dualalgorithmis proposedfor theminimizationof non-convex objective functions

subjectto simpleboundsandlinearequalityconstraints.Themethodalternatesbetweena classicalprimal-

dual stepanda Newton-like modifiedbarrierstepin orderto ensuredescenton a suitablemerit function.

Convergenceof a well-definedsubsequenceof iteratesis proved from arbitrarystartingpoints.Preliminary

numericalresultsfor quadraticprogrammingproblemsarepresented.

A. R. Conn,N. I. M. Gould,D. Orban,andPh.L. Toint. A primal-dualtrust-region algorithmfor non-convex nonlinearprogramming.MathematicalProgramming, 87(2), 215–249,2000.


Abstract. A new primal-dualalgorithmis proposedfor theminimizationof non-convex objective functions

subjectto generalinequality and linear equality constraints.The methodusesa primal-dual trust-region

modelto ensuredescenton a suitablemerit function.Convergenceis provedto second-ordercritical points

from arbitrarystartingpoints.Numericalresultsarepresentedfor generalquadraticprograms.

G.C.Contaxis,C.Delkis,andG.Korres.Decoupledoptimalloadflow usinglinearor quadraticprogramming.IEEETransactionson PowerSystems, PWRS-1(2), 1–7,1986.

Abstract. Theauthorsproposeaniterative schemefor theoptimal loadflow (OLF) problem,a constrained

optimizationproblemof large sizeandgreatcomplexity. For online implementation,fastexecutiontimes

andminimumcomputerstoragearerequired.Theiterative schemedecomposestheoptimal loadflow prob-

lem into realandreactive subproblems.Therealandreactive subproblemsaresolvedalternatelyuntil they

converge.Quadraticor linearprogrammingis utilized to solve thetwo subproblems.If thecostcurveof each

generatoris approximatedasa quadraticfunctionthenthecostfunctionof theOLF problemis in quadratic

form andquadraticprogrammingis appliedfor the solution. If the effect of valve point loading is to be

examined,thecostfunctionof theOLF problembecomeslinearandlinearprogrammingis applied.

G. C. Contaxis,C. Delkis, E. Glytsis,andB. C. Papadias.Economicpower dispatchwith linesecuritylimits usingZ-matrix techniquesandquadraticprogramming.In E. LaugerandJ. Moltoft, eds,‘Reliability in ElectricalandElectronicComponentsandSystems.FifthEuropeanConferenceon ElectrotechnicsEUROCON’82’, pp.709–713,NorthHolland,Amsterdam,theNetherlands,1982.

Abstract. Presentsan efficient algorithmfor solving the economicpower dispatchproblem.The general

problemof the economicpower dispatchis formulatedasa non-linearconstrainedoptimizationproblem,

recognizingsystemlosses,operatinglimits on thegenerationunitsandline securitylimits. Theoptimization

problemis solved by an iterative scheme.At eachiteration the optimizationproblemis transformedto a

quadraticprogrammingproblemby utilizing Z-matrixtechniquesandthegeneralizedgenerationdistribution

factors.

G. C. Contaxis,C. Delkis,B. Paraskevopoulos,andB. C. Papadias.Economicpower dispatchusingquadraticprogrammingandZ-matrix. In ‘Electrotechnologyfor Development.Pro-ceedingsof MELECON’81 theFirst MediterraneanElectrotechnicalConference.IEEE,New York, NY, USA’, pp.67–70,1981.

Abstract. Describesanalgorithmdevelopedat theNationalTechnicalUniversityof Athensfor solvingthe

problemof economicpower dispatch.This algorithmusesquadraticprogrammingandZ-matrix techniques

to minimizethetotal operationalcostof thethermalunits.Thealgorithmrecognizesoperatinglimits on the

generationunitsandcaneasilybemodifiedto accountfor theline securitylimits also.

B. L. Contesse.Une caracterisationcompletedesminima locauxen programmationquadra-tique. NumerischeMathematik, 34(3), 315–332,1980.

Abstract. The generalsecondorder conditionsfor the characterizationof local minima areshown to be

necessaryandsufficient in thecaseof aquadraticobjective functionsubjectto linearconstraints.Specifically,

it is shown that the well known secondorderhand,that the well known secondordersufficient condition

for an isolatedlocal minimum is, actually, necessary. Theseresultsare proved from scratchand include

severalexistingones.In theparticularcaseof equalityconstrainedquadraticproblems,Orden’ssecondorder

conditionsfor globalminimaarerecovered.

F. Cotiu. On the quadraticprogrammingproblem. In ‘4th conferenceon probability theory.Abstracts.AcadSocialistRepublicof Rumania,Bucharest,Romania’,p. 42,1971.

Abstract. Quadraticprogrammingis consideredin thecasein which conditionsarelinearandthefunction

to beoptimizedis of a positively definedquadraticform. A methodof solvingsucha programis discussed

andanalgorithmfor solvingquadraticprogramswith mutuallyexclusive constraintsis alsogiven.


J. S. CotnerandR. R. Levary. A quadraticprogrammingmodel for determiningshort-termmultiple currency portfolios. Opsearch, 24(4), 218–227,1987.

Abstract. Basedon Eurocurrency depositinterestratesandcurrency exchangerates,a quadraticprogram-

ming model is developedfor defining optimal short-termmultiple currency-denominatedportfolios. It is

shown thatthis modelcanbeappliedin definingefficient portfoliosat variouslevelsof expectedreturn,and

that investingin thesemulticurrency portfolios will provide higher risk-adjustedreturnsthancomparable

single-currency holdings.

R.W. Cottle.Symmetricdualquadraticprograms.Quarterlyof AppliedMathematics, 21, 237–

243,1963.

R. W. Cottle. A fundamentaltheoremin quadraticprogramming. Journal of the Societyof

Industrial andAppliedMathematics, 12, 663–665,1964.

R. W. Cottle. On the convexity of quadraticforms over convex sets. OperationsResearch,

15, 170–173,1967.

R. W. Cottle. The principal pivoting methodof quadraticprogramming. In G. B. Dantzig

andA. F. Veinott,eds,‘Lectutesin Applied MathematicsII’, number1 in ‘Mathematics

of the DecisionSciences’,pp. 144–162,AmericanMathematicalSociety, Providence,

RhodeIsland,USA, 1968.

R. W. Cottle. Threeremarksabouttwo paperson quadraticforms. Zeitschrift fur Operations

Research, 19, 123–124,1975.

R. W. Cottle. Fundamentalsof quadraticprogrammingand linear complementarity. In

M. Z. CohnandG. Maier, eds,‘EngineeringPlasticityby MathematicalProgramming’,

pp.293–323.PergamonPress,New York, USA, 1979.

R. W. Cottle andA. Djang. Algorithmic equivalencein quadraticprogramming.I. A least-distanceprogrammingproblem. Journal of Optimization Theory and Applications,28(3), 275–301,1979.

Abstract. It is demonstratedthat Wolfe’s algorithmfor finding the point of smallestEuclideannorm in a

given convex polytopegeneratesthe samesequenceof feasiblepointsasdoesthe van dePanne-Whinston

symmetricalgorithmappliedto the associatedquadraticprogrammingproblem.Furthermore,it is shown

how thelatteralgorithmmaybesimplifiedfor applicationto problemsof this type.

R.W. CottleandM. S.Goheen.A specialclassof largequadraticprograms.In O.L. Mangasar-

ian, R. R. Meyer andS. M. Robinson,eds,‘Nonlinear Programming,3’, pp. 361–390,

AcademicPress,LondonandNew York, 1977.

R. W. Cottle and W. C. Mylander. Ritter’s cutting planemethodfor nonconvex quadraticprogramming. In J. Abadie, ed., ‘Integer and nonlinearprogramming’,pp. 257–283,North Holland,Amsterdam,theNetherlands,1970.

Abstract. Reviews the methodsof Ritter (1964,1965,1966)for obtaininga global solutionto a linearly-

constraintedquadraticminimizationproblem.Theproblemconcernedcanbestatedwithoutlossof generality

asminimizeφ x% cTx 1 2xT Dx subjectto Ax b, x 0,wherethematrixA is of orderm nandD DT .

R. W. Cottle andS. Schaible. On pseudoconvex quadraticforms. In E. F. Beckenbach,ed.,

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R. W. Cottle, G. J. Habetler, andC. E. Lemke. Quadraticforms semi-definiteover convex

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R. W. Cottle, J.-S.Pang,andR. E. Stone. TheLinear ComplementarityProblem. Academic

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Abstract. Let A be a real symmetricpositive definite n n andb a real column n-vector, then find real

columnn-vectorsx andy suchthatAx b y, andxTy 0 x 0 y 0, Problemsof this typeoccurwhen

themethodof Christophersonis usedto solve freeboundaryproblemsfor journalbearings.In suchcases,A

is a ’finite difference’matrix.A methodfor solvingtheabove problemis presentedwhich is a modification

of systematicover- relaxation.This methodis particularlysuitablewhenA is afinite differencematrix.

W. Cui andD. I. Blockley. Decisionmakingwith fuzzy quadraticprogramming.Civil Engi-neeringSystems, 7(3), 140–147,1990.

Abstract. Decisionmakingin civil engineeringsystemsnecessarilyinvolves imprecisedata.Fuzzylinear

programming(FLP) hasbeenusedto deal with imprecisionin parameterdefinitions.In this paper, it is

arguedthatpresentmethodsof FLP canbe improveduponby usinga new methodof fuzzy quadraticpro-

gramming(FQP)whichenablesthemodellingof independentfuzzyparameters.Severalnumericalexamples

arepresentedandcomparedwith othertechniques.

J.W. Daniel. Stability of thesolutionof definitequadraticprograms.MathematicalProgram-

ming, 5(1), 41–53,1973.

N. P. Danilkin, P. F. Denisenko,andV. V. Sotskiy. Computationof ionosphericNh profilesby

quadraticprogramming.GeomagnetismandAeronomy, 15(2), 292–293,1975.

Abstract. Proposesa methodmaking it possibleto obtain from two componentsa nonmonotonousN hprofileunderadverseconditions,i.e.whenthevirtual heightsareinsufficiently accuratelydetermined,in the

presenceof horizontalN gradients,andin computationsfrom averagedionograms.

G. DanningerandI. M. Bomze. Using copositivity for global optimality criteria in concavequadratic-programmingproblems.MathematicalProgramming, 62(3), 575–580,1993.

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quadraticminimizationproblems.Using this condition,it follows that, from the perspective of worst-case

complexity of concave quadraticproblems,the differencebetweenlocal andglobal optimality conditions

is not as large as in general.As an essentialingredient,we hereusethe ε-subdifferential calculusvia an

approachof Hiriart-Urruty andLemarechal(1990).

G. B. Dantzig. Quadraticprogramming:A variantof the Wolfe-Markowitz algorithms. Re-

searchreport2, OperationsResearchCenter, Universityof California,Berkeley, Califor-

nia,USA, 1961.

G. B. Dantzig. Quadraticprogramming. In ‘Linear ProgrammingandExtensions’,chapter

12-4,pp.490–498.PrincetonUiversityPress,Princeton,USA, 1963.


M. D’Apuzzo, V. De Simone,andG. Toraldo. A parallelalgorithmfor box constrainedcon-

vex quadraticprogramming.TechnicalReportTR-96-8,Centerfor Researchon Parallel

ComputingandSupercomputers,Universityof Naples”FedericoII”, Italy, 1996a.

M. D’Apuzzo, V. De Simone,andG. Toraldo. Parallel softwarefor box constrainedconvex

quadraticprogramming. TechnicalReportTR-96-13,Centerfor Researchon Parallel

ComputingandSupercomputers,Universityof Naples”FedericoII”, Italy, 1996b.

M. S.Daskin,J.L. Schofer, andA. E. Haghani.A quadraticprogrammingmodelfor designingandevaluatingdistance-basedandzonefaresfor urbantransit.TransportationResearch,Part B (Methodological), 22(1), 25–44,1988.

Abstract. A microcomputerbasedsystemfor designingandevaluatingdistance-basedandzonefaresfor

transitpropertiesis described.At theheartof thesystemis anoptimizationmodelthatfindsthefixedcharge,

mileagecharge,andtransfercharge thatmaximizegrossrevenuesubjectto constraintson ridershipandthe

form of the fareequation.A linear approximationto the demandcurve at thebasecasevaluesresultsin a

quadraticprogrammingproblem.Threealternative modesof usingthemodelsystemaredemonstratedusing

selecteddatafrom theChicagoTransitAuthority. Modelextensionsandproposedfuturework areoutlined.

V. A. Daugavet andA. V. Lazarev. Developmentof Dantzig’s methodin quadraticprogram-

ming.USSRComputationalMathematicsandMathematicalPhysics, 26(2),67–72,1986.

V. A. DaugavetandT. M. Salikh. A complementarybasisprocedurefor convex quadraticpro-gramming.ZhurnalVychislitel’noi Matematikii Matematicheskoi Fiziki, 32(12), 1981–1992,1992.

Abstract. Thecomplementarybasisprocedure,usedfor thesolutionof quadraticprogrammingproblems,is

basedon ideasof theLemke, Wolfe andDantzigmethods.It canbeappliedto anarbitrarycaseof convex

quadraticprogramming,without reducingit to any kind of astandardform.

A. Dax. Thegradientprojectionmethodfor quadraticprogramming.Technicalreport,Insitute

of Mathematics,TheHebrew Universityof Jerusalem,Isreal,1978.

A. Dax. An active setalgorithmfor convex quadraticprogramming.Technicalreport,Hydro-

logical Service,Jerusalem,Israel,1982.

G. Dayal, L. L. Grigsby, and L. Hasdorff. Quadraticprogrammingfor optimal active andreactivepowerdispatchusingspecialtechniquesfor reducingstoragerequirements.IEEEPower EngineeringSocietySummerMeeting. (Text of A papers) IEEE, New York, NY,USA, pp.A76 388–389/1–5,1976.

Abstract. This paperpresentssomeof the techniqueswhich canbeeffectively employed for reducingthe

computationaleffort andstoragerequirements,whenlinear andquadraticprogrammingmethodsareused

for solving large size power systemproblems.The effectivenessof theseprocedureshasbeentestedby

implementingthemin a new computercode,speciallydevelopedfor solving economicdispatchproblem.

Theapproved IEEE testsystemsup to 118buseswereusedfor thetestpurposes.Theresultsshow that the

improvementin thecaseof 118bussystemis ratherdramatic.Numberof nonzeroentriesin thebasisinverse,

while usingthenew code,arefoundto beonly 25%of thosereportedearlier.

O. de Donatoand A. Franchi. A modified gradientmethodfor finite elementelastoplasticanalysisby quadraticprogramming.ComputerMethodsin AppliedMechanicsandEn-gineering, 2(2), 107–131,1973.


Abstract. Thefinite analysisproblemwith piecewise linearconstitutive laws is formulatedasa linearcom-

plementarityanda quadraticprogrammingproblem.The solution techniquesusing the well-known opti-

mizationmethodsof mathematicalprogrammingarediscussedanda procedure,belongingto the classof

gradientmethods,is proposedwhichovercomesthecomputationaldifficultiesthatarisewhenthereis a large

numberof variables.Throughaphysicalinterpretationof thegradientof theobjective function,eachmathe-

maticalstepof theproposedoptimizationtechniqueis translatedinto acorrespondingphysicaloperationon

thestructureandamechanicalsolutionprocedurewith afinite numberof stepsis derived.Finally, for thein-

crementalanalysisproblemundernon-holonomicconstitutive laws thesameprocedureis adoptedcombined

with themultistageloadingtechnique.Illustrative examplesfor eachof theprecedingproblemsaregiven.

G. Debreu.Definiteandsemidefiniteforms. Econometrica, 20, 295–300,1952.

H. Dejonghe. A quadraticprogrammingtechniquefor modelinggravitating systems.Astro-physicalJournal, 343(1), 113–124,1989.

Abstract. Thepaperpresentsa generalmethodthatproducesanalyticdistribution functionsfor gravitating

systems.A typical modelfits photometricandkinematicaldataandis a superpositionof components.The

coefficients of the linear combinationaredeterminedby a quadraticprogrammingtechnique.The author

givesanexplicit examplefor systemswith sphericalsymmetry, andsubsequentlyappliesthemethodto the

ComaClusterof galaxies.Statisticallysignificantfits areobtainedusingonly asmallnumberof components.

This techniqueis alsoa dynamicalmassestimator. The authorobtainsmassesbetween1 7 & 1015h' 150 M (

and5 & 1015h' 150 M ( within 5h' 1

50 Mpc, andfinds someevidencefor luminosity segregation in the orbital

structure.

C. L. DeMarcoandD. M. Divan. Synthesisof optimaldiscretepulsemodulationwaveformsthroughanintegerquadraticprogram. In ‘1988 IEEE InternationalSymposiumon Cir-cuits and Systems.Proceedings.IEEE, New York, NY, USA’, Vol. 2, pp. 1377–1380,1988.

Abstract. Theauthorsintroduceaphysicallybasedobjective functionfor determiningoptimaldiscretepulse

modulationwaveformsin power electronicsapplications.Using this objective function, it is demonstrated

that theproblemof selectinganoptimalwaveformreducesto a quadratic0–1 integer program.It is shown

thatthestructureof this 0–1programlendsitself to a boundingalgorithmthatallows many of thebranches

in thesearchtreeto beeliminated,considerablyreducingthecomputationalcost.

R. S. DemboandU. Tulowitzki. On the minimizationof quadraticfunctionssubjectto box

constraints.Schoolof OrganizationandManagementWorking paperSeriesB no. 71,

YaleUniversity, Yale,USA, 1983.

R. S.DemboandU. Tulowitzki. Computingequilibriaon largemulticommoditynetworks: anapplicationof truncatedquadraticprogrammingalgorithms.Networks, 18(4), 273–284,1988.

Abstract. Presentsa generalschemefor improving theasymptoticbehavior of a given nonlinearprogram-

ming algorithmwithout incurringa significantincreasein storageoverhead.To enhancetherateof conver-

gencetheauthorscomputesearchdirectionsby partially solvingasequenceof quadraticprogramming(QP)

problems.Theideais illustratedonaclassof extremelylargenonlinearprogrammingproblemsarisingfrom

traffic equilibrium calculationsusingboth the Frank-Wolfe andPARTAN algorithmsto partially solve the

QP subproblems.Computationalresultsindicatethat the convergencerate of the underlyingalgorithm is

indeedenhancedsignificantlywhenFrank-Wolfe is usedto solve theQPsubproblemsbut only marginally

soin thecaseof PARTAN. It is conjectured,andsupportedby thetheory, thatwith betteralgorithmsfor the

QPsubproblemstheimprovementsdueto theproposedframework would bemoremarked.

N. DemokanandA. H. Land. A parametricquadraticprogramto solve a classof bicriteriadecisionproblems.Journalof theOperationalResearch Society, 32(6), 477–488,1981.


Abstract. Presentsanextensionof Beale’s quadraticprogrammingalgorithmto performparametricanalysis

on theright handsideelementof a singleconstraint.Specificexamplesaregiven to illustratetheutilization

of themethodfor a classof bi-criteriadecisionproblems,whereoneof thecriteriacanbe formulatedasa

quadraticfunction,suchasrisk, sumof squareddeviationsor a utility function,andtheotheroneis usually

in linearcostor returnfunctionwhichcanbetreatedasaparametricconstraint.

D. denHertog. Interior point approach to linear, quadratic, and convex programming: al-

gorithmsand complexity. Kluwer AcademicPublishers,Dordrecht,The Netherlands,

1994.

D. denHertog,C. Roos,andT. Terlaky. A polynomialmethodof weightedcentersfor convex

quadraticprogramming.Journal of Information& OptimizationSciences, 12(2), 187–

205,1991.

M. A. Diamond.Thesolutionof aquadraticprogrammingproblemusingfastmethodsto solvesystemsof linear equations. InternationalJournal of SystemsScience, 5(2), 131–136,1974.

Abstract. Let A bea realsymmetricpositive definiten & n matrixandb a realcolumnn-vector. Theproblem

of finding n-vectorsx and y suchthat Ax b y, xT y 0 x 0andy 0, occurswhen the methodof

Christophersonis usedto solve free boundaryproblemsfor journal bearings.In this caseA is a ’finite

difference’matrix. Theauthorpresentsa directmethodfor solvingtheabove problemby solvinga number

of linear systemsAkx bk. Eachsystemcanbe solved usingoneof the recentlydevelopedfastdirect or

iterative procedures.Heconsidersthesolutionby factorizationtechniques.

C. R. Dietrich. Unit graphestimationandstabilizationusingquadraticprogramminganddif-

ferencenorms—reply. Water ResourcesResearch, 31(10),2635,1995.

C. R. Dietrich andT. G. Chapman. Unit graphestimationandstabilizationusingquadraticprogramminganddifferencenorms.WaterResourcesResearch, 29(8),2629–2635,1993.

Abstract. For a rainfall-runoff event,quadraticprogrammingis particularlysuitedto theestimationof unit

graphaslinear andpositivity constraintson unit graphordinatescanbenaturallyimplemented.In this pa-

per, thequadraticprogrammingframework is invoked in a novel way to stabilizetheunit graphestimation

procedurevia theuseof differencenorms.Theadvantageof thelatterover standardridgeregressionis that

penaltiesareplacedon oscillationsof theunit graphratherthanon thesizeof its ordinates.Applicationof

themethodologyto realrainfall-runoff datais providedandcomparisonswith existingapproachesaremade.

Thelatterindicatethatourapproachgenerallyyieldsunit graphestimatesof morerealisticshapeandsmaller

varianceresultingin abetterfit to therunoff data.

I. I. Dikin. Iterative solution of problemsof linear and quadraticprogramming. Doklady

AkademiiaNauk USSR, 174, 747–748,1967. Seealso, Soviet MathematicsDoklady

volume8, pages674–675,1967.

V. V. Dikusar. Extendedquadraticprogrammingproblem.DokladyAkademiiNauk, 349(1),29–

31,1996.

M. A. Diniz-Ehrhardt,M. A. Gomes-Ruggiero,andS.A. Santos.Numericalanalysisof leaving-faceparametersin bound-constrainedquadraticminimization. TechnicalReport52/98,Departmentof AppliedMathematics,IMECC-UNICAMP, Campinas,Brasil,1998.

Abstract. In this work, we focusour attentionon the quadraticsubproblemof trust-region algorithmsfor

large-scalebound-constrainedminimization.An approachthatcombinesa mild active setstrategy with gra-

dientprojectiontechniquesis employedin thesolutionof large-scalebound-constrainedquadraticproblems.


To fill in somegapsthathave appearedin previouswork, we propose,testandanalyzeheuristicswhich dy-

namicallychoosetheparametersin chargeof thedecisionof leaving or notthecurrentfaceof thefeasibleset.

Thenumericalanalysisis basedon problemsfrom CUTE collectionandrandomlygeneratedconvex prob-

lemswith controlledconditioninganddegeneracy. Thepracticalconsequencesof anappropriatedecisionof

suchparametersareshown to becrucial,particularlywhendualdegenerateandill-conditionedproblemsare

solved.

A. Djang. Algorithmic equivalencein quadratic programming. PhD thesis,Departmentof

OperationsResearch,StanfordUniversity, Stanford,California,USA, 1979.

M. Dormany. A parametricmethodfor thesolutionof indefinitequadraticprogrammingprob-lems.Szigma, 13(4), 285–303,1980.

Abstract. Thesolutionof indefinitequadraticprogrammingproblemsraisesseriousdifficulties.Parametric

methodsyield a way of solutionwhena partof programvariablesaretreatedasparametersanda seriesof

multiparametricconvex quadraticor linearprogrammingproblemsof reducedsizecanbe formulated.The

procedurepresentedin this paperdraws attentionto a new possibilityof parametrizationmakinguseof the

minor matricestaken from themaindiagonalof an indefinitematrix D. Primal-dualmultiparametriclinear

programmingis dealtwith in detail; furthermore,it is shown how the efficiency of the algorithmcanbe

raisedby usingBranchandBoundtechniques.

W. Dorn. Duality in quadraticprogramming.Quarterlyof AppliedMathematics, 18, 155–162,

1960a.

W. S. Dorn. Self-dualquadraticprograms.Journal of the Societyfor Industrial andApplied

Mathematics, 9, 51–54,1960b.

W. S.Dorn. A symmetricdualtheoremfor quadraticprogramming.Journalof theOperations

Research Societyof Japan, 2, 93–97,1960c.

Z. Dostal. On the penaltyapproximationof quadraticprogrammingproblem. Kybernetika,27(2), 151–154,1991.

Abstract. An upperboundfor the differenceof the exact solution of the problemof minimization of a

quadraticfunctionalon thesubspaceandits penaltyapproximationis given.Thepapersuppliesanumerical

example.

Z. Dostal. Box constrainedquadraticprogrammingwith controlled precisionof auxiliaryproblemsand applications. Zeitschrift fur AngewandteMathematikund Mechanik,76(S3),413–414,1996.

Abstract. We review our recentresultson the solution of quadraticprogrammingproblemswith simple

boundsby meansof theconjugategradientmethodwith inexactsolutionof auxiliary subproblemsandpro-

jections.Precisionof the solutionof auxiliary problemsis controlledby the productof a positive constant

Gammawith thenormof violationof theKuhn-Tuckercontactconditions.Theresultingalgorithmconverges

for any positive Gammaandreachesthe solutionin a finite numberof stepsprovided the problemis non-

degenerate.A lower boundon Gammais given sothat thefinite terminationpropertyis preserved even for

degenerateproblems.Thealgorithmmaybeimplementedwith projectionssothatit candropandaddmany

constraintswhenever theactive setis changed.Applicationsto thesolutionof innerobstacleproblemsand

contactproblemsof elasticityarereported.

Z. Dostal. Box constrainedquadraticprogrammingwith proportioningandprojections.SIAMJournalon Optimization, 7(3), 871–887,1997.


Abstract. Two new closelyrelatedconceptsareintroducedthatdependon a positive constantGamma. An

iterationis proportionalif thenormof violation of theKuhn-Tucker conditionsat active variablesdoesnot

excessively exceedthenormof thepartof thegradientthatcorrespondsto freevariables,while aprogressive

directiondeterminesa descentdirection that enablesthe releasedvariablesto move far enoughfrom the

boundaryin astepcalledproportioning.An algorithmthatusestheconjugategradientmethodto explorethe

faceof theregion definedby thecurrentiterateuntil a disproportionaliterationis generatedis proposed.It

thenchangesthe faceby meansof theprogressive direction.It is proved that for strictly convex problems,

the proportioningis a spaceriteration so that the algorithm converges to the solution. If the solution is

nondegeneratethenthe algorithmfinds the solutionin a finite numberof steps.Moreover, a simplelower

boundon Gammais given to ensurefinite terminationeven for problemswith degeneratesolutions.The

theorycoversaclassof algorithms,allowing many constraintsto beaddedor droppedatatimeandaccepting

approximatesolutionsof auxiliaryproblems.Preliminarynumericalresultsarepromising.

Z. Dostal, A. Friedlander, andS. A. Santos.Adaptive precisioncontrol in quadraticprogram-

ming with simpleboundsand/orequalityconstraints.In R. D. Leone,A. Murli, P. M.

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1998.

Z. Dostal, A. Friedlander, andS. A. Santos.AugmentedLagrangianswith adaptive precisioncontrol for quadraticprogrammingwith equalityconstraints.ComputationalOptimiza-tion andApplications, 14(1), 37–53,1999.

Abstract. In this paperwe introduceanaugmentedLagrangiantypealgorithmfor strictly convex quadratic

programmingproblemswith equalityconstraints.The new featureof the proposedalgorithmis the adap-

tive precisioncontrolof thesolutionof auxiliary problemsin the inner loop of thebasicalgorithm.Global

convergenceandboundednessof thepenaltyparameterareproved andanerrorestimateis given thatdoes

not have any term thataccountsfor the inexact solutionof the auxiliary problems.Numericalexperiments

illustrateefficiency of thealgorithmpresented.

I. Doukhovni andD. Givoli. Quadraticprogrammingalgorithmsfor obstacleproblems.Com-municationsin NumericalMethodsin Engineering, 12(4), 249–256,1996.

Abstract. The numericalsolutionof problemsinvolving frictionlesscontactbetweenan elasticbody and

a rigid obstacleis considered.The elasticbody may undergo small or large deformation.Finite element

discretizationandrepetitive linearizationleadto a sequenceof quadraticprogramming(QP) problemsfor

incrementaldisplacement.The performancesof several QP algorithms,including two new versionsof a

modifiedsteepestdescentalgorithm,arecomparedin this context. Numericalexamplesincludea string,a

membraneandanEuler-Bernoulli beam,in contactwith flat andnon-flatrigid obstacles.

J. P. Dussault,J. A. Ferland,andB. Lemaire. Convex quadraticprogrammingwith onecon-straintandboundedvariables.MathematicalProgramming, 36(1), 90–104,1986.

Abstract. Theauthorsproposeaniterative algorithmfor solvingaconvex quadraticprogramwith oneequal-

ity constraintandboundedvariables.At eachiteration,aseparableconvex quadraticprogramwith thesame

constraintsetis solved.Two variantsareanalyzed:onethatusesanexact line search,andtheothera unit

stepsize.Preliminarytestingsuggeststhat this approachis efficient for problemswith diagonallydominant

matrices.

N. Dyn andW. E.Ferguson.Thenumericalsolutionof equality-constrainedquadraticprogram-ming problems.Mathematicsof Computation, 41(163),165–170,1983.

Abstract. Thispaperprovesthata largeclassof iterative schemescanbeusedto solve acertainconstrained

minimizationproblem.The constrainedminimizationproblemconsideredinvolves the minimizationof a

quadraticfunctionalsubjectto linearequalityconstraints.Amongthisclassof convergentinteractive schemes

aregeneralizationsof therelaxedJacobi,Gauss-Seidel,andsymmetricGauss-Seidelschemes.


B. C. Eaves.Onquadraticprogramming.ManagementScience, 17(11),698–711,1971.

Abstract. A procedurebasedonLemke’salgorithmis developedwhicheithercomputesstationarypointsfor

generalquadraticprogramsor elseshows that theprogramhasno optimum.If a generalquadraticprogram

hasanoptimumandsatisfiesanondegeneracy conditionthenit is demonstratedthatthereareanoddnumber

of stationarypoints.

B. C. Eaves. A finite procedurefor determiningif a quadraticform is boundedbelow on a

closedpolyhedralconvex set.MathematicalProgramming, 14(1), 122–124,1978.

J.G. Ecker. Computationalproceduresin quadraticprogrammingand ) p- approximation.Bul-letin of theOperationsResearch Societyof America, 17(2), B237–238,1969.

Abstract. Abstractonly given, substantiallyas follows. In the framework of a duality theoryof extended

geometricprogramming,computationalproceduresaregiven for obtainingoptimal solutionsto quadratic

programswith quadraticconstraintsandprogramswhich minimizethe * p-normof thedifferencebetweena

fixedvectorandvariablelinearcombinationof otherfixedvectors,subjectto inequalityconstraintsexpressed

by meansof * p-norms.To apply theprocedures,onebasicallyneedonly solve a finite ’Duffin sequence’of

linearly-constrained,concave, uppersemi-continuousprograms.Thegeneralclassof programsto which the

computationalproceduresapplyproperlycontainsavarietyof specialproblems.

U. Eckhardt.Quadraticprogrammingby successiveoverrelaxation.TechnicalReportJul-1064-MA, Kernforschungsanlage,Juelich,WestGermany, 1974.

Abstract. Theideaof solvingthedefinitelinearcomplementarityproblemby successive overrelaxationwas

originally proposedby Cryer. A detaileddiscussionof Cryer’s methodappliedto quadraticprogramming

problemsis given.Theconvergencebehaviour is treatedwithout assumptionsonsolvability of theproblem.

Numericalexamplesindicatetheefficiency of themethod.

U. Eckhardt.Iterative losungquadratischeroptimierungsaufgaben.Zeitschrift fur Angewandte

MathematikundMechanik, 55, T236–T237,1975.

U. Eckhardt.Semi-infinitequadraticprogramming.ORSpektrum, 1(1), 51–55,1979.

Abstract. A methodis presentedfor minimizing a definitequadraticfunction underan infinite numberof

linearinequalityrestrictions.Specialfeaturesof themethodarethatit generatesasequenceof feasiblesolu-

tionsandasequenceof basicsolutionssimultaneouslyandthat it hasvery favourablepropertiesconcerning

numericalstability.

U. Eckhardt. Linear inequalitiesandquadraticprogramming-someapplications.MethodsofOperationsResearch, 53;, 67–81,1986.

Abstract. Someapplicationsof linear inequality systemsand quadraticprogrammingproblemsare pre-

sented.It is not intendedto go into theoreticaldetailsor to investigatenumericalspecialities.

M. M. El Metwally andZ. M. Al Hamouz. Transmissionnetworks planningusingquadraticprogramming.ElectricMachinesandPowerSystems, 18(2), 137–148,1990.

Abstract. Theproblemof transmissionnetworksexpansionhasbeensolvedby consideringthecostof losses

aswell asthecostof investmentin theobjective function.Theproblemis solvedusinganexactquadratic

programmingtechnique.Thisnew formulationhasbeenappliedto a6-bussystem.Thefinal configurations,

which arecharacterizedby minimum costof lossesshow that as the costof the kWh increases,the total

systemcostdecreases.

M. M. El Metwally andA. M. Harb. Transmissionplanningusingadmittanceapproachandquadraticprogramming.ElectricMachinesandPowerSystems, 21(1), 69–83,1993.


Abstract. A methodfor transmissionnetworks planningis proposedusingquadraticprogrammingandthe

admittanceapproach.Thecostof investmentandcostof losses,load flow andsecurityconstraintsandthe

interestandinflation ratesareincluded.Thedevelopedmethodcanbeusedfor staticanddynamicmodesof

transmissionplanning.

M. A. H. El Sayed,T. M. Abdel Rahman,andM. O. Mansour. A fastquadraticprogrammingapproachfor large-scalereactivepoweroptimization.Electric MachinesandPowerSys-tems, 20(1), 17–23,1992. Seealso,EuropeanTransactionson ElectricalPower Engi-neering,volume2, number4, pages253–257,1992.

Abstract. A fastquadraticprogrammingapproachhasbeendevelopedto control thereactive power (VAR)

generationin leaddispatchingcenters.The main objective of VAr control is to minimize the transmission

lossestakinginto considerationthevoltageconstraintsateachnodeof thenetwork. Thedevelopedapproach

utilizes the decompositionof the whole systeminto smallersubsystemsandthe coordinationof different

subsystemsolutionsto obtainoptimalVAr control in the largescalesystems.Thesmallestpracticalsizeof

thedecomposedsubsystemsis theone-generator-bussubsystem.Thefastandnon-iterative VAr optimization

of the decomposedsubsystemsis obtainedbasedon consideringonly one active constraintat a time by

solving the quadraticprogrammingsub-problems.The numericaltest resultsof IEEE-118and200 buses

power systemhave indicatedthat the speedandaccuracy of the proposedapproachareadequatefor real-

time applicationson large-scalesystems.This approachis alsoapplicablefor largesystemdeviationsfrom

its normalconditionsandneedssmallmemorysize.

M. A. El Shibini andE. S. Ibrahim. Quadraticprogrammingapproachto reactive power opti-mizationon theprimaryfeeders.Archiv fur Elektrotechnik, 68(4), 267–271,1985.

Abstract. Suggeststheuseof thequadraticprogrammingtechniqueto determinetheoptimumsizeandlo-

cationof shuntcapacitorsonradialdistribution feederssoasto maximizeoverall savings,includingthecost

of capacitors.The saving function which is of quadraticform is maximizedfor a setof linear inequality

constraintsby usingquadraticprogramming.For quadraticprogramming,efficient algorithmshave beende-

velopedwhichcaneasilybeimplementedondigital computers.Theapproachis illustratedby anapplication

to a typicaldistribution feederof 23kV.

N. Elia andM. A. Dahleh. A quadraticprogrammingapproachfor solving the ) 1 multiblockproblem. IEEE Transactionson AutomaticControl, 43(9), 1242–1252,1998. Seealso,Proceedingsof the 35th IEEE Conferenceon DecisionandControl, IEEE, New York,NY, USA, volume4, pages4028–4033,1996.

Abstract. We presenta new methodto computesolutionsto the generalmulti-block l1 control problem.

Themethodis basedon solvinga standardH2 problemanda finite-dimensionalsemidefinitequadraticpro-

grammingproblemof appropriatedimension.The new methodhasmostof the propertiesthat separately

characterizemany existing approaches,in particular, asthedimensionof thequadraticprogrammingprob-

lemincreases,thismethodprovidesconverging upperandlowerboundsontheoptimal * 1 normand,for well

posedmulti-block problems,ensurestheconvergencein normof thesuboptimalsolutionsto anoptimal * 1solution.Thenew methoddoesnot requirethecomputationof theinterpolationconditions,andit allows the

directcomputationof thesuboptimalcontroller.

D. EndresandP. Foldiak. Quadraticprogrammingfor learningsparsecodes.In ‘Proceedings

of ICANN99, Edinburgh,1999’,p. (to appear),1999.

S. S. ErengucandH. P. Benson.An algorithmfor indefiniteintegerquadraticprogramming.ComputersandMathematicswith Applications, 21(6–7),99–106,1991.

Abstract. Presentsanalgorithmfor finding theglobalminimumof anindefinitequadraticfunctionover the

integerscontainedin acompact,convex set.To find thisminimum,thealgorithmfirst transformstheproblem

into anequivalentproblemwith aseparableobjective function.It thenusesabranchandboundsearchonthe

valuesof theconstraints,ratherthanthevariables,of thetransformedproblem.


S. E. EriksenandP. D. Berger. A quadraticprogrammingmodel for productconfigurationoptimization. Zeitschrift fur Operations Research, SerieB (Praxis), 31(6), 143–159,1987.

Abstract. The variousfeaturesof any productare differentially appealingto the variousportionsof the

consumerpopulationandhavedifferentialcostsof productionandmarketing.Thispaperconsiderstheprob-

lemof ’overall productoptimization’,or morespecifically, ’optimal productconfiguration’.Productpriceis

includedasapartof thisconfiguration.

S. M. Faber. Quadraticprogrammingappliedto the problemof galaxypopulationsynthesis.AstronomyandAstrophysics, 20(3), 361–374,1972.

Abstract. Thetechniqueof quadraticprogrammingasappliedto theproblemof galaxypopulationsynthesis

is described.Themethodoffers significantadvantagesover the trial-and-errorapproachusuallyemployed.

This techniqueis appliedto 38-colordataon the nuclei of M 31, M 32 andM 81 and to integrated10-

colorphotometryfor elliptical galaxies.Theresultsindicatethatestimatesof meanline strengthsin external

galaxiesby meansof populationsynthesisarewell determined.Agesbasedon the main-sequenceturnoff

point are uncertainby a factor of two. The methodis not sensitive to the numberof starson the main

sequencebetweenK 0 V andM 7 V. Consequentlytheslopeof theluminosityfunctionbelow turnoff cannot

bedetermined.Themass-to-lightratioof thecomputedpopulationfor M 31is uncertainby a factorof 4. The

correspondingfactorsfor M 32 andM 81 are50 and16 respectively. Modelsfor elliptical galaxiessuggest

thatthemeanmetalabundanceof thestellarpopulationincreaseswith increasinggalaxyluminosity.

B. J. Falkowski. Risk analysisusingperceptronsandquadraticprogramming. In ‘Computa-tional Intelligence.TheoryandApplications.InternationalConference,6th FuzzyDays,Springer-Verlag,Berlin, Germany’, pp.543–547,1999.

Abstract. A heuristicmethod(computationof weightedaverages)is consideredfor risk analysis.It is im-

proved uponby utilizing the perceptionlearningtheoremandquadraticprogramming.Experimentalwork

shows that both techniquesgive good results,the former onebeingsomewhat moreefficient in termsof

CPU-timeused.In spiteof certaintheoreticalshortcomingsit is arguedthat the familiar paradigmoffers

considerablepotentialfor practicalapplications.

J. Y. FanandL. Zhang. Real-timeeconomicdispatchwith line flow andemissionconstraintsusingquadraticprogramming. IEEE Transactionson Power Systems, 13(2), 320–325,1998.

Abstract. Thepresenceof multiple constraintsdueto network line flow limits andemissionallowancesin

theeconomicdispatchof modernpowersystemsmakestheconventionalLambda-Deltaiterative approachno

longereffective. This paperproposesa practicalstrategy basedon quadraticprogramming(QP) techniques

to solve thereal-timeeconomicdispatchproblem.It formulatestheproblemwith aquadraticobjective func-

tion basedon the unit’s cost curves in quadraticor piecewise-quadraticforms. The operationconstraints

aremodeledaslinear equality/inequality equations,resultingin a typical QPproblem.Goal programming

techniquesarealsoincorporatedin theformulationwhich guaranteesthebestavailablesolutionevenunder

infeasibleconditions.In addition,theproposedstrategy formulatestheproblemin thesecondphasedispatch

in real-timeby includingasetof emergency controlvariablesto provideeffective controlstrategiesfor prop-

erly relieving constraintviolationsif they exist. Theeffectivenessof theproposedstrategy is demonstrated

by anexamplepower dispatchproblem.

L. Fan,I. Chatterton,andL. Walker. Quadraticprogrammingfor grid supplypointmw demandcompositionanalysis.In ‘34-th UniversitiesPower EngineeringConference.Universityof Leicester, UK’, Vol. 1, pp.245–248,1999.

Abstract. This paperreportswork of analysingthe grid supplypoint (GSP)load compositionacrossthe

entireNationalGrid system.A quadraticprogrammingbasedalgorithmhasbeendevelopedfor the nodal

load compositionanalysis.The static load compositionat GSPlevel is the primary informationcrucial to


further load analysis,including load dynamics,reactive load composition,and GSPload modelling and

forecasting.

S.C. FangandS.C. Puthenpura.Affinescalingfor convex quadraticprogramming.In ‘Linear

OptimizationandExtensions:TheoryandAlgorithms’, chapter9. PrenticeHall, Engle-

woodCliffs, New Jersey, USA, 1993.

S. C. FangandH. S. J. Tsao. An unconstrainedconvex programmingapproachto solvingconvex quadraticprogrammingproblems.Optimization, 27(3), 235–243,1993.

Abstract. Derivesan unconstrainedconvex programmingapproachto solving convex quadraticprogram-

ming problemsin standardform. Relatedduality theoryis establishedby usingtwo simpleinequalities.An

ε-optimalsolutionis obtainedby solvinganunconstraineddualconvex program.A dual-to-primalconver-

sion formula is alsoprovided.Somepreliminarycomputationalresultsof usinga curved searchmethodis

included.

S.C. Fang,T. M. Huang,C. H. Lin, andW. W. Lin. A relaxedinterior pathfollowing primal–

dualalgorithmfor convex quadraticprogramming.MathematicsToday, SpecialIssueon

MathematicalProgramming, XII-A , 115–144,1994.

H. FarhatandS. From. A quadraticprogrammingapproachto estimatingthe testabilityandrandomor deterministiccoverageof a VLSI circuit. VLSIDesign, 2(3), 223–231,1994.

Abstract. The testabilitydistribution of a VLSI circuit is modeledasa seriesof stepfunctionsover the in-

terval (0, 1). Themodelgeneralizespreviousrelatedwork on testability. Unlike previouswork, however, we

includeestimatesof testabilityby randomvectors.Quadraticprogrammingmethodsareusedto estimatethe

parametersof the testabilitydistribution from fault coveragedata(randomanddeterministic)on a sample

of faults.The estimatedtestabilityis thenusedto predictthe randomanddeterministicfault coveragedis-

tributionswithout theneedto employ testgenerationor fault simulations.Thepredictionof fault coverage

distribution cananswerimportantquestionsaboutthe”goodness”of a designfrom a testingpoint of view.

Experimentalresultsaregivenon thelargeISCAS-85andISCAS-89circuits.

H. Farhat,S.From,andA. Lioy. A quadraticprogrammingapproachto estimatingthetestabilityandcoveragedistributionsof a VLSI circuit. MicroprocessingandMicroprogramming,35(1–5),479–483,1992.

Abstract. Thetestabilitydistributionof aVLSI circuit is modeledasaseriesof stepfunctionsover theinter-

val (0,1).The modelgeneralizesprevious relatedwork on testability. Quadraticprogrammingmethodsare

usedto estimatetheparametersof thetestabilitydistribution from fault coveragedataona sampleof faults.

Theestimatedtestability is thenusedto predictthe fault coveragedistribution without the needto employ

testgenerationor fault simulations.Thepredictionof fault coveragedistribution cananswerimportantques-

tionsaboutthe’goodness’of a designfrom testingpoint of view. Experimentalresultson threeof thelarge

ISCAS-89benchmarkcircuitsreflecttheaccuracy of themodel.

A. M. FaustinoandJ. J.Judice. A slcpalgorithmfor bilinearandconcave quadraticprogram-

ming. InvestigacionOperacional, 8(2), 67–87,1988.

A. M. FaustinoandJ.J.Judice.Solutionof large-scaleconvex quadraticprogramsby Lemke’s

method. In M. A. TurkmanandM. L. Carvalho,eds,‘Actasda 1aConferenciaem Es-

tatisticae Optimizaaao’, pp.681–695,1992.

A. M. FaustinoandJ. J. Judice. Minimization of a concave quadraticfunctionsubjectto box

constraints.InvestigacionOperativa, 4, 49–68,1994.


L. Y. Faybusovich. Wolfe’s algorithmfor infinite-dimensionalquadraticprogrammingprob-lems.EngineeringCybernetics, 20(3), 20–30,1982.

Abstract. Wolfe’s algorithmis generalizedto infinite-dimensionalquadraticprogrammingproblemswith a

finite numberof inequalityconstraints.A coordinate-freeapproachis developedenablingoneto consider

infinite-dimensionalanaloguesof computationalproceduresof thesimplex type.

L. Y. Faybusovich andJ.B. Moore. Long-steppath-following algorithmfor convex quadraticprogrammingproblemsin a Hilbert space.Journal of OptimizationTheoryand Appli-cations, 95(3), 615–635,1997. Seealso,Proceedingsof the 34th IEEE ConferenceonDecisionandControl,IEEE,New York, NY, USA, volume2, pages1109–1114,1995.

Abstract. We developaninterior-point techniquefor solvingquadraticprogrammingproblemsin a Hilbert

space.As an example,we consideran applicationof theseresultsto the linear-quadraticcontrol problem

with linear inequalityconstraints.It is shown that the newton stepin this situationis basicallyreducedto

solvingthestandardlinear-quadraticcontrolproblem.

K. A. Fegley, S.Blum, J.O. Bergholm,A. J.Calise,J.E. Marowitz, G. Porcelli,andL. P. Sinha.Stochasticanddeterministicdesignandcontrol via linear andquadraticprogramming.IEEE Transactionson AutomaticControl, AC-16(6), 759–766,1971.

Abstract. Theapplicationof linearandquadraticprogrammingto optimalcontrolproblemsandto stochastic

or deterministicsystemdesignproblemsis discussedandillustratedwith examples.

L. Fernandes,A. Fischer, J.J.Judice,C.Requejo,andJ.Soares.A blockactivesetalgorithmforlarge-scalequadraticprogrammingwith boxconstraints.Annalsof OperationsResearch,81, 75–95,1998.

Abstract. An algorithmfor computingastationarypointof aquadraticprogramwith boxconstraints(BQP)

is proposed.Eachiterationof thisprocedurecomprisesaguessingstrategy whichforecaststheactive bounds

at a stationarypoint, thedeterminationof a descentdirectionby meansof solvinga reducedstrictly convex

quadraticprogramwith box constraintsandan exact line search.Global convergenceis establishedin the

sensethatevery accumulationpoint is stationary. Moreover, it is shown that thealgorithmterminatesafter

a finite numberof iterations,if at leastoneiterateis sufficiently closeto a stationarypoint which satisfiesa

certainsufficientoptimalitycondition.Thealgorithmcanbeeasilyimplementedfor sparselarge-scaleBQPs.

Furthermore,it simplifiesfor concave BQPs,asit is not requiredto solvestrictly convex quadraticprograms

in this case.Computationalexperiencewith large-scaleBQPsis includedandshows theappropriatenessof

this typeof methodology.

M. C. Ferris. Parallel constraintdistribution in convex quadraticprogramming.Mathematicsof OperationsResearch, 19(3), 645–658,1994.

Abstract. We considerconvex quadraticprogramswith large numbersof constraints.We distribute these

constraintsamongseveral parallelprocessorsandmodify theobjective function for eachof thesesubprob-

lemswith Lagrangemultiplier informationfrom theotherprocessors.New Lagrangemultiplier information

is aggregatedin a masterprocessorandthewholeprocessis repeated.Linearconvergenceis establishedfor

stronglyconvex quadraticprogramsby formulatingthe algorithmin an appropriatedual space.The algo-

rithm correspondsto astepof aniterative matrix splittingalgorithmfor asymmetriclinearcomplementarity

problemfollowedby aprojectionontoa subspace.

J. A. Filar. Quadraticprogrammingand the single-controllerstochasticgame. Journal of

MathematicalAnalysisandApplications, 113(1), 136–147,1986.

J.A. Filar. TheHamiltoniancycleproblem,controlledMarkov chainsandquadraticprogram-

ming. In ‘Proceedingsof ASOR12thNationalConference’,Vol. 15,pp.263–281,1993.


J. A. Filar, M. Oberije,andP. M. Pardalos. Hamiltoniancycle problem,controlledMarkov

chainsandquadraticprogramming.In ‘Proceedingsof the12thNationalConferenceof

theAustralianSocietyfor OperationsResearch’,pp.263–281,1993.

F. L. Filippelli, M. Forti, andS.Manetti.New linearandquadraticprogrammingneuralnetwork.ElectronicsLetters, 30(20),1693–1694,1994.

Abstract. A neuralnetwork is proposedfor solving linearandquadraticprogrammingproblems.Themain

featureis thattherequiredconditionsof symmetryandasymmetryin theinterconnectionsareautomatically

metin practicalimplementations,sothatstability is guaranteed.

B. FinkbeinerandP. Kall. Direct algorithmsin quadraticprogramming.Zeitschrift fur Opera-tionsResearch, SerieA (Theorie), 17(1), 45–54,1973.

Abstract. Accordingto theliteraturethereseemto besomedifficultiesin thesemidefinitecase.A numerical

exampleof asemicomplementarysolutionis presented,whereCottle’s algorithmfails. It will beprovedthat

at leastin Zangwill’s versionof Cottle’s algorithmthis situationcannotoccurtheoretically. Since,however,

in practicalcomputationssuchcasesmayoccurdueto roundoff errors.aslightmodificationof thealgorithm

is proposed,for whichmonotonicityandfinitenessareproved.

F. D. Fischer. To the solutionof the contactproblemof elasticbodieswith extendedcontactareasby quadraticprogramming.Computing, 13(3–4),353–384,1974.

Abstract. An elasticbodyin contactwith anelasticor rigid subgradeis representedby finite elements.The

total potentialenergy of thesystemunderconsiderationof linearly elasticmaterialandsmalldeformations

is now a quadraticfunctionof thenodaldeformationsandthenodalvaluesof thecontactpressurewhich is

approximatedby a polynomial.Only a partof thesurfaceof thebodymustbeproposedwhich includesthe

realcontactsurface.After evaluationof theequilibriumequationsandthecontactconditionin aninequality

anda linear transformationof thenodalvariables,all relationsarenow so formulated,theminimisationof

totalpotentialenergy canbeexpressedasaquadraticprogramin theunknown nodaldeformationsandnodal

valuesof contactpressure.Standardprogramscannow beusedfor solution.

J. Fischer. Stackelberg solutions in constrainedquadratic programmingproblems. InK. ReinischandM. Thoma,eds,‘LargeScaleSystems:TheoryandApplications1989.SelectedPapersfrom the5thIFAC/IFORS/IMACSSymposium.Pergamon,Oxford,Eng-land’, pp.241–246,1990.

Abstract. Dealswith two-personstatic Stackelberg gameswith quadraticobjectives and commonlinear

constraintsfor the leaderandthe follower. Importantpropertiesof therationalreactionfunctionof the fol-

lowercanbederivedfrom theoutlinedrelationsto thetheoryof parametricoptimization.Fromtheresultthe

solutionof thespecifiedStackelberg problemis reducedto thesequentialsolutionof afinite numberof stan-

dardconstrainedquadraticprogrammingproblemsandthe iterative selectionof solutionstructuresfor the

follower reaction.The proposedalgorithmhasbeenimplementedon a large-scalenonlinearprogramming

solver.

N. J.Fisher. Gravity interpretationwith theaidof quadraticprogramming—reply. Geophysics,

46(3), 341–342,1981.

N. J. Fisher. A quadraticprogrammingalgorithmfor geophysicalgravity inversionandotherapplications.Bulletin of theAustralian MathematicalSociety, 25(1), 159–160,1982.

Abstract. The geophysicalgravity methodis describedandthe correspondinginverseproblemstateasan

integral equation.A survey of previous work associatedwith gravity datainversionis given followed by

anaccountof the inherentdifficulties associatedwith gravity inversion;this is doneby a discussionof the

integralequationalreadygiven,theinversionof whichconstitutesanill-posedproblem.Theintegralequation

givesriseto anill-conditionedsystemof linearequations;existingmethodsfor thesolutionsof suchsystems


are reviewed. Then follows an accountof the previous useof quadraticprogrammingtechniquesin the

solutionof ill-posedproblems.

N. J. FisherandL. E. Howard. Gravity interpretationwith theaid of quadraticprogramming.Geophysics, 45(3), 403–419,1980.

Abstract. The inversegravity problemis posedasa linear least-squaresproblemwith the variablesbeing

densitiesof two-dimensionalprisms.Upperandlowerboundsonthedensitiesareprescribedsothattheprob-

lem becomesa linearly constrainedleast-squaresproblem,which is solvedusinga quadraticprogramming

algorithmdesignedfor upperandlowerbound-typeconstraints.Thesolutionto any problemis smoothedby

damping,usingthesingularvaluedecompositionof thematrix of gravitationalattractions.If thesolutionis

requiredto bemonotonicallyincreasingwith depth,thenthis featurecanbeincorporated.Themethodis ap-

plied to bothfield andtheoreticaldata.Theresultsareplottedfor (1) undamped,nonmonotonic,(2) damped,

nonmonotonic,and(3) damped,monotonicsolutions;theseconditionsillustratethecompositeapproachof

interpretationwherebothdampingtechniquesandlinearconstraintsareusedin refininga solutionwhich at

first is unacceptableongeologicgroundswhile fitting theobserveddatawell.

R. Fletcher. A FORTRAN subroutinefor generalquadraticprogramming. Technicalreport,UKAEA, Harwell,Berks,England,1970.

Abstract. A FORTRAN subroutineis describedandlistedfor minimizingaquadraticfunctionof n variables

subjectto m linear equalityand inequality constraints.The methodis a generaloneso that thereare no

restrictionson the typesof quadraticfunction which canbeminimized.The solutionis usuallyfound in a

moderatemultiple of n3 (or at worst n2m computeroperations.Thesubroutineis very flexible andallows

optionswhichcanconsiderablyshortenthetime takento solve any particularproblem.

R. Fletcher. A generalquadraticprogrammingalgorithm.Journalof theInstituteof Mathemat-icsandits Applications, 7, 76–91,1971.

Abstract. An effective algorithmis presentedfor quadraticprogrammingwhich is of generalapplicability,

but whichis notdependentupontheavailability of a linearprogrammingcodefor its implementation.It is an

algorithmof exchangetype,theexchangesbeingchosensoasto avoid theaccumulationof errorto aslarge

anextentaspossible.

R. Fletcher. Quadraticprogramming. In ‘Practical Methodsof Optimization’, chapter10,

pp.229–258.J.Wiley andSons,Chichester, England,secondedn,1987a.

R. Fletcher. Recentdevelopmentsin linear and quadraticprogramming. In A. IserlesandM. J. D. Powell, eds, ‘State of the Art in Numerical Analysis. Proceedingsof theJoint IMA/SIAM Conference’,pp.213–243.Oxford UniversityPress,Oxford,England,1987b.

Abstract. Describesrecentdevelopmentsin linearprogramming,includingtheellipsoidalgorithm,theKar-

markaralgorithm,new strategiesfor updatingLU factorsin thesimplex method,andmethodswith guaran-

teedterminationin the presenceof degeneracy andround-off errors.Variousnew algorithmsfor quadratic

programmingarediscussed,andthechoiceof matrix factorizationsandtheirupdatesis considered.Theuse

of * 1 penaltyfunctionsin linearandquadraticprogrammingis alsomentionedbriefly.

R. Fletcher. Resolvingdegeneracy in quadraticprogramming.Annalsof OperationsResearch,46–47(1–4),307–334,1993.

Abstract. A techniquefor theresolutionof degeneracy in anactive setmethodfor quadraticprogramming

is described.TheapproachgeneralisesFletcher’s method(1988)which appliesto theLP case.Themethod

is describedin termsof a linearcomplementarityproblemtableau,which is seento provide usefulinsights.

It is shown thatthedegeneracy procedureonly needsto operatewhenthedegenerateconstraintsarelinearly

dependenton thosein theactive set.No significantoverheadsareincurredby thedegeneracy procedure.It is


readily implementedin a null spaceformat,andno complicationsin thematrix algebraareintroduced.The

guaranteesof terminationprovidedby Fletcher’s method,extendingin particularto thecasewhereround-off

erroris present,arepreservedin theQPcase.It is arguedthatthetechniquegivesstrongerguaranteesthanare

availablewith otherpopularmethodssuchasWolfe’s method(1963)or themethodof GoldfarbandIdnani

(1983).

R. Fletcher. Stablereducedhessianupdatesfor indefinitequadraticprogramming.Mathemati-cal Programming, 87(2), 251–264,2000.

Abstract. Stabletechniquesareconsideredfor updatingthereducedHessianmatrixthatarisesin anull-space

active setmethodfor quadraticprogrammingwhenthe Hessianmatrix itself may be indefinite.A scheme

for defining andupdatingthe null-spacebasismatrix is describedwhich is adequatelystableandallows

advantageto be taken of sparsityin the constraintmatrix. A new canonicalform for the reducedHessian

matrix is proposedthat canbe updatedin a numericallystableway. Someconsequencesfor the choiceof

minor iterationsearchdirectionaredescribed.

R. FletcherandM. P. Jackson.Minimization of a quadraticfunction of many variablessub-

ject only to lower and upperbounds. Journal of the Institute of Mathematicsand its

Applications, 14(2), 159–174,1974.

O. E. Flippo andA. H. G. Rinnooy Kan. A noteon Bendersdecompositionin mixed-integer

quadraticprogramming.OperationsResearch Letters, 9(2), 81–83,1990.

C. A. FloudasandV. Visweswaran. Quadraticoptimization. In R. Horst andP. M. Pardalos,eds,‘Handbookof GlobalOptimization’,Kluwer AcademicPublishers,Dordrecht,TheNetherlands,1995.

Abstract. Quadraticoptimizationcomprisesone of the most importantareasof nonlinearprogramming.

Numerousproblemsin realworld applications,includingproblemsin planningandscheduling,economies

of scale,andengineeringdesign,andcontrolarenaturallyexpressedasquadraticproblems.Moreover, the

quadraticproblemis known to be NP-hard,which makesthis oneof the most interestingandchallenging

classof optimizationproblems.In this chapter, we review variouspropertiesof thequadraticproblem,and

discussdifferenttechniquesfor solvingvariousclassesof quadraticproblems.Someof themoresuccessful

algorithmsfor solvingthespecialcasesof boundconstrainedandlargescalequadraticproblemsareconsid-

ered.Examplesof variousapplicationsof quadraticprogrammingarepresented.A summaryof theavailable

computationalresultsfor thealgorithmsto solve thevariousclassesof problemsis presented.

F. Forgo. Relationshipbetweenmixed zero-oneinteger linear programmingand certain

quadraticprogrammingproblems. Studia ScientiarumMathematicarumHungarica,

4, 37–43,1969.

A. ForsgrenandG. Sporre.Onweightedlinearleast-squaresproblemsrelatedto interiormeth-odsfor convex quadraticprogramming.ReportTRITA-MAT-2000-OS11,DepartmentofMathematics,Royal Instituteof Technology, Stockholm,Sweden,2000.

Abstract. It is known that thenormof thesolutionto a weightedlinear least-squaresproblemis uniformly

boundedfor the setof diagonallydominantsymmetricpositive definiteweight matrices.This result is ex-

tendedto weightmatricesthatarenonnegative linearcombinationsof symmetricpositive semidefinitema-

trices.Further, resultsare given concerningthe strongconnectionbetweenthe boundednessof weighted

projectiononto a subspaceand the projectiononto its complementarysubspaceusing the inverseweight

matrix. In particular, explicit boundsaregiven for the Euclideannorm of the projections.We apply these

resultsto theNewton equationsarisingin a primal-dualinterior methodfor convex quadraticprogramming

andprove boundednessfor thecorrespondingprojectionoperator.


A. L. Forsgren,P. E. Gill, andW. Murray. On theidentificationof local minimizersin inertia-controllingmethodsfor quadraticprogramming.SIAMJournal on Matrix AnalysisandApplications, 12(4), 730–746,1991.

Abstract. Theverificationof a localminimizerof ageneral(i.e.,nonconvex) quadraticprogramis in general

anNP-hardproblem.Thedifficulty concernstheoptimality of certainpoints(which we call deadpoints)at

which the first-ordernecessaryconditionsfor optimality aresatisfied,but strict complementaritydoesnot

hold. Inertia-controllingquadraticprogramming(ICQP) methodsform an importantclassof methodsfor

solvinggeneralquadraticprograms.Wederive acomputationalschemefor proceedingatadeadpoint thatis

appropriatefor ageneralICQPmethod.

M. Forti andA. Tesi.New conditionsfor globalstabilityof neuralnetworkswith applicationtolinearandquadraticprogrammingproblems.IEEETransactionsonCircuitsandSystemsI: FundamentalTheoryandApplications, 42(7), 354–366,1995.

Abstract. In this paper, we presentnew conditionsensuringexistence,uniqueness,andGlobalAsymptotic

Stability (GAS) of the equilibrium point for a large classof neuralnetworks.The resultsareapplicableto

bothsymmetricandnonsymmetricinterconnectionmatricesandallow for theconsiderationof all continuous

nondecreasingneuronactivation functions.Suchfunctionsmay be unbounded(but not necessarilysurjec-

tive), mayhave infinite intervalswith zeroslopeasin a piece-wise-linearmodel,or both.Theconditionson

GAS rely on the conceptof Lyapunov DiagonallyStable(or Lyapunov DiagonallySemi-Stable)matrices

andareproved by employing a classof Lyapunov functionsof thegeneralizedLur’e-Postnikov type.Sev-

eralclassesof interconnectionmatricesof applicative interestareshown to satisfyour conditionsfor GAS.

In particular, the resultsareappliedto analyzeGAS for the classof neuralcircuits introducedfor solving

linear andquadraticprogrammingproblems.In this application,the principal resulthereobtainedis that

thesenetworks areGAS alsowhenthe constraintamplifiersaredynamical,as it happensin any practical

implementation.

M. FrankandP. Wolfe. An algorithmfor quadraticprogramming.Naval Research Logistics

Quarterly, 3, 95–110,1956.

P. D. Frank,M. J.Healy, andR. A. Mastro. A range-spaceimplementationfor largequadratic

programswith small active sets. Journal of OptimizationTheory and Applications,

69(1), 109–127,1991.

R. M. Freund. Dual gaugeprograms,with applicationsto quadraticprogrammingand theminimum-normproblem.MathematicalProgramming, 38(1), 47–67,1987.

Abstract. A gaugefunction f + is a nonnegative convex functionthat is positively homogeneousandsatis-

fiesf(0)=0. Normsandpseudonormsarespecificinstancesof a gaugefunction.Thepaperpresentsa gauge

duality theoryfor a gaugeprogram,which is theproblemof minimizing thevalueof a gaugefunction f + over a convex set.The gaugedual programis alsoa gaugeprogram,unlike the standardLagrangedual.

Theauthorpresentssufficient conditionson f + thatensuretheexistenceof optimalsolutionsto thegauge

programandits dual, with no duality gap.Thesesufficient conditionsarerelatively weakandareeasyto

verify, andareindependentof any qualificationsontheconstraints.Thetheoryis appliedto aclassof convex

quadraticprograms,andto theminimum * p normproblem.Thegaugedualprogramis shown to provide a

smallerduality thanthestandarddual,in acertainsensediscussed.

A. FriedlanderandJ.M. Martınez. On themaximizationof a concave quadraticfunctionwith

box constraints.SIAMJournalonOptimization, 4(1), 177–192,1994.

A. Friedlander, J.M. Martınez,andM. Raydan.A new methodfor large-scalebox constrained

convex quadraticminimizationproblems.OptimizationMethodsandSoftware, 5(1), 57–

74,1995.


K. R. Frisch. Quadraticprogrammingby the multiplex methodinthe generalcasewherethe

quadraticform maybesingular. Memorandum,UniversityInstitutefor Economics,Oslo,

1960.

M. Fu, Z. Q. Luo, andY. Ye. Approximationalgorithmsfor quadraticprogramming.Journalof CombinatorialOptimization, 2(1), 29–50,1998.

Abstract. We considerthe problemof approximatingtheglobal minimum of a generalquadraticprogram

(QP)with n variablessubjectto mellipsoidalconstraints.For m 1,werigorouslyshow thatanε-minimizer,

whereerrorε , 0 1 , canbeobtainedin polynomialtime,meaningthatthenumberof arithmeticoperations

is a polynomialin n, m, andlog 1 ε . For m 2, we presenta polynomial-time 1 1 m2 -approximation

algorithmaswell asasemidefiniteprogrammingrelaxationfor thisproblem.In addition,wepresentapprox-

imationalgorithmsfor solvingQPundertheboxconstraintsandtheassignmentpolytopeconstraints.

S. T. Fu andR. T. Wang. Power-systemsecurityanalysisandenhancementusingFletcher’s

quadratic-programmingmethod. In ‘Proceedingsof theEighthPower SystemsCompu-

tationConference’,pp.439–445,1984.

S. T. Fu, E. Yu, andX. Zhang. A decoupledoptimal power flow approachusingFletcher’squadraticprogrammingmethod.Proceedingsof theChineseSocietyof Electrical Engi-neering, 6(1), 1–11,1986. Seealso,BridgeBetweenControlScienceandTechnology.Proceedingsof the Ninth Triennial World Congressof IFAC, PergamonPress,Oxford,England,volume4, pages2145–2150,1985.

Abstract. A decoupledmodelwith active andreactive optimizationis derived. In a P-model,a very sparse

schemeis usedastheequalityconstraints.In a Q-model,transmissionlossesareexpressedasdeviation of

swing generation.The active power at generatorbuses(exceptswing generator)is chosenasindependent

variablein P-optimization.Thevoltageamplitudeof PV nodesandV θ node,andthe ratio of transforma-

tion of all tap-changingtransformersarechosenasindependentvariablesin Q-optimization.Theprocedure

is carriedout only in thesubspacedeterminedby independentvariables.Theequalityconstraintsareelim-

inated,and the relationshipsbetweendependentand independentvariablesareestablishedby solutionof

a fastdecoupledload flow method.During P-optimization,all bus voltagesandreactive injectionsareas-

sumedconstantandtheobjective function is to minimize the total fuel consumptionin thesystem.During

Q-optimization,all active generations(except swing generator)andall voltagephaseanglesareassumed

constantandthe objective function is to minimize total transmissionlossesin the network. This approach

hasbeenprogrammedin FORTRAN languageandtestedon a CLASSIC7835computer. Thesparsematrix

programmingtechniquebeingused,thememoryrequirementof a network with 150busesis only 100kB.

For comparisonpurposes,the computationtime andresultsof the sameIEEE 30 bus test systemusinga

differentmethodaretabulated.A Chinese133bussystemwith 164 lineswassimulatedusingthis method.

Therewere2 linesoverloadedand3 busvoltagesout of limit in theinitial state.An OPFresultis tabulated.

All line overloadsarerelievedandall busvoltagesarewithin limit.

T. Fujie andM. Kojima. Semidefiniteprogrammingrelaxationfor nonconvex quadraticpro-grams.TechnicalReportResearchReporton InformationSciencesB-298,Tokyo Insti-tuteof Technology, 1995.

Abstract. Any quadraticinequalityin then-dimensionalEuclideanspacecanberelaxedinto a linearmatrix

inequalityin 1 n- 1 n symmetricmatrices.Basedon this principle,we extendtheLovasz-Schrijver

SDP(semidefiniteprogramming)relaxationdevelopedfor a 0–1 integer programto a generalnonconvex

QP(quadraticprogram),andpresentsomefundamentalcharacterizationof theSDPrelaxationincludingits

equivalenceto a relaxationusingconvex-quadraticvalid inequalitiesfor thefeasibleregion of theQP.

Y. FujimotoandA. Kawanura.Bipedwalking controlwith optimal foot forcedistribution byquadraticprogramming.IEEE/ASMEInternationalConferenceon AdvancedIntelligentMechatronics’97. Final ProgramandAbstractsIEEE,New York, NY, USA, p. 108,1997.


Abstract. Summaryform only given.This paperdescribesa novel bipedwalking systembasedon optimal

foot forcedistribution by quadraticprogramming(QP).Thehierarchicalcontrolsystemconsistsof a robust

force controller for the supportingleg, a robust positioncontroller for the non-supportingleg, an attitude

controllerfor thebody, andafree-leg motionplanner. Theattitudeof thebodyof thebipedrobotis controlled

usingthereactive forceasthe input. Themainresultof this paperis an introductionof QPto distribute the

forcerequiredby theattitudecontrollerto eachfoot. Themethodcanstabilizetheattitudeof therobotand

thecontactbetweenthefoot andtheground,in spiteof theexistenceof thelimitation of thefrictional force

betweenthefoot andtheground.Also anew walkingpatterngeneratorbasedonaninvertedpendulummodel

is proposed,which realizesgloballystabletrackingof thecenterof massof thebipedrobot.

K. FukudaandA. Kawanaka. Adaptive processingwith quadraticprogrammingandlpf forreducingblockingartifactsin DCT imagecoding. Transactionsof the Instituteof Elec-tronics,InformationandCommunicationEngineers, J82-A(1), 142–150,1999.

Abstract. This paperpresentsanadaptive post-processingmethodto reducetheblockingartifactsin DCT

imagecoding.In thismethod,theblocksareclassifiedinto thesmoothgroupandthevariationgroupaccord-

ing to thereceived DCT coefficients.For theblocksin thesmoothgroup,we estimatetheDCT coefficients

by solving a quadraticprogrammingproblemconsideringthe continuity acrossthe block boundariesand

thequantizationmatrixusedon thecoder. Furthermore,theDCT coefficientsaresequentiallyobtainedfrom

lower order to higher. For the blocks classifiedinto the variation group, the low passfilters are applied

for smoothingtheblock boundaries.Experimentalresultsfor someimagescodedat low bit ratesshow the

proposedmethodimprovesthereconstructedimagequalityobjectively aswell assubjectively.

R. Gabasov, F. M. Kirillo va, andV. M. Raketskii. Methodsfor solving the generalconvexquadraticprogrammingproblem. Doklady AkademiiNauk SSSR, 258(6), 1289–1293,1981.

Abstract. A theoreticaland numericalinvestigationis reportedof 4 methods(Bil’ s, the direct, dual and

adaptive methods)for solvingthe(mxn) convex quadraticprogrammingproblem: F x. cTx xT Dx 2 to

min, Ax b, d/ x d / , whereD D J J 0 O, A A I J , rankA m, I 1 21 m$ J 1 2 n .E. Galligani and V. Ruggiero. Numerical solution of equality-constrainedquadraticpro-

grammingproblemson vector-parallelcomputers.OptimizationMethodsandSoftware,

2(3), 233–247,1993.

E. Galligani and L. Zanni. Error analysisof elimination methodsfor equality constrained

quadratic-programmingproblems. In ‘MathematicalResearch’,Vol. 89, pp. 107–112,

1996.

E. Galligani andL. Zanni. Error analysisof an algorithmfor equality-constrainedquadraticprogrammingproblems.Computing, 58(1), 47–67,1997.

Abstract. Thenumericalstabilityof theorthogonalfactorizationmethod(R.Fletcher, 1987)for linearequal-

ity constrainedquadraticprogrammingproblemsis studiedusinga backwarderroranalysis.A perturbation

formula for theproblemis analyzed;theconditionnumbersof this formulaareexaminedin orderto com-

parethemwith the conditionnumbersof the two matricesof the problem.A classof testproblemsis also

consideredin orderto show experimentallythebehaviour of themethod.

E. Galligani,V. Ruggiero,andL. Zanni. Splitting methodsfor constrainedquadraticprograms

in dataanalysis.ComputersMath.Applic., 32, 1–9,1996.

E. Galligani,V. Ruggiero,andL. Zanni. Splitting methodsandparallelsolutionof constrained

quadraticprograms.Rend.Circ. Matem.PalermoSeriesII , 48, 121–136,1997a.


E. Galligani,V. Ruggiero,andL. Zanni. Splitting methodsfor quadraticoptimizationin data

analysis.InternationalJournalof ComputerMathematics, 63, 289–307,1997b.

E. Galligani,V. Ruggiero,andL. Zanni. Parallel solutionof large-scalequadraticprograms.

In R. D. Leone,A. Murli, P. M. PardalosandG. Toraldo,eds,‘High PerformanceAlgo-

rithms andSoftwarein NonlinearOptimization’, pp. 189–205,Kluwer AcademicPub-

lishers,Dordrecht,TheNetherlands,1998.

G. Gallo, P. L. Hammer, andB. Simeone.Quadraticknapsackproblems.MathematicalPro-

grammingStudies, 12, 132–149,1980.

C. E. GarciaandA. M. Morshedi.Quadratic-programmingsolutionof dynamicmatrixcontrol

(QDMC). ChemicalEngineeringCommunications, 46(1–3),73–87,1986.

M. K. Gavurin andV. N. Malozemov. Foundationsof quadraticprogrammingtheory. Vest-nik LeningradskogoUniversiteta,SeriyaMatematikaMekhanikai Astronomiya, 1, 9–16,1980.

Abstract. A compactdescriptionof quadraticprogrammingtheoryis given. It includestheexistencetheo-

rem different formsof optimality criterionandduality theorems.Only compatibility conditionfor a linear

algebraicsystemanda necessaryoptimality conditionin the linear programmingproblemaresupposedto

beknown. Thegeneralconvex programmingtheoryis not used.

J. A. George. Assessmentof errorsin approximatingthe objective function of a quadraticprogrammingproblem.AsiaPacificJournalof OperationalResearch, 5(1),21–36,1988.

Abstract. Dealswith the effectivenessof variousdecisionrulesto approximatetheobjective functionof a

quadraticprogram(QP).It is assumedthat theconstraintsetof theQPis known preciselybut theobjective

functionis known only from thedataof observationsmadeof thatfunction.Decisionrules,somelinearand

somenonlinear, areusedto derive functionsto approximatethe true unknown function from the data.A

seriesof problemsaregeneratedto testthesedecisionrules.In additionto comparingthemwith eachother,

thestudyalsoconsiderstheeffectsof thetypeof quadraticfunction (matrix type), thesizeof theproblem,

thenumberof observations,thescatterpatternof theobservationsandthecurvature(degreeof nonlinearity)

of thequadratic.

V. Georgescu. Estimationof fuzzy regressionmodelswith possibilistic constraints,usingquadraticprogramming.EconomicComputationandEconomicCyberneticsStudiesandResearch, 31(1–4),105–123,1997.

Abstract. We introducea formal criterion, the decouplingprinciple, which allows us to naturallyextend

the projectiontheoreminto a fuzzy regressionframework. This criterion justifiesa radical revision of the

usualestimationmethods.Actually dominantin the fuzzy regressionanalysisis anotherapproach,which

resortsto linearprogrammingandfuzzy arithmeticin orderto minimizethefuzzinessof themodel,subject

to somepossibilisticconstraints.Our criticism relatingto this approachis concludedby the proposalof a

new strategy, wherewe give up usingfuzzy arithmetic(identifiedasa sourceof estimationdistortions)and

we insist in the reestablishmentof thenaturalframework for solvinga minimum normproblem.Sincethe

orthogonalityconceptis relatedto thechoiceof a norm inducedby an innerproduct,our fuzzy estimation

procedureinevitably leadsto a quadraticprogrammingproblemandnot to a linear one.The decoupling

principle consistsin a set of rules,which allow us to expressthe fuzzy regressionmodelasa systemof

two classicalequationsandto choosethe correspondingprojectionsubspaces.We explore the algorithmic

consequences,whichfollow from suchtheoreticalconsiderationsfor varioussituationsandconcretetypesof

problems,andwe proposesomeMATLAB implementations.


F. Giannessiand G. Tomasin. Nonconvex quadraticprogramming,linear complementarity

problemsandintegerprogramming.In F. GiannessiandG. Tomasin,eds,‘Linear com-

plementarityproblemsandintegerProgramming’,pp.162–201,North-Holland,Amster-

dam,1974.

P. E. Gill andW. Murray. Numericallystablemethodsfor quadraticprogramming.Mathemati-cal Programming, 14(3), 349–372,1978.

Abstract. Numericallystablealgorithmsfor quadraticprogrammingarediscussed.A new algorithmis de-

scribedfor indefinitequadraticprogrammingwhich utilizes methodsfor updatingpositive-definitefactor-

izationsonly. Consequentlyall the updatingproceduresrequiredare commonto algorithmsfor linearly-

constrainedoptimization.Thenew algorithmcanbeusedfor thepositive-definitecasewithout lossof effi-

ciency.

P. E. Gill, N. I. M. Gould,W. Murray, M. A. Saunders,andM. H. Wright. Range-spacemethods

for convex quadraticprogrammingproblems.TechnicalReportSOL 82-9,Department

of OperationsResearch,StanfordUniversity, California,USA, 1982a.

P. E. Gill, N. I. M. Gould, W. Murray, M. A. Saunders,andM. H. Wright. A range-space

quadraticprogrammingalgorithm for problemswith a mixture of boundsandgeneral

constraints.TechnicalReportSOL 82-10,Departmentof OperationsResearch,Stanford

University, California,USA, 1982b.

P. E. Gill, N. I. M. Gould, W. Murray, M. A. Saunders,and M. H. Wright. A weightedGramm-Schmidtmethodfor convex quadraticprogramming. MathematicalProgram-ming, 30(2), 176–195,1984.

Abstract. Range-spacemethodsfor convex quadraticprogrammingimprove in efficiency asthenumberof

constraintsactive at the solutiondecreases.In this paperthe authorsdescribea rangespacemethodbased

uponupdatingweightedGram-Schmidtfactorizationof theconstraintsin theactive set.Theupdatingmeth-

odsdescribedareapplicableto bothprimalanddualquadraticprogrammingalgorithmsthatuseanactive-set

strategy. Many quadraticprogrammingproblemsincludesimpleboundson all thevariablesaswell asgen-

eral linear constraints.A featureof theproposedmethodis that it is ableto exploit the structureof simple

boundconstraints.This allows themethodto retainefficiency whenthenumberof generalconstraintsactive

at thesolutionis small.Furthermore,theefficiency of themethodimprovesasthenumberof active bound

constraintsincreases.

P. E. Gill, W. Murray, andM. A. Saunders.User’sguidefor QPOPT1.0: A Fortranpackagefor

quadraticprogramming.TechnicalReportSOL ??,Departmentof OperationsResearch,

StanfordUniversity, California,USA, 1995.

P. E. Gill, W. Murray, andM. H. Wright. Quadraticprogramming.In ‘PracticalOptimization’,

chapter5.3.2–5.4.1,pp.177–184.AcademicPress,London,England,1981.

P. E. Gill, W. Murray, D. B. Ponceleon, andM. A. Saunders.Solving reducedKKT–systems

in barriermethodsfor linear andquadraticprogramming.TechnicalReportSOL 91-7,

Departmentof OperationsResearch,StanfordUniversity, California,USA, 1991a.

P. E. Gill, W. Murray, M. A. Saunders,andM. H. Wright. User’s guidefor SOL/QPSOL:A

Fortranpackagefor quadraticprogramming.TechnicalReportSOL86-2,Departmentof

OperationsResearch,StanfordUniversity, California,USA, 1986.


P. E. Gill, W. Murray, M. A. Saunders,and M. H. Wright. A Schur-complementmethod

for sparsequadraticprogramming.In M. G. Cox andS. J. Hammarling,eds,‘Reliable

ScientificComputation’,pp.113–138,OxfordUniversityPress,Oxford,England,1990.

P. E. Gill, W. Murray, M. A. Saunders,andM. H. Wright. Inertia-controllingmethodsforgeneralquadraticprogramming.SIAMReview, 33(1), 1–36,1991b.

Abstract. Active-setquadraticprogramming(QP)methodsusea working setto definethesearchdirection

andmultiplier estimates.In themethodproposedby Fletcherin 1971,andin severalsubsequentmathemat-

ically equivalent methods,the working set is chosento control the inertia of the reducedHessian,which

is never permittedto have more than one nonpositive eigenvalue. (Suchmethodswill be called inertia-

controlling.) This paperpresentsan overview of a genericinertia-controllingQP method,including the

equationssatisfiedby the searchdirectionwhenthe reducedHessianis positive definite,singularand in-

definite.Recurrencerelationsare derived that definethe searchdirectionand Lagrangemultiplier vector

throughequationsrelatedto the Karush-Kuhn-Tucker system.Discussionis includedof connectionswith

inertia-controllingmethodsthatmaintainanexplicit factorizationof thereducedHessianmatrix.

D. Givoli andI. Doukhovni. Finiteelement-quadraticprogrammingapproachfor contactprob-lemswith geometricalnonlinearity. ComputersandStructures, 61(1), 31–41,1996.

Abstract. Thefinite element-quadraticprogramming(FE-QP)approachfor problemsinvolving frictionless

contactbetweenan elasticbody anda rigid obstacleis presentedin a generalsetting.The validity of this

approachis first provedin thecontext of smalldeformationproblems.Thenaway to extendit to thecaseof

largedeformationproblemsis proposed,andthevalidity andimplementationof thisprocedurearediscussed.

Numericalexamplesinvolving Euler-Bernoulli beamsarepresentedto demonstratetheseissues,andto show

theregimeswherethemethodis successfulor failing.

C. R. Glassey. Analysisof the federalenergy agency programby a quadraticprogrammingmodel. In ‘Proceedingsof LawrenceSymposiumon SystemsandDecisionSciences.WesternPeriodicalsCo,NorthHollywood,CA, USA’, pp.146–150,1977.

Abstract. A 21 sectorinput-outputmodel of the United Stateseconomy, in which exports, imports and

consumerdemandsaredeterminedendogenouslyby linearprice-demandfunctions,is usedto examinethe

impact of increasesin world market oil prices.Equilibrium pricesand flows in the model economyare

computedby maximizing a quadraticprogram.Policiesof the FederalEnergy Agency to mitigate these

impactsareanalysedby suitablemodificationsof themodel.

C.R. Glassey. Pricesensitiveconsumerdemandsin energy modeling-aquadraticprogrammingapproachto theanalysisof somefederalenergy agency policies. ManagementScience,24(9), 877–886,1978a.

Abstract. A 21 sectorinput-outputmodelof the 1972U.S. economyis extendedto includeconsumerde-

mands,imports,andexportsasendogeneousvariables,andusedto analyzesomeconsequencesof theprice

controlpolicy adoptedto mitigatetheimpactof thequadruplingof world crudeoil pricesin 1973–1974.It is

assumedthathouseholdconsumptionof goodsis linearlyrelatedto prices.An equilibriumof themodelecon-

omy is thencomputedby solvingaquadraticprogram.It is shown thatthepricecontrolpolicy is equivalent

to subsidizingimportedoil with revenuesfrom a taxondomesticcrudeproduction.Equilibriaarecomputed

underthis policy andin its absence.Thecomparisonindicatesthepolicy waseffective in reducingtheprice

index increaseandGNPreductionthatwould otherwisehave occurred,but at thecostof adverselyaffecting

thebalanceof payments.

C.R.Glassey. A quadraticnetwork optimizationmodelfor equilibriumsinglecommoditytrade

flows. MathematicalProgramming, 14(1), 98–107,1978b.

J. H. Glick, P. M. Pardalos,and J. B. Rosen. Global minimization of indefinite quadratic

problems.Computing, 39, 281–291,1987.


M. S.Goheen.Largescaleboundedvariablequadratic programming. PhDthesis,Department

of OperationsResearch,StanfordUniversity, Stanford,California,USA, 1976.

R. Goldbach.Somerandomizedalgorithmsfor convex quadraticprogramming.AppliedMath-ematicsandOptimization, 32, 121–142,1999.

Abstract. We adaptsomerandomizedalgorithmsof Clarkson[3] for linearprogrammingto theframework

of so-calledLP-typeproblems,whichwasintroducedby SharirandWelzl [10]. This framework is quitegen-

eralandallows aunifiedandelegantpresentationandanalysis.Wealsoshow thatLP-typeproblemsinclude

minimizationof a convex quadraticfunction subjectto convex quadraticconstraintsasa specialcase,for

which thealgorithmscanbeimplementedefficiently, if only linearconstraintsarepresent.Weshow thatthe

expectedrunningtimesdependonly linearlyonthenumberof constraints,andillustratethisby somenumer-

ical results.Eventhoughtheframework of LP-typeproblemsmayappearratherabstractat first, application

of themethodsconsideredin this paperto a givenproblemof that type is easyandefficient. Moreover, our

proofsarein fact rathersimple,sincemany technicaldetailsof moreexplicit problemrepresentationsare

handledin a uniform mannerby our approach.In particular, we do not assumeboundednessof thefeasible

setasrequiredin relatedmethods.

D. Goldfarb. Analogsof Newton’smethodfor quadraticprogramming.Noticesof theAmerican

MathematicsSociety, 15(2), 400,1968.

D. Goldfarb. Extensionsof Newton’s methodandSimplex methodsfor solvingquadraticpro-grams.In F. A. Lootsma,ed.,‘Numericalmethodsfor nonlinearoptimization’,pp.255–263.AcademicPress,London,England,1972.

Abstract. Two closely relatedmethods,that may be viewed either asextensionsof Newtons’ methodto

handlelinear equalitiesand inequalitiesor as quadraticanaloguesof the gradientprojectionmethod,are

presentedfor solvingstrictly convex quadraticprograms.Oneof thesemethodsandtheSimplex methodfor

quadraticprogrammingareshown to follow thesamesolutionpathif startedatavertex. A usefulrelationship

betweentheLagrangemultiplier variablesfor anestedsetof constraintbasesis alsoshown

D. Goldfarb. Efficient primal algorithmsfor strictly convex quadraticprograms.In J.P. Hen-

nart,ed., ‘NumericalAnalysis’, LectureNotesin Mathematics,p. ???,SpringerVerlag,

Heidelberg, Berlin, New York, 1985.

D. Goldfarb. Strategiesfor constrintdeletionin active setalgorithms. In D. F. Griffiths and

G. A. Watson,eds, ‘Numerical Analysis’, number140 in ‘Pitman ResearchNotes in

MathematicsSeries’,pp.66–81,LongmanScientificandTechnical,Harlow, Essex, Eng-

land,1986.

D. Goldfarb and A. U. Idnani. Dual and primal-dualmethodsfor solving strictly convex

quadraticprograms.In J.P. Hennart,ed.,‘NumericalAnalysis’,number909 in ‘Lecture

Notesin Mathematics’,pp. 226–239,SpringerVerlag,Heidelberg, Berlin, New York,

1982.

D. Goldfarb andA. U. Idnani. A numericallystabledual methodfor solving strictly convex

quadraticprograms.MathematicalProgramming, 27(1), 1–33,1983.

D. Goldfarb andS. C. Liu. Interior point potentialfunction reductionalgorithmfor solving

convex quadraticprogramming.Technicalreport,Departmentof IndustrialEngineering

andOperationsResearch,ColumbiaUniversity, New York, USA, 1990.


D. Goldfarb andS. C. Liu. An On3L primal interior point algorithmfor convex quadratic

programming.MathematicalProgramming, 49(3), 325–340,1991.

Abstract. Wepresentaprimal interiorpointmethodfor convex quadraticprogrammingwhichis basedupon

a logarithmicbarrierfunctionapproach.This approachgeneratesa sequenceof problems,eachof which is

approximatelysolvedby takingasingleNewtonstep.It is shown thatthemethodrequiresO nL iterations

andO n3 ( 5L arithmeticoperations.By usingmodifiedNewton stepsthe numberof arithmeticoperations

requiredby thealgorithmcanbereducedto O n3L .D. GoldfarbandS. C. Liu. An O

n3L primal dual potentialreductionalgorithmfor solving

convex quadraticprograms.MathematicalProgramming, 61(2), 161–170,1993.

D. GolovashkinandI. E. Grossmann.Interior cuts—anew type of cutting planesfor indefi-

nite quadraticprogramming.Technicalreport,Centerfor AdvancedProcessDecision-

Making, Departmentof ChemicalEngineering,Carnegie Mellon University, Pittsburgh,

PA, USA, 1998.

G.H. GolubandM. A. Saunders.Linearleastsquaresandquadraticprogramming.In J.Abadie,ed.,‘Integerandnonlinearprogramming’,pp.229–256,North Holland,Amsterdam,theNetherlands,1970.

Abstract. Oneof themostcommonproblemsin any computationcenteris thatof findinglinearleastsquares

solutions.Theseproblemsarisein a varietyof areasandin a varietyof contexts. For instance,thedatamay

bearriving sequentiallyfrom asourceandtheremaybesomeconstrainton thesolution.Linearleastsquares

problemsareparticularlydifficult to solve becausethey frequentlyinvolve largequantitiesof data,andthey

areill-conditionedby their very nature.This paperpresentsseveralnumericalalgorithmsfor solving linear

leastsquaresproblemsin ahighly accuratemanner. In addition,it givesanalgorithmfor solvinglinearleast

squaresproblemswith linearinequalityconstraints.

M. A. Gomez.An on2 activesetalgorithmfor thesolutionof aparametricquadraticprogram.

NumericalAlgorithms, 22(3–4),305–316,1999.

Abstract. In this paper, an O n2 active setmethodis presentedfor minimizing the parametricquadratic

function f rac12xT Dx aTx λmax c γTx 0 subjectto l x b, for all nonnegative valuesof the pa-

rameterlambda. Here,D is a positive diagonalnxnmatrix, a andγ arearbitraryn-vectors,c is anarbitrary

scalar, l andb arearbitraryn-vectors,suchthat l b. An extensionof this algorithmis presentedfor min-

imizing theparametricfunction 12xT Dx aTx λ 2 γT x c 2 subjectto l x b. It is alsoshown that these

problemsarisenaturallyin a tax programmingproblem.

A. S.Goncalves.A generalizedprimal-dualtechniquefor quadraticprogrammingin parametric

form. RevistaFaculdadeCienciasUniversidadeCoimbra, 47, 1–23,1971.

A. S. Goncalves. A primal-dualmethodfor quadraticprogrammingwith boundedvariables.In F. A. Lootsma,ed., ‘Numerical methodsfor non linear optimization’, pp. 255–263.AcademicPress,London,England,1972.

Abstract. A primal-dualalgorithmfor quadraticprogrammingwith boundedvariablesis establishedhere,

whichhasthesamecharacteristicsasthemethodby Eisemann(1963)for thelinearprogrammingcase,and

which may be consideredasan extensionof that method.However, distinct from the linear programming

case,therestrictedproblemsconstructedherehave no boundsin their variables.Thecomputationsmaybe

performedby pivotal operationsona tableau.Programmingof thealgorithmis facilitatedby its efficient use

of theproductform of theinversemechanismavailablein mostcommerciallinearprogrammingsystems.

A. S. Goncalves. A L.P. algorithmfor non-convex quadraticprogrammingandits economicinterpretation.Bulletin of theOperationsResearch Societyof America, 22, B95,1974.


Abstract. Considersthe problemof maximizinga convex quadraticobjective. The algorithmdevisedfirst

’normalizes’ the given problem,i.e., substitutesit by anotherequivalent problemwhosesolutionscanbe

found very easilyby a L.P. columngenerationprocedure.Extensionsto the generalnon-convex quadratic

programmingarealsoconsidered.

J.Gondzio.Hopdmversion2.30: Benchmarkresults:Solvingconvex quadraticprogramming

problems. Logilab technicalnote,Departmentof ManagementSciences,Universityof

Geneva,Geneva,Switzerland,1998.

V. N. Gordeev. The finitenessof methodsof solving a problemin quadraticprogramming.

Cybernetics, 7(1), 110–114,1971.

V. N. Gordeev. Simplemethodof solving an auxiliary quadraticprogrammingproblem. Cy-

bernetics, 16(4), 616–619,1980.

N. I. M. Gould. Numericalmethodsfor linear and quadratic programming. D. Phil. thesis,

Oxford University, England,1982.

N. I. M. Gould. Thestability of thesolutionof generalquadraticprograms.TechnicalReport

CORR83-11,Departmentof CombinatoricsandOptimization,Universityof Waterloo,

Ontario,Canada,1983.

N. I. M. Gould.Onpracticalconditionsfor theexistenceanduniquenessof solutionsto thegen-eral equalityquadraticprogrammingproblem. MathematicalProgramming, 32(1), 90–99,1985.

Abstract. The authorpresentspracticalconditionsunderwhich the existenceand uniquenessof a finite

solution to a given equality quadraticprogrammay be examined.Different setsof conditionsallow this

examinationto take placewhennull-space,range-spaceor Lagrangianmethodsareusedto find stationary

pointsfor thequadraticprogram.

N. I. M. Gould. An algorithmfor large-scalequadraticprogramming.IMA Journalof Numeri-cal Analysis, 11(3), 299–324,1991.

Abstract. The paperdescribesa methodfor solving large-scalegeneralquadraticprogrammingproblems.

Themethodis basedupona compendiumof ideaswhich have their originsin sparsematrix techniquesand

methodsfor solvingsmallerquadraticprograms.A discussionis includedonresolvingdegeneracy, onsingle

phasemethodsandonsolvingparametricproblems.Somenumericalresultsareincluded.

N. I. M. GouldandPh.L. Toint. A quadraticprogrammingbibliography. NumericalAnalysis

Group InternalReport2000-1,RutherfordAppletonLaboratory, Chilton, Oxfordshire,

England,2000.See”http://www.numerical.rl.ac.uk/qp/qp.html”.

N. I. M. Gould and Ph. L. Toint. Numericalmethodsfor large-scalenon-convex quadratic

programming.TechnicalReportRAL-TR-2001–017,RutherfordAppletonLaboratory,

Chilton,Oxfordshire,England,2001.

N. I. M. Gould,M. E. Hribar, andJ.Nocedal.Onthesolutionof equalityconstrainedquadratic

problemsarisingin optimization.TechnicalReportRAL-TR-98–069,RutherfordApple-

ton Laboratory, Chilton,Oxfordshire,England,1998.


F. Granotand J. Skorin-Kapov. Someproximity andsensitivity resultsin quadraticinteger

programming.MathematicalProgramming, 47(2), 259–268,1990a.

F. GranotandJ. Skorin-Kapov. Towardsa stronglypolynomialalgorithmfor strictly convex

quadraticprograms—anextensionof Tardos’algorithm. MathematicalProgramming,

46(2), 225–236,1990b.

F. Granot,J. Skorin-Kapov, andA. Tamir. Usingquadraticprogrammingto solve high multi-plicity schedulingproblemson parallelmachines.Algorithmica, 17(2), 100–110,1997.

Abstract. We introduceandanalyzeseveral modelsof schedulingn different types(groups)of jobs on m

parallelmachines,wherein eachgroupall jobsareidentical.Our maingoal is to exhibit the usefulnessof

quadraticprogrammingapproachesto solve theseclassesof highmultiplicity schedulingproblems,with the

total weightedcompletiontime astheminimizationcriterion.We develop polynomialalgorithmsfor some

models,andstronglypolynomialalgorithmsfor certainspecialcases.In particular, themodelin which the

weightsare job independent,as well as the generallyweightedmodel in which processingrequirements

are job independent,can be formulatedas an integer convex separablequadraticcost now problem,and

thereforesolvedin polynomialtime.Whenwespecializefurther, stronglypolynomialboundsareachievable.

Specifically, for theweightedmodelwith job-independentprocessingrequirementsif werestricttheweights

to bemachineindependent(while still assumingdifferentmachinespeeds),anO mn nlogn algorithmis

developed.If it is alsoassumedthatall themachineshave thesamespeed,thecomplexity of thealgorithm

canbe improved to O mlogm nlogn . Theseresultscanbeextendedto relatedunweightedmodelswith

variableprocessingrequirementsin whichall themachinesareavailableat timezero.

R. L. Graves. A principal pivoting Simplex algorithmfor linear andquadraticprogramming.

OperationsResearch, 15, 482–494,1967.

R. L. Graves. Quadraticprogrammingin Hilbert space.Bulletin of the OperationsResearchSocietyof America, 20, B–78,1972.

Abstract. Problemsareanalysedwhich arisein optimisingover infinite horizonswith discretetime, while

satisfyingafinite numberof inequalityconstraintsin eachtime period.Theseproblemsaretransformedinto

quadraticprogramson a separableHilbert spaceandtheexistenceof solutionsto the resultingprogramsis

established.Waysto calculatedapproximatesolutionsby solvingproblemsover a finite horizonareshown.

Someresults(not complete)for theexistenceof dualvariablesaregiven.

R. L. GravesandL. G. Telser. An infinite-horizondiscrete-timequadraticprogramasapplied

to a monopolyproblem.Econometrica, 35(2), ??,1967.

M. GrigoriadisandK. Ritter. A parametricmethodfor semidefinitequadraticprogramming.

SIAMJournalon Control andOptimization, 7(4), 559–577,1969.

L. GrippoandS.Lucidi. A differentiableexactpenaltyfunctionfor boundconstrainedquadraticprogrammingproblems.Optimization, 22(4), 557–578,1991.

Abstract. Definesacontinuouslydifferentiableexactpenaltyfunctionfor thesolutionof boundconstrained

quadraticprogrammingproblems.The authorsprove that thereexists a computablevalue of the penalty

parametersuchthat global and local minimizersof the penaltyfunction yield global and local solutions

to the original problem.This permitsthe constructionof Newton-typealgorithmsbasedon consistentap-

proximationsof theNewton’s directionof thepenaltyfunction.Conditionsthatensurefinite terminationare

established.

N. Grudinin. Combinedquadratic-separableprogrammingOPFalgorithmfor economicdis-patchandsecuritycontrol. IEEE Transactionson Power Systems, 12(4), 1682–1688,1997.


Abstract. Thispaperpresentsanew algorithmfor optimalpowerflow (OPF),whichis basedonP/Qdecom-

positionof OPFproblemandon combinedapplicationof quadraticandseparableprogrammingmethods.

Initially theunit costcurvesareapproximatedby thequadraticfunctionsandthequadraticprogrammingal-

gorithmis appliedasastartingmethod.This givesagoodinitial point for optimizationandreducesthetotal

computationtime.Themodificationof a separableprogramming(SP)algorithmwith generationof approx-

imatingintervals is considered.A new quadratic-separablealgorithmfor OPFis proposed,which combines

themainadvantagesof quadraticandseparableprogrammingmethods.A bi-criterionformulationof security

controlproblemon thebaseof economicandsecurityobjective functions(OF) is proposed.Thenumerical

resultsof OPFfor large-scalepower systemsaregivenfor differentmethods.

L. Guan. An optimal neuronevolution algorithmfor constrainedquadraticprogramminginimagerestoration.IEEETransactionsonSystemsManandCyberneticsPart A—SystemsandHumans, 26(4), 513–518,1996.

Abstract. An optimalneuronevolutionalgorithmfor therestorationof linearlydistortedimagesis presented

in thispaper. Theproposedalgorithmismotivatedby thesymmetricpositive-definitequadraticprogramming

structureinherentin restoration.Theoreticalanalysisandexperimentalresultsshow that the algorithmnot

only significantlyincreasesthe convergencerateof processing,bat alsoproducesgoodrestorationresults.

In addition,the algorithmprovidesa genuineparallelprocessingstructurewhich ensurescomputationally

feasiblespatialdomainimagerestoration.

L. GuanandX. L. Zhou. Real-timeimagefiltering: from optimalneuronevolution to vectorquadraticprogramming. 1994 IEEE International Conferenceon Systems,Man, andCybernetics.Humans,InformationandTechnology IEEE,New York, NY, USA, pp.694–699,1994.

Abstract. In this paper, a new schemeis introducedfor thepartitioningof imagesin imagefiltering using

neuralnetworks.Theproposedschemetakesinto accountthephysicalnatureof theimageformationprocess,

andthussimplifiesthe orderingschemeassociatedwith the imageprocessingframework basedon neural

networkswith hierarchicalclusterarchitecture.Also presentedin thepaperis avectorprocessingalgorithm.

By utilizing this algorithm,the pixels in the samerow/columnof an imageareprocessedsimultaneously.

Comparedwith thescalarneuronevolution algorithms,thevectoralgorithmprovidesgoodvisualquality in

imagefiltering. Visualexamplesareprovidedto demonstratetheperformanceof thenew approach.

J.Guddat.Stability in convex quadraticparametricprogramming.MathematischeOperations-

forschungundStatistik, 7(2), 223–245,1976.

F. Guder. Pairwisereactive SORalgorithmfor quadraticprogrammingof net import spatial

equilibrium-models.MathematicalProgramming, 43(2), 175–186,1989.

F. GuderandJ. G. Morris. Optimal objective function approximationfor separableconvexquadraticprogramming.MathematicalProgramming, 67(1), 133–142,1994.

Abstract. Wepresentanoptimalpiecewise-linearapproximationmethodfor theobjective functionof sepa-

rableconvex quadraticprograms.Themethodprovidesguidelineson how many grid pointsto useandhow

to position them for a piecewise-linearapproximationif the error inducedby the approximationis to be

boundedapriori.

I. Guevski. An analytic form of solution in quadraticprogramming. TekhnicheskaMisul,9(2), 7–21,1972.

Abstract. A parametricmethodis proposedto find a non-negative vectorx*, maximizing(minimizing) the

goal function F x pTx xTCx, whereA is a diagonalnegatively definedmatrix having dimensionsof

n & n, providedthatdx b p 3 0 d 3 0 b 0 . Themethodis usedto expresstherelationx f p A d bin aclosedform.


I. Guevski. Parametricmethodfor analyticalsolution of quadraticprogrammingproblems.Izvestiyana InstitutapoTekhnickeskaKibernetika, 15, 15–28,1973.

Abstract. A parametricmethodfor solutionof a quadraticsquareprogrammingproblem,arising in opti-

mizationof resourcesystems,is used.Thedependencebetweentheoptimumvectorandthecoefficientsin

theaim functionsandthe limitations is obtainedin apparentform. Thepossibility for solutionof problems

with a largenumberof variableswithout significantcomputingdifficulties, is shown. A numericalexample

is given.

T. D. GuoandS.Q. Wu. Predictor-correctoralgorithmfor convex quadraticprogrammingwithupperbounds.Journalof ComputationalMathematics, 13(2), 161–171,1995.

Abstract. Thepredictor-correctoralgorithmfor linearprogramming,proposedby S. Mizuno et al. (1990),

becamethemostwell known of theinteriorpointmethods.Thepurposeof thepaperis to extendtheseresults

in two directions.First,wemodify thealgorithmin orderto solveconvex quadraticprogrammingwith upper

bounds.Second,we replacethe correctorstepwith an iterationof R. C. Monteiro andAdler’s algorithm

(1989).With thesemodifications,thedualitygapis reducedby aconstantfactoraftereachcorrectorstepfor

convex quadraticprogramming.It is shown thatthenew algorithmhasaO nL iterationcomplexity.

T. D. GuoandS. Q. Wu. Interior ellipsoid methodfor convex quadraticprogramming.ActaMathematicaeApplacataeSinica, 19(1), 46–50,1996a.

Abstract. In this paper, a new variantof theinterior ellipsoidmethodfor convex quadraticprogrammingis

developed.Comparedto thealgorithmproposedby YeandTse(1989),thesub-problemof thenew algorithm

is a linearprogrammingproblemwhich is easyto solve,althoughbothhave thesamecomplexity.

T. D. Guo andS. Q. Wu. Propertiesof primal interior point methodsfor QP. Optimization,37(3), 227–238,1996b.

Abstract. Studiesthe propertiesof a primal interior point algorithmfor convex quadraticprograms.The

algorithmcanbeviewedasamodificationof thealgorithmspresentedby MonteiroandAdler (1989),Gold-

farbandLiu (1991),etc.In eachiteration,it computesanapproximatelyprojectedNewton directionhk and

needsO n2 4 5 arithmeticoperationsin average.Moving alonghk with a fixedstepsizewhich will make the

log-barrierfunction a significantdecrease.The algorithmterminatesafter O nL iterations.So the total

complexity of this algorithmis O n3L .Z. GuoandG. Xu. Calculationof economicdispatchof interconnectedsystemusingquadratic

programming.Automationof ElectricPowerSystems, 22(1), 40–44,1998.

Abstract. Basedon the principle of decompositioncoordinationandsensitivity analysis,a new modelof

the economicdispatchof interconnectedpower systemsis presentedin this paper, in which active powers

of tie lines areconsideredascoordinatingvariables.The primal problemis decomposedinto N individual

subproblemsof optimizationandonesimplecoordinationproblem.Thealgorithmis basedonquadraticpro-

grammingwith theideaof parametricprogrammingandrelaxationtechniques.Thefeaturesof theproposed

algorithmare little computationand reliableconvergencefor economicdispatchof interconnectedpower

systemsconcerningsafetyconstraints.Numericaltestshavebeencarriedout for a realinterconnectedpower

systemcomposedof threeindividual systems.The test resultsshow that the modelandthe algorithmare

bothfeasibleandeffective.

A. K. GuptaandJ. K. Sharma. A generalizedSimplex techniquefor solving quadraticpro-grammingproblem.IndianJournalof Technology, 21(5), 198–201,1983.

Abstract. A finite iterationprocedureis presentedfor solvingthequasi-concavequadraticfunctionsubjectto

linearconstraints.In eachiterationtwo basicvariablesarereplacedby two non-basicvariables.Theproblem

is solvedstartingwith a basicfeasiblesolutionandshowing theconditionsunderwhich thesolutioncanbe

improved.An illustrative exampleis given.


O.K. Gupta.Applicationsof quadraticprogramming.Journalof InformationandOptimizationSciences, 16(1), 177–194,1995.

Abstract. Quadraticprogramming(QP) haslong beenstudiedasan importantOR technique.Many algo-

rithmshave beendevelopedfor solvingQPproblems.QPhasalsobeenvery successfulfor modelingmany

real-life problems.This paperreviews applicationareaswhereQPhasbeeneffectively applied.Most appli-

cationsof QP have beenin finance,agriculture,economics,productionoperations,marketing, andpublic

policy. Applicationsin eachof theseareasarebriefly described.

P. GuptaandD. Bhatia. Multiparametricanalysisof themaximumtolerancein quadraticpro-grammingproblems.Opsearch, 37(1), 36–46,2000.

Abstract. In this paper, we usean alternative approachto study multiparametricsensitivity analysisin

quadraticprogrammingproblemsby usingtheconceptof maximumvolumein thetoleranceregion. A nu-

mericalexampleis givento illustratetheresultsof thepaper.

R. P. Gupta. Symmetricdualquadraticprogramin complex space.Proceedingsof theIndianAcademyof Sciences,SectionA, 72(2), 74–87,1970.

Abstract. Symmetricdualquadraticprogramin complex spaceis presentedandsomeduality theoremsare

proved. Self-duallinear andquadraticprogramsin complex spaceare formedandself-duality theoremis

extendedto thesecases.

C. D. Ha. An algorithmfor structured,large-scalequadraticprogrammingproblems.Technical

report,VirginiaCommonwealthUniversity, Virginia,USA, 1981.

W. W. HagerandD. W. Hearn. Thedualactive setmethodandquadraticnetworks. Research

Report90-7, Departmentof Industrial andSystemsEngineering,Gainesville,Florida,

USA, 1990.

W. W. HagerandY. Krylyuk. Graphpartitioningandcontinuousquadraticprogramming.SIAMJournalon DiscreteMathematics, 12(4), 500–523,1999.

Abstract. A continuousquadraticprogrammingformulation is given for min-cut graphpartitioningprob-

lems. In theseproblems,we partition the verticesof a graphinto a collection of disjoint setssatisfying

specifiedsizeconstraints,while minimizing the sumof weightsof edgesconnectingverticesin different

sets.An optimal solution is relatedto an eigenvector (Fiedlervector)correspondingto the secondsmall-

esteigenvalueof thegraph’s Laplacian.Necessaryandsufficient conditionscharacterizinglocal minimaof

thequadraticprogramaregiven.Theeffect of diagonalperturbationson thenumberof local minimizersis

investigatedusinga testproblemfrom theliterature.

W. W. Hager, T. A. Davis, Y. Krylyuk, andS.-C. Park. Graphpartitioningusingquadraticprogramming. Technicalreport, Computerand Information Scienceand EngineeringDepartment,Gainesville,Florida,USA, 1997.

Abstract. A continuousquadraticprogrammingformulationsaregivenfor min-cutgraphpartitioningprob-

lems.An optimalsolutionis relatedto aneigenvector(Fiedlervector)correspondingto thesecondsmallest

eigenvalueof thegraph’s Laplacian.Necessaryandsufficient conditionscharacterizinglocal minimaof the

quadraticprogramare given. A generalizationof the Kernighanand Lin exchangealgorithm to a block

exchangealgorithmcanbeusedto escapefrom alocalminimumof thecontinuousquadraticprogram.Com-

parisonsto theoptimalcut obtainedby otherapproachesto graphpartitioningarepresented.

W. W. Hager, P. M. Pardalos,I. M. Roussos,andH. D. Sahinoglou. Active constraints,in-

definitequadraticprogramming,andtestproblems.Journalof OptimizationTheoryand

Applications, 68(3), 499–511,1991.


H. H. Hall, E. O. Heady, A. Stoecker, andV. A. Sposito.Spatialequilibriumin usagriculture:a quadraticprogramminganalysis.SIAMReview, 17(2), 323–338,1975.

Abstract. A spatialcompetitiveequilibriumfor thecropandlivestocksectorsof theUSagriculturaleconomy

is approximated.Agricultural commoditiesare classifiedinto threemutually exclusive classes:primary,

intermediateanddesired.Primarycommoditiesrepresentavailableresources;intermediatecommoditiesare

producedonly asinputsfor furtherproduction;desiredcommoditiesarewantedeitherfor consumptionor

for otherusesoutsidethe system(export). Productionof cropsand livestockand intermarket commodity

shipmentsarerepresentedby linearactivities.Theobjective functionmaximizesaggregateproducerprofits.

Sincethedemandfunctionsarelinear, total revenueis quadraticin thepricesof desiredcommodities,hence

thequadraticprogrammingformulation.In the results,estimatedpricesfor desiredcommoditiesarelower

thanobservedpricesand,with minorexceptions,estimatedquantitiesexceedobservedquantities.

P. L. HammerandA. A. Rubin. Someremarkson quadraticprogrammingwith 0–1variables.RevueFrancaised’Informatiqueet deRechercheOperationnelle, 4(3), 67–79,1970.

Abstract. The aim of this paperis to show that every bivalent (0,1) quadraticprogrammingproblemis

equivalentto onehaving apositive (negative) semi-definitematrix in theobjective function,to establishcon-

ditionsfor differentclassesof localoptimality, andto show thatany problemof bivalent(0,1)programmingis

equivalent(a) to theproblemof minimizing a realvaluedfunction,partly in (0,1)andpartly in non-negative

variables,(b) to theproblemof finding theminimaxof a realvaluedfunctionin bivalent(0,1)variables.

P. L. Hammer, P. Hansen,andB. Simeone.Roof duality, complementationandpersistency in

quadratic0–1optimization.MathematicalProgramming, 28(2), 121–155,1984.

C. G. Han,P. M. Pardalos,andY. Ye. Computationalaspectsof aninterior point algorithmfor

quadratic-programmingproblemswith box constraints.In T. F. ColemanandY. li, eds,

‘Large-ScaleNumericalOptimization’,pp.92–112,SIAM, Philadelphia,USA, 1990a.

C. G. Han, P. M. Pardalos,andY. Ye. Interior point algorithmsfor quadraticprogramming

problems.In ‘Proceedingsof theConferenceon OptimizationMethodsandtheir Appli-

cations,Nauka,USSR’,pp.194–213,1990b. (In Russian).

C. G. Han, P. M. Pardalos,and Y. Ye. Algorithms for the solution of quadraticknapsack

problems.LinearAlgebra andits Applications, 152, 69–91,1991.

C. G. Han,P. M. Pardalos,andY. Ye. Onthesolutionof indefinitequadraticproblemsusingan

interiorpoint method.Informatica, 3(4), 474–496,1992.

S. P. Han. Solvingquadraticprogramswith anexactpenaltyfunction. In O. L. Mangasarian,

R.R.MeyerandS.M. Robinson,eds,‘NonlinearProgramming,4’, pp.25–55,Academic

Press,LondonandNew York, 1981.

S. P. Han. On the hessianof the Lagrangianandsecond-orderoptimality conditions. SIAM

Journalon Control andOptimization, 24, 339–345,1985.

S. P. Han andO. Fujiwara. An inertia theoremfor symmetricmatricesandits applicationto

nonlinearprogramming.LinearAlgebra andits Applications, 72, 47–58,1985.

S.P. HanandO.L. Mangasarian.Characterizationof positivedefiniteandsemidefinitematricesvia quadraticprogrammingduality. SIAM Journal of Algebraic and DiscreteMethods,5(1), 26–32,1984.


Abstract. Positive definiteandsemidefinitematricesinducewell-known duality resultsin quadraticpro-

gramming.Theconverseis establishedhere.Thusif certainduality resultshold for a pair of dualquadratic

programs,thentheunderlyingmatrix mustbepositive definiteor semidefinite.For example,if a strict local

minimumof aquadraticprogramexceedsor equalsastrictglobalmaximumof thedual,thentheunderlying

symmetricmatrixQ is positivedefinite.If aquadraticprogramhasalocalminimum,thentheunderlyingma-

trix Q is positive semidefiniteif andonly if theprimalminimumexceedsor equalsthedualglobalmaximum

andxT Qx 0 implies Qx 0. A significantimplication of theseresultsis that theWolfe dual maynot be

meaningfulfor nonconvex quadraticprogramsandfor nonlinearprogramswithout locally positive definite

or semidefiniteHessians,evenif theprimal secondordersufficient optimality conditionsaresatisfied.

M. T. HannaandM. Simaan. A closedform solution to a quadraticprogrammingproblemin complex variables. In ‘Proceedingsof the 23rd IEEE Conferenceon DecisionandControl.IEEE,New York, NY, USA’, Vol. 2, pp.1087–1092,1984.

Abstract. A specialquadraticprogrammingproblemin complex variablesis investigatedfor a closedform

solution.Two differentapproachesareused.Thefirst is a directapproachthat leadsto a family of solutions

becauseof a singularmatrix encounteredin thesolutionprocess.Thesecondis anindirectapproachbased

on parameterizingtheobjective function. It leadsto a solutionwhich is a memberin theabove family and

which is shown to bebounded.

P. Hansen.Quadratic0–1programmingby implicit enumeration.In F. A. Lootsma,ed.,‘Nu-

mericalmethodsfor non linear optimization’, pp. 265–278.AcademicPress,London,

England,1972.

P. HansenandB. Jaumard.Reductionof indefinitequadraticprogramsto bilinear programs.

Journalof GlobalOptimization, 2, 41–60,1992.

P. Hansen,B. Jaumard,M. Ruiz, andJ. Xiong. Global minimizationof indefinitequadratic

functionssubjectto box constraints. ReportG-91-54,GERAD, Ecole Polytechnique,

UniversiteMcGill, Montreal,Quebec,Canada,1991.

F. HarriganandI. Buchanan.A quadratic-programmingapproachto input-outputestimation

andsimulation.Journalof RegionalScience, 24(3), 339–358,1984.

K. HassanandF. Mahmoud.An incrementalapproachfor thesolutionof quadraticprogram-ming problems.MathematicalModelling, 8, 34–36,1987.

Abstract. An approachfor thesolutionof quadraticprogrammingproblemsis introduced.It is basedon an

incrementalmethod,andthemaximumnumberof incrementsis equalto thenumberof constraintsplus1.

E.J.Haug,R.Chand,andK. Pan.Multibody elasticcontactanalysisbyquadraticprogramming.Journalof OptimizationTheoryandApplications, 21(2), 189–198,1977.

Abstract. A quadraticprogrammingmethodfor contactproblemsis extendedto ageneralprobleminvolving

contactof n elasticbodies.Sharpresultsof quadraticprogrammingtheoryprovide anequivalencebetween

the original n-body contactproblemand the simplex algorithmusedto solve the quadraticprogramming

problem.Two multibodyexamplesaresolvedto illustratethetechnique.

B. S.He. A projectionandcontractionmethodfor a classof linearcomplementarity-problemsandits applicationin convex quadraticprogramming. AppliedMathematicsand Opti-mization, 25(3), 247–262,1992.

Abstract. In this paperwe proposea new iterative methodfor solving a classof linear complementarity

problems:u 0, Mu q , uT Mu q5 0, whereM is a given l l positive semidefinitematrix (not

necessarilysymmetric)andq is agiven l-vector. Themethodmakestwo matrix-vectormultiplicationsanda


trivial projectionontothenonnegative orthantat eachiteration,andtheEuclideandistanceof theiteratesto

thesolutionsetmonotonouslyconvergesto zero.Themainadvantagesof themethodpresentedareits sim-

plicity, robustness,andability to handlelargeproblemswith any startpoint. It is pointedout thatthemethod

maybeusedto solve generalconvex quadraticprogrammingproblems.Preliminarynumericalexperiments

indicatethatthismethodmaybevery efficient for largesparseproblems.

G. L. Hefley andM. E. Thomas. A comparisonof two quadraticprogrammingalgorithms.Technicalreport,FloridaUniv , Gainesville,FL, USA, 1970.

Abstract. ThepapercomparesWolfe’squadraticprogrammingalgorithmwith CottleandDantzig’sprinciple

pivot method.It is shown thatWolfe’s algorithmrequiresmoreoperations.

R. Helgason,J. L. Kennington,andH. Lall. A polynomially boundedalgorithmfor a singlyconstrainedquadraticprogram.MathematicalProgramming, 18(3), 338–343,1980.

Abstract. Presentsa characterizationof thesolutionsof a singly constrainedquadraticprogram.This char-

acterizationis thenusedin thedevelopmentof apolynomiallyboundedalgorithmfor thisclassof problems.

C. Helmberg. Fixing variablesin semidefiniterelaxations.SIAMJournal on Matrix AnalysisandApplications, 21(3), 952–969,2000.

Abstract. Thestandardtechniqueof reducedcostfixing from linearprogrammingis not trivially extensible

to semidefiniterelaxationsbecausethecorrespondingLagrangemultipliersareusuallynotavailable.Wepro-

posea generaltechniquefor computingreasonableLagrangemultipliers for constraintsthatarenot partof

theproblemdescription.Its specializationto thesemidefinite67 1 18 relaxationof quadratic0–1program-

ming yieldsanefficient routinefor fixing variables.Theroutineoffers thepossibilityof exploiting problem

structure.Weextendthetraditionalbijective mapbetween6 0 18 and 67 1 18 formulationsto theconstraints

sothat thedualvariablesremainthesameandstructuralpropertiesarepreserved.Consequently, thefixing

routinecanbe appliedefficiently to optimal solutionsof the semidefinite6 0 18 relaxationof constrained

quadratic0–1programmingaswell. Weprovidenumericalresultsshowing theefficacy of this approach.

M. P. HelmeandT. L. Magnanti.Designingsatellitecommunicationnetworksby 0–1quadraticprogramming.Networks, 19(4), 427–450,1989.

Abstract. In satellitecommunicationnetworks,distinctive facilitiescalledhomingstationsperformspecial

transmissionfunctions.Local demandnodesclusteredaroundeachhomingstationcommunicatewith each

othervia a local switch at the homingstation;demandnodesin differentclusterscommunicatewith each

othervia satelliteearthstationsat thehomingstations.Designingsucha communicationnetwork requires

choiceson the locationsof theearthstationsandon theassignmentsof demandnodesto the local clusters

at the earthstations.The authorsformulatethis problemasa 0–1 quadraticfacility locationproblemand

transformit into anequivalent0–1integerlinearprogram.Computationalexperienceonrealdatashows that

a branchandboundprocedureis effective in solving problemswith up to 40 demandnodes(major cities)

andthatthesolutionsthatthis algorithmfindsimprove considerablyuponmanagementgeneratedsolutions.

It is alsoshown thata greedyaddheuristic,asimplementedon anIBM PC,consistentlygeneratesoptimal

or near-optimalsolutions.

C. T. Herakovich andP. G. Hodge.Elastic-plastictorsionof hollow barsby quadraticprogram-ming. InternationalJournalof MechanicalSciences, 11(1), 53–63,1969.

Abstract. A numericalmethodwhichprovidesthecompletehistoryof thestressfunctionfor elastic-plastic

torsionof hollow barsduringquasistatic,monotonictwist is presented.Resultsaregiven for severalcross-

sectionsandarecomparedto otheravailableresults.Plasticunloadingis explicitly shown.

G. T. HermanandA. Lent. A family of iterative quadraticoptimizationalgorithmsfor pairs

of inequalities,with applicationin diagnosticradiology. MathematicalProgramming

Studies, 9, 15–29,1978.


C. Hildreth. A quadraticprogrammingprocedure.NavalResearch LogisticsQuarterly, 4, 79–

85,1957.Erratum,ibid. p. 361.

K. Hitomi andH. Nagasawa. Note on “Solution of the aggregateproductionplanningprob-lem in a multi-stage-multi-productmanufacturingsystemusingfunctionalspaceanaly-sisandquadraticprogrammingapproaches”.InternationalJournal of SystemsScience,15(11),1257–1262,1984.

Abstract. The paperof Teny andKochhar(seeibid., vol. = 14, num. 3, p. 325, 1983)concludedthat the

solutioncalculatedby using the functional spaceanalysistechniquepresentedby Hitomi andNakamura

(1976)is inferior to thatrenderedby thequadraticprogrammingapproach(Beale,1968).This papershows

thecontradictionincludedin their discussionanddeniestheabove conclusion.Functionalspaceanalysisis

alsodevelopedto obtainamixed-integersolution.

D. S. HochbaumandS. P. Hong. About stronglypolynomial-timealgorithmsfor quadratic

optimizationoversubmodularconstraints.MathematicalProgramming, 69(2), 269–309,

1995.

D. S. Hochbaum,R. Shamir, andJ. G. Shanthikumar. A polynomialalgorithmfor an integer

quadraticnonseparabletransportationproblem.MathematicalProgramming, 55(3),359–

371,1992.

P. G. Hodge,T. Belytschko, and C. T. Herakovich. Plasticity and quadraticprogramming.

ComputerApproachesin AppliedMechanicsASME, pp.73–84,1969.

V. R. Hoffner. A quadraticprogrammingformulation of the Markov studentflow problem.TIMS/ORSA-Bulletin, 1, 91,1976.

Abstract. Discussesthe problemsof forecastingeducationalenrollmentswith the useof a Markov flow

model.In many cases,thestudenttransitiondataneededto estimatethetransitionprobabilitiesby themaxi-

mumlikelihoodtechniqueis notavailable.Theaggregateenrollmentsareknown, thetransitionprobabilities

canbe estimatedby usinga quadraticprogrammingformulation of the problem.This methodis usedto

forecastelementary, secondary, andhighereducationenrollmentsin acontinuousflow process.

W. HollensteinandH. Glavitsch. Constraintsin quadraticprogrammingtreatedby switchingconcepts(power systems). In ‘Proceedingsof the TenthPower SystemsComputationConference’,pp.551–558.Butterworths,London,England,1990.

Abstract. When applying standardquadraticprogrammingmethodsto large power systemsthe size of

tableausis a handicap.Hence,an approachwas chosenwherebypenaltytermsas usually employed for

constrainingvariablesareconvertedto straight inequalitiesthusyielding linear programmingtechniques.

The methoddevelopedworks in two steps.The first solvesthe unconstrainedproblemwhich is linear, the

secondsolvessuperimposedLP problem.For computationalreasonstheviolationof constraintsis monitored

andtheLP tableauis built up asviolationsaredetected.Sparsity, forward-backwardsubstitutionandupdat-

ing techniquesareexploited.Economicloaddispatchingandlossminimizationareexamplesto illustratethe

effectivenessof themethod.

S.P. HongandS.Verma.A noteonthestrongpolynomialityof convex quadraticprogramming.MathematicalProgramming, 68(2), 131–139,1995.

Abstract. Weprove thatageneralconvex quadraticprogram(QP)canbereducedto theproblemof finding

thenearestpointonasimplicial conein O(n(3)+ n log L) steps,wheren andL are,respectively, thedimen-

sion andthe encodinglengthof QP. The proof is quite simpleandusesduality andrepeatedperturbation,

The implication,however, is nontrivial sincethe problemof finding the nearestpoint on a simplicial cone

hasbeenconsidereda simplerproblemto solve in the practicalsensedueto its specialstructure,Also we


show that,theoretically, this reductionimpliesthat(i) if analgorithmsolvesQPin a polynomialnumberof

elementaryarithmeticoperationsthatis independentof theencodinglengthof datain theobjective function

thenit canbeusedto solve QPin stronglypolynomialtime,and(ii) if L is boundedby a ’first order’ expo-

nentialfunctionof n then(i) canbestatedeven in strongerterms:to solve QPin stronglypolynomialtime,

it sufficesto find analgorithmrunningin polynomialtime that is independentof theencodinglengthof the

quadratictermmatrixor constraintmatrix.Finally, basedon theseresults,weproposeaconjecture.

R. Horst andN. VanThoai. A new algorithmfor solving thegeneralquadraticprogrammingproblem.ComputationalOptimizationandApplications, 5(1), 39–48,1996.

Abstract. For thegeneralquadraticprogrammingproblem(includinganequivalentform of thelinearcom-

plementarityproblem)a new solutionmethodof branchandboundtype is proposed.The branchingpro-

cedureusesa well-known simplicial subdivision andtheboundestimationis performedby solvingcertain

linearprograms.

H. S.Houthakker. TheCapacitymethodof quadraticprogramming.Econometrica, 28, 62–87,

1960.

S. Hoyle. A single-phasemethodfor quadraticprogramming. TechnicalReort SOL 86-9,

Departmentof OperationsResearch,StanfordUniversity, California,USA, 1986.

W. S.Hsia.A methodfor strictly convex quadraticprogrammingproblems.Opsearch, 14, 118–

124,1977.

G. H. HuangandB. W. Baetz. Grey quadraticprogrammingand its applicationto munic-ipal solid-wastemanagementplanningunderuncertainty. EngineeringOptimization,23(3), 201–223,1995.

Abstract. Thispaperintroducesagrey quadraticprogramming(GQP)methodasameansfor decisionmak-

ing underuncertainty. The methodimproves upon existing grey linear programming(GLP) methodsby

allowing theconsiderationof theeffectsof economiesof scaleoncostcoefficientsin theobjective function.

The approachalsohasadvantagesover a grey nonlinearprogrammingmethod,sincea global optimumis

obtainableandthe model is moderatelyeasyto solve throughcommerciallyavailablequadraticprogram-

mingpackages.Themodellingapproachis appliedto ahypotheticalproblemof wasteflow allocationwithin

a municipal solid wastemanagementsystem.The resultsindicatethat, comparedwith the GLP method,

GQPprovidesamoreeffective meansfor reflectingsystemcostvariationsandmaythereforegeneratemore

realisticandapplicablesolutions.

G.H. Huang,B. W. Baetz,andG.G.Patry. Wasteflow allocationplanningthroughagrey fuzzyquadratic-programmingapproach.Civil EngineeringSystems, 11(3), 209–243,1994.

Abstract. This paperproposesa grey fuzzy quadraticprogramming(GFQP)approachasa meansfor op-

timization analysisunderuncertainty. The methodcombinesthe ideasof grey fuzzy linear programming

(GFLP)andfuzzyquadraticprogramming(FQP)within ageneraloptimizationframework. It improvesupon

thepreviousGFLPmethodby usingn grey controlvariables,x λ i $ i 1 2 n , for n constraintsinstead

of onex λ for n constraintsin orderto incorporatethe independentpropertiesof thestipulationuncertain-

ties; it alsoimprovesupontheFQPmethodby furtherintroducinggrey numbersfor coefficientsin A andC

to effectively reflectthe lefthandsideuncertainties.Comparedwith theGFLPmethod,theGFQPapproach

is helpful for bettersatisfyingmodelobjective/constraints andproviding grey solutionswith highersystem

certaintyandlower systemcost;comparedwith theFQPmethod,moreinformationof the independentun-

certainfeaturesof not only the stipulationsbut also the lefthandsidecoefficients areeffectively reflected

in theGFQPmethod.TheGFQPmodellingapproachis appliedto a hypotheticalcasestudyof wasteflow

allocationplanningunderuncertainty, with the input model stipulationsfluctuatingwithin wide intervals

andhaving independentuncertaincharacteristics.Theresultsindicatedthat reasonablesolutionshave been

generated.ComparisonsbetweentheGFQPandFQP/GFLPsolutionsarealsoprovided,whichdemonstrate


that the GFQPmethodcould betterreflectsystemuncertaintiesandprovide morerealisticandapplicable

solutionswith lower systemuncertaintiesandhighersystembenefits.

M. Huang. Linearandquadraticprogrammingmethodsfor solvingsecurity-constrainedeco-

nomicdispatchproblems.Master’s thesis,Departmentof ElectricalandComputerEngi-

neering,Universityof Waterloo,Ontario,Canada,1991.

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image.First theextrapolationproblemis formulatedasanerrorminimizationproblemandthenit is itera-

tively solvedasastandardquadraticprogrammingproblem.Two examplesgivenin this papershow thatthe

proposedmethodoutperformsanaverageextrapolationmethodandthePOCS-basedextrapolationmethod

in anexchangeof thecalculationtime.

A. U. Idnani.Numericallystabledualprojectionmethodsfor solvingpositivedefinitequadratic

programs. PhDthesis,Departmentof ComputerScience,City Collegeof New York,New

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M. R. Irving andM. J.H. Sterling. Economicdispatchof active power by quadraticprogram-ming usinga sparselinearcomplementaryalgorithm(transmissionnetworks). Interna-tional Journalof ElectricalPowerandEnergySystems, 7(1), 2–6,1985.

Abstract. A new methodfor theeconomicdispatchof active power in transmissionnetworks is presented.

Formulationof the problemas a quadraticprogramallows the inclusion of a linear network model and

operationalconstraintstogetherwith anexplicit representationof the costincurredby transmissionlosses.

The problemis solved by a linear complementarypivoting algorithm which takes full advantageof the

sparseform of theobjective functionandconstraints.Computationalexperienceindicatesthatthemethodis

highly efficient andis capableof solvingrealisticproblemsin elapsedtimesthatarecompatiblewith online

applicationsusingminicomputerhardware.

H. Isoda. Decisionof optimumsupplyvoltageof substationtransformersby quadraticpro-gramming.I. Searchof a feasiblesolutionusingCrout’smethod.ElectricalEngineeringin Japan, 95(2), 33–39,1975a.

Abstract. Proposesadigital computingmethodto determinetheoptimumsupplyvoltageof substationtrans-

formers.Thesupplyvoltageof substationtransformersis regulatedsothatthesupplyvoltageto consumersis

keptwithin anallowablerangewithoutoverloadingsubstationtransformersanddistribution lines.Thisprob-

lem canbeformulatedinto quadraticprogramming,which canbesolvedmosttypically by Beale’s method.

Crout’s methodis useful for solving large-scalesimultaneouslinear equationsand for finding a feasible

solution.

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solutionof thequadraticprogramming.A practicaldigital computerprogramis alsodevelopedandapplied

to a large-scaledistribution network containingsix transformerbanks,34 feedersand177 loadpoints.The

total computingtime is about4.0sec.Therequiredmemorysizeis assmallas20kilowords.


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years(Markowitz’s efficient portfolioswith minimumrisk) andtheotheramorerecentinnovation(Sharpe’s

style analysiswhich estimatesan implied assetallocationfor an investmentfund). We show how, in the

presenceof inequalityconstraints,Excel’s Solver canbeusedto find theoptimalweightsin bothquadratic

programmingapplications.We alsoimplementa directanalyticsolutionfor generatingtheefficient frontier

whenthereareno inequalityconstraintsusingthe matrix functionsin Excel.Both applicationsuseonly a

smallnumberof assetclassesandrequirerepeateduseof theminimisationtask.Weshow how VisualBasic

for Applications(Microsoft’s macrolanguagefor Excel)canbeusedto programsuchtasks,confirmingthat

techniquesthatwerethe preserve of dedicatedsoftwareonly a few yearsagocannow beeasilyreplicated

usingExcelto solve realproblems.

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with quadraticperformanceindex. Themethodis basedon parametrizingthestatevariablesby Chebyshev

series.Thecontrolvariablesareobtainedfrom thesystemstateequationsasa functionof theapproximated

statevariables.In this method,thereis no needto integratethesystemstateequations,andtheperformance

index is evaluatedby an algorithmwhich is alsoproposedin this paper. This converts the optimal control

probleminto a small sizeparameteroptimizationproblemwhich is quadraticin the unknown parameters,

thereforethe optimal value of theseparameterscanbe obtainedby usingquadraticprogrammingresults.

Somenumericalexamplesarepresentedto show theusefulnessof theproposedalgorithm.

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vex quadraticprogrammingproblems.The basicsimplex algorithmfor convex quadraticprogrammingis

described.We thenshow how thesimplex methodfor linearprogrammingcanbeusedin a decomposition

crashprocedureto obtaina goodinitial basicsolutionfor the quadraticprogrammingalgorithm.We show

how this solutionmay beusedasa startingsolutionfor thesimplex-basedalgorithm.Besidesits ability to

obtaingoodstartingsolutions,thisprocedurehasseveraladditionalproperties.It canbeuseddirectly to find

anoptimalsolutionto aquadraticprograminsteadof simply finding a goodinitial solution;it providesboth


upperandlower boundson theobjective functionvalueasthealgorithmproceeds;it reducesthecomplex-

ity of intermediatecalculations;it avoidscertainnumericaldifficulties thatarisein quadratic,but not linear

programming.

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minimumof bilinearprogrammingproblem(BLP) or aconcave quadraticprogram(CQP)is examined.The

algorithmconsistsof solving a sequenceof linear complementarityproblems(LCP). A branch-and-bound

methodis alsoconsidered.This algorithmis basedon thereformulationof a BLP into anLCP with a linear

functionto minimize.Computationalexperiencewith smallandmediumscaleBLPsandCQPsindicatesthat

the SLCPalgorithmis quite efficient in finding a global minimum but it is, in general,unableto establish

that sucha solutionhasbeenfound. An algorithmto find a lower-boundfor the BLP canovercomethis

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solutionsto two personnonzerosummatrix (i.e.,bimatrix) gamesandto obtainsymmetricmatrix solutions

to vectorquadraticprogrammingproblems,andto obtainsolutionsto linearinterval programmingproblems.

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in the literature,its multivariategeneralizationsappearto have beenneglected.The paperpresentssome

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quadraticminimizationproblemis thensolvedusingthetheoryof generalizedinversesof matricesthereby

obtainingthe explicit equationsof motion of constrained,discretemechanicalsystems.The approachex-

tendsthe boundariesof Lagrangianmechanicsin thatwe provide a generalformulationfor describingthe

constrainedmotion of suchsystemswithout either the useof Lagrangemultipliers or the useof quasi-

coordinates.An importantfeatureof theapproachis thatwe do not requireprior knowledgeof thespecific

setof constraintsto accomplishthis formulation.This makesthe equationspresentedheremoregenerally

useful,andperhapsmoreaesthetic,than the Gibbs-Appellequationswhich requirea felicitous choiceof

problem-specificquasi-coordinates.Thenew equationsof motionpresentedhereareapplicableto both the

holonomicandnonholonomicconstraintsthatLagrangianmechanicsdealswith. They areobtainedin terms

of the usualgeneralizedcoordinatesusedto describethe constrainedsystem.Furthermore,they canbe in-

tegratedby any of the currently available numericalintegration methods,thus yielding analyticaland/or

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which attainsits global minimum at a specifiedvertex of a polytopein IRn9 k. The constructedfunction is

strictly concave in thevariablesx IRn andis linear in thevariablesy IRk. Thenumberof linearvariables

k maybemuchlargerthann, sothatlarge-scaleglobalminimizationtestproblemscanbeconstructedby the

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y d IRk, b IRm, andwhereA andB arem & n andm & k matrices,respectively, andQ is ann & n symmetric

positive definitematrix.Theauthorsfirst considerthecasewherek 0 andconstructa ’ tight’ parallelepiped

R, containingOmegaby usinganarbitrarysetof Q-conjugatedirectionsandby solving2n linearprograms.

They thendescribea branchandboundalgorithmin which thelinearity of theconvex envelopeallows effi-

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andpossiblymuchlargerthann. Preliminarycomputationalresultsarealsopresented.

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onthesolutionsto suchoptimizationproblems.Wetransformtheprobleminto aproblemfor minimizing the

traceof amatrixsubjectto positive definitenesscondition.Wethenproposeaninterior-point methodto solve

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quadraticprogramming.Technicalreport,Departmentof ComputerScience,University

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Abstract. In this paper, we proposean applicationof ”quadraticprogrammingbasedon binary search

(QPBS)” to ELD (economicload dispatch)calculationfor the purposeof determiningthe optimal gener-

ationallocationof thermalunitsaccuratelyandquickly. Most conventionalequalincrementalmethodscan-

not alwaysobtaintheoptimalsolutionfor ELD problemswhentransmissionlossesand/orupperandlower

boundsof generatoroutputsaretakeninto account.Numerousstudieshavebeenmadeto solve theproblems

accuratelywith considerablecomplexities.Theformulationof theproblemconsideredasstandardfor QPBS

is very similar to that of the ELD problemwhentransmissionlossesareneglected.Thereforeto the ELD

calculationwe apply the QPBSalgorithm,especiallythe Pardalos-Kovoor method,whosecomputingtime

orderis O n wheren is thenumberof variables.

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We find themaximumtolerancefor the simultaneousperturbationin right-hand-sidetermsof the problem

suchthat theoriginal andtheperturbedproblemshave thesameoptimalbasis.Themaximumtolerancefor

thecoefficientsin thelinearpartof theobjective functionof theproblemis alsocalculated.

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Abstract. Cell placementcanbeperformedusinga combinationof mathematicalprogramming,graphpar-

titioning anditerative improvementtechniques.Mathematicalprogrammingprovidesthe relative positions

of cells throughouttheplacementareawhile ignoringseveralplacementrestrictions.We describequadratic

andlinearprogramformulationsfor finding relative cell positions.Moreover, we demonstratethatbothfor-

mulationscanbesolvedusinganinterior point method.Numericalresultsarepresentedto demonstratethe

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A. KenningsandA. Vannelli. VLSI placementusingquadraticprogrammingandnetwork par-titioning techniques.InternationalTransactionsin Operational Research, 4(5–6),353–364,1997.

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mizing the interconnectingwire lengthandplacementarea.In this paper, the placementproblemis solved

usinga combinationof quadraticprogramming,circuit partitioning,clusteringandgreedycell interchange

heuristics.Thesolutionof a sequenceof quadraticprogramsandcircuit partitioningproblemsprovidesthe

generalpositionsof cellsin ahighqualityplacement.Computationalefficiency is achievedby usinganinte-

rior pointmethodfor solvingthesequenceof quadraticprograms.A veryefficientclusteringheuristicis used

to keepimportantgroupsof cellstogetherasthecellsarespreadthroughouttheplacementarea.Numerical

resultsonasetof benchmarkcircuitsillustratethatthisnew approachproducesstandardcell placementsthat

areup to 17%betterin wire length,14%betterin row lengthandup to 24 timesfasterthana well known

simulatedannealingplacementheuristic.

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Abstract. The problemmaxcT x subjectto Ax b, andx integer with componentseither 0 or 1, canbe

reformulatedas,maxQ x aT x x;Cx subjectto Ax b, andx < 0 1= , whereQ x is strictly convex. The

work of Ritterhasbeenspecializedto solve thisquadraticproblem.Initial computationalresultsindicatethat

theapproachis not competitive time-wisewith theimplicit enumerationalgorithmof Geoffrion.

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fuzzycontrolsystemsbasedonoptimizationtechniques.First, it demonstratesthatasingleton-typelinguistic

fuzzy logic controller(FLC) canbeconvertedinto a region-wisesector-boundedcontrolleror, moregener-

ally, a polytopic systemby quadraticprogramming(QP). Next, the convex optimizationtechniquecalled

linearmatrix inequalities(LMI) is usedto analyzetheclosedloopof theconvertedpolytopicsystem.Finally,

theapplicabilityof thesuggestedmethodologyis highlightedvia simulationresults.

P. Y. Kim andC. W. Yang. Advertisingdecision-model—aquadratic-programmingapproach.

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Abstract. Highlights the fundamentallimitation of the linearprogrammingmediaselectionmodelthrough

numerouscomputersimulations.As analternative, a quadraticprogrammingmodelwasproposedto over-

comethelimitation of linearprogrammingsolutionsto mediaselectionproblems.

Y. H. Kim, S. Y. Kim, andJ. B. Kim. Adaptive-controlof a binarydistillation columnusing

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K. C.Kiwiel. A dualmethodfor certainpositivesemidefinitequadraticprogrammingproblems.SIAMJournalon ScientificandStatisticalComputing, 10(1), 175–186,1989.

Abstract. Presentsadualactive setmethodfor minimizingasumof piecewiselinearfunctionsandastrictly

convex quadraticfunction,subjectto linearconstraints.It maybeusedfor directionfinding in nondifferen-

tiableoptimizationalgorithmsandfor solvingexactpenaltyformulationsof (possiblyinconsistent)strictly

convex quadraticprogrammingproblems.An efficient implementationis describedextendingtheGoldfarb

andIdnanialgorithm,which includesPowell’s refinements.Numericalresultsindicateexcellentaccuracy of

theimplementation.

K. C. Kiwiel. A Cholesky dual methodfor proximal piecewise linear programming. Nu-merischeMathematik, 68, 325–340,1994.

Abstract. A quadraticprogrammingmethodis given for minimizing a sumof piecewise linear functions

anda proximal quadraticterm, subjectto simpleboundson variables.It may be usedfor searchdirection

finding in nondifferentiableoptimizationalgorithms.An efficient implementationis describedthatupdatesa

Cholesky factorizationof active constraintsandprovidesgoodaccuracy via iterative refinement.Numerical

experienceis reportedfor somelargeproblems.

E. Klafszky and T. Terlaky. Orientedmatroids,quadraticprogrammingand the criss-crossmethod.AlkalmazottMatematikaiLapok, 14(3–4),365–375,1989.

Abstract. Quadraticprogramming,symmetry, andpositive (semi)definitenessweregeneralizedby Morris

andToddto orientedmatroids.Theauthorsslightly modify their definitionsin orderto getmoresymmetric

structures.Somegeneralizationsof Terlaky’s criss-crossmethodarepresentedfor orientedmatroidquadratic


programming.Thesealgorithmsarebasedon thesmallsubscriptrule andon signpatterns,anddo not pre-

serve feasibility on any subsets.Finally two specialcases(positive definitecaseandorientedmatroidlinear

programming)anda modificationarepresented.

E. Klafszky andT. Terlaky. Somegeneralizationsof thecriss-crossmethodfor quadraticpro-gramming.Optimization, 24(1–2),127–139,1992.

Abstract. Three generalizationsof the criss-crossmethod for quadratic programmingare presented.

Tucker’s, Cottle’s andDantzig’s principal pivoting methodsarespecializedasdiagonalandexchangepiv-

ots for the linearcomplementarityproblemobtainedfrom a convex quadraticprogram.A finite criss-cross

method,basedon least-index resolution,is constructedfor solving the LCP. In proving finiteness,orthog-

onality propertiesof pivot tableausandpositive semidefinitenessof quadraticmatricesareused.In the last

sectionsomespecialcasesandtwo furthervariantsof thequadraticcriss-crossmethodarediscussed.If the

matrixof theLCPhasfull rank,thenasurprisinglysimplealgorithmfollows, whichcoincideswith Murty’s

’Bard typeschema’in theP matrix case.

D. Klatte. OntheLipschitzbehaviour of optimalsolutionsin parametricproblemsof quadratic

programmingandlinearcomplementarity. Optimization, 16, 819–831,1985.

J.M. Kleinhans,G. Sigl, F. M. Johannes,andK. J.Antreich. GORDIAN: VLSI placementbyquadraticprogrammingandslicingoptimization.IEEETransactionsonComputerAidedDesignof IntegratedCircuitsandSystems, 10(3), 356–365,1991.

Abstract. In this paperwe presenta new placementmethodfor cell-basedlayoutstyles.It is composedof

alternatingandinteractingglobaloptimizationandpartitioningstepsthatarefollowedby anoptimizationof

theareautilization.Methodsusingthedivide-and-conquerparadigmusuallylosetheglobalview by gener-

atingsmallerandsmallersubproblems.In contrast,GORDIAN maintainsthesimultaneoustreatmentof all

cellsover all globaloptimizationsteps,therebyconsideringconstraintsthatreflectthecurrentdissectionof

the circuit. The global optimizationsareperformedby solving quadraticprogrammingproblemsthat pos-

sessuniqueglobalminima.Improvedpartitioningschemesfor thestepwiserefinementof theplacementare

introduced.Theareautilization is optimizedby anexhaustive slicing procedure.Theplacementmethodhas

beenappliedto realworld problemsandexcellentresultsin termsof bothplacementqualityandcomputation

timehave beenobtained.

B. Klummer. Globalestabilitat quadratischeroptimierungsauprobleme.Wissenschaftlische

Zeitschrift derrHumboldtUniversitat zuBerlin, 5, 565–569,1977.

T. Koch. Algorithm for resolving manipulatorredundancy—the compactQP method inNewton-Raphsoniterationscheme.In V. ChundyandE. Kurekova, eds,‘ISMCR ’95.Proceedingsof the Fourth InternationalSymposiumon Measurementand Control inRobotics.SlovakTech.Univ, Bratislava,Slovakia’, pp.357–362,1995.

Abstract. Kinematicredundancy occurswhena manipulatorpossessesmoredegreesof freedomthanthe

minimumnumberrequiredto executea given task.Thekinematicredundantrobotoffersgreaterflexibility,

versatility anddexterity. It cansimultaneouslyrealizethe main taskandadditionalformulatedtasks,like

for examplekeepingthe joints movementswithin a given limits, obstacleavoidance,singularconfiguration

avoidance,etc. The principal problemof redundantrobot applicationis to find suitablecontrol algorithm

(kinematicinversetransformationscheme).First, the papersetsup the requirementsthat suchalgorithms

shouldfulfil for industrialapplications.Next, thepaperpresentsanalgorithmthatmeetstheserequirements.

It is the compactQP methodin Newton-Raphsoniteration schemewith someelementsof the so called

configurationcontrol.Finally, simulationresultsfrom testingthis algorithmin simulationsystemfor robot

workcell aregiven.

T. Koch and P. Kowalczewski. Control of redundantrobot using the quadraticprogram-ming method.In ‘PraceNaukowe InstytutuCybernetykiTechnicznejPolitechnikiWro-clawskiej,Seria:Konferencjenumber43’, pp.411–418,1996.


Abstract. Thealgorithmfor controlof redundantrobotsis presented.It is thecompactquadraticprogram-

mingmethodin aNewton-Raphsoniterationschemewith someelementsof socalledconfigurationcontrol.

Two optionsof thealgorithmaredescribed.Finally theresultsfrom movementssimulationof therobotswith

openkinematicmultiple redundantchainswith 11 jointsarepresented.

M. Kojima and L. Tuncel. Discretizationand localization in successive convex relaxation

methodsfor nonconvex quadraticoptimizationproblems. TechnicalReportCORR98-

34, Departmentof Combinatoricsand Optimization,University of Waterloo,Ontario,

Canada,1998.

H. Konno.Maximizationof aconvex quadraticfunctionunderlinearconstraints.Mathematical

Programming, 11(2), 117–127,1976.

H. Konno.Maximizingaconvex quadraticfunctionoverahypercube.Journalof theOperations

Research Societyof Japan, 23(2), 171–189,1980.

P. KorhonenandG.Y. Yu. A referencedirectionapproachto multipleobjectivequadratic-linearprogramming.EuropeanJournalof OperationalResearch, 102(3), 601–610,1997.

Abstract. In this paper, weproposeaninteractive procedurefor solvingmultiplecriteriaproblemswith one

quadraticobjective,severallinearobjectives,andasetof linearconstraints.Theprocedureis basedontheuse

of referencedirectionsandweightedsums.Referencedirectionsfor thelinearfunctions,andweightedsums

for combiningthequadraticfunctionwith thelinearonesareusedasparametersto implementthefreesearch

of nondominatedsolutions.The idealeadsto the parametriclinear complementarityproblemformulation.

An approachto dealwith this typeof problemsis givenaswell. Theapproachis illustratedwith anumerical

example.

P. Korhonenand G. Y. Yu. On computingobjective function valuesin multiple objectivequadratic-linearprogramming.EuropeanJournalof OperationalResearch, 106(1),184–190,1998.

Abstract. We considerthe computationof objective function valueswhen a nondominatedfrontier is

searchedin multipleobjective quadratic-linearprogramming(MOQLP).Referencedirectionsandweighted-

sumsconstituteamethodologicalbasisfor thesearch.This idealeadsto aparametriclinearcomplementarity

model formulation.A critical task of making a searchprocedureefficient, is to computethe changesin

quadraticandlinearobjective functionsefficiently whena searchdirectionis changedor a basischangeis

performed.Thosechangesin objective functionscanbecomputedby a so-calleddirector indirectmethod.

The direct methodis a straightforward oneand basedon the useof unit changesin basicdecisionvari-

ables.Instead,theindirectmethodutilizessomeotherbasicvariablesof themodel.Weintroducetheindirect

methodandmake theoreticalandempiricalcomparisonsbetweenthemethods.Basedon the comparisons,

wepointout thattheindirectmethodis clearlymuchmoreefficient thanthedirectone.

F. Korner. Remarkson second-orderconditionsin connectionwith thealgorithmof Bealeforquadraticprogramming.EuropeanJournalof OperationalResearch, 40(1),85–89,1989.

Abstract. Thealgorithmof Bealeis considered.It is discussedfor thecasein which the’solution’ point of

Beale’s algorithmis a localminimizer. In thisalgorithmageneralform of second-orderconditionsarises.

F. Korner. Onthenumericalrealizationof theexactpenaltymethodfor quadraticprogrammingalgorithms.EuropeanJournalof OperationalResearch, 46(3), 404–408,1990.

Abstract. Optimizationproblemswith quadraticnonconvex objective functionsandlinear inequalitiesare

considered.An activeconstraintstrategy isusedto defineasequenceof equalityconstrainedproblems.Under

discussionis how thesubproblemscanbesolvedefficiently.


F. Korner. A noteon weakly active constraintsin connectionwith nonconvex quadraticpro-gramming.EuropeanJournalof OperationalResearch, 57(3), 409–411,1992.

Abstract. In generalit is very difficult to determinea true local minimizer in nonconvex quadraticpro-

gramming.Themainproblemsariseif wehave so-calledweaklyactive constraints.Theauthordiscussesan

efficient methodfor checkingthelocal optimality or determiningadirectionof descent.

F. KornerandB. Luderer. On the implementationof quadraticprogrammingalgorithms.Sys-temsAnalysisModellingSimulation, 6(9), 699–707,1989.

Abstract. Sequencesof systemsof linearequationsareconsideredwhich ariserecursively by addingor re-

moving oneequation.It is shown how asolutioncanbeefficiently obtainedfrom thesolutionof theprevious

system.Systemsof this kind originatefor examplein thesolutionprocessof generalquadraticoptimization

problemswith linear inequality constraintsvia reducingthemto equality constrainedproblems.Together

with thesolutionof thesystems,thenumberof positive, negative andzeroeigenvaluesis determined.Thus,

thevalidity of second-orderKuhn-Tucker conditionsmaybeeasilychecked.

F. KornerandC. Richter. On the efficient treatmentof the booleanquadraticprogrammingproblem.NumerischeMathematik, 40(1), 99–109,1982.

Abstract. Thedualof theBooleanquadraticprogrammingproblemis considered.Theoptimalvalueof the

objective functiongivesboundsfor thebranchandboundprocesswhichseemto bebetterthanthoseknown

from otherauthors.

L. V. Korsi andV. G. Sokolov. Synthesisof a systemof isotropic radiatorsby the methodof quadraticprogramming.RadioEngineeringandElectronic Physics, 17(3), 358–364,1972.

Abstract. An effective methodof synthesizinga systemof isotropicradiatorsis proposed.Theoptimizing

algorithmis basedon themethodof projectedgradient.Thesolutionsof theproblemsof mixedandphase

synthesisof an array of radiatorsare obtainedby this method.Examplesof numericalcomputationare

presentedillustratingdifferentapplicationsof theproposedmethod.

M. M. Kostreva. Generalizationof Murty’s direct algorithm to linear andconvex quadraticprogramming.Journalof OptimizationTheoryandApplications, 62(1), 63–76,1989.

Abstract. Murty’s algorithmfor the linearcomplementarityproblemis generalizedto solve theoptimality

conditionsfor linear andconvex quadraticprogrammingproblemswith both equalityand inequalitycon-

straints.An implementationis suggestedwhich providesbothefficiency andtight errorcontrol.Numerical

experimentsaswell asfield testsin variousapplicationsshow favorableresults.

O. I. Kostyukova andV. M. Raketskii. A methodof reducingthe suboptimalityestimatein

quadraticprogramming.DokladyAkademiiNaukBelarusi, 29(12),1072–1075,1985.

M. KotaniandI. Nemoto.Theinverseproblemin magnetopneumography—useof hypotheticaldistributionsandquadraticprogramming.NuovoCimentoD, 2(2), 594–607,1983.

Abstract. Theinverseproblemin themagneticmeasurementof thelungwasstudiedwith amethodusinghy-

potheticaldistributionsof magneticdust.Both unconstrainedandconstrainedminimizationsof anobjective

functionwereperformed.Simulationsandanalysisshowedtheefficacy of themethod.

P. F. Kough.Globalsolutionto theindefinitequadraticprogrammingproblem.Technicalreport,WashingtonUniv., St Louis,MO, USA, 1974.

Abstract. Theglobal solutionto the indefinitequadraticproblemis obtainedvia a generalizedBendercut

procedure.TheBendercut methoditeratively addscutsto a masterproblem.Maximising themasterprob-

lem is shown to be equivalent to maximisingseveral convex quadraticsubproblems.Eachcut createsa


subproblem.Thisdecompositionprovidesfor thesolutionof themasterproblemby animplicit enumeration

algorithmcombinedwith Tui cuts.In orderto accelerateconvergenceonly a subsetof the subproblemsis

solved. The formal methodis thencombinedwith the acceleratedprocedureto insureconvergenceto the

globaloptimum.

P. F. Kough.Theindefinitequadraticprogrammingproblem.OperationsResearch, 27(3),516–533,1979.

Abstract. Developsseveral algorithmsthatobtaintheglobaloptimumto the indefinitequadraticprogram-

mingproblem.A generalizedBenderscutmethodis employed.Thesealgorithmsall possessε-finite conver-

gence.To obtainfinite convergencetheauthordevelopsexactcuts,whicharelocally preciserepresentations

of a reducedobjective. A finite algorithmis thenconstructed.Introductorycomputationalresultsarepre-

sented.

I. S.Kourtev andE. G. Friedman.Clockskew schedulingfor improvedreliability via quadraticprogramming.1999IEEE/ACM InternationalConferenceon ComputeridedDesignDi-gestof TechnicalPapers.IEEE,Piscataway, NJ, USA, pp.239–243,1999.Also appearedin the Proceedingsof the Twelfth Annual IEEE InternationalASIC/SOCConference,IEEE,Piscataway, NJ,USA, 1999,pp 210–215.

Abstract. Thispaperconsiderstheproblemof determininganoptimalclockskew schedulefor asynchronous

VLSI circuit. A novel formulationof clock skew schedulingasa constrainedquadraticprogramming(QP)

problemis introduced.Theconceptof a permissiblerange,or avalid interval, for theclockskew of eachlo-

caldatapathis key to this QPapproach.Fromareliability perspective, theidealclockschedulecorresponds

to eachclock skew within thecircuit beingat thecenterof the respective permissiblerange.However, this

ideal clock scheduleis nor practically implementablebecauseof limitations imposedby the connectivity

amongthe registerswithin thecircuit. To evaluatethe reliability, a quadraticcostfunction is introducedas

theEuclideandistancebetweentheidealscheduleanda givenpracticallyfeasibleclock schedule.This cost

function is the minimizationobjective of the describedalgorithmsfor the solutionof the previously men-

tionedquadraticprogram.Furthermore,thework describedheresubstantiallydiffersfrom previousresearch

in that it permitscompletecontrolover specificclock signaldelaysor skews within thecircuit. Specifically,

thealgorithmsdescribedherecanbeemployedto obtainresultswith explicitly specifiedtargetvaluesof im-

portantclock delays/skews with a circuit, suchasfor example,theclock delays/skews for I/O registers.An

additionalbenefitis apotentialreductionin clock periodof up to 10%.An efficient mathematicalalgorithm

is derived for thesolutionof theQPproblemwith O r3 run time complexity andO r2 storagecomplex-

ity, wherer is the numberof registersin the circuit. The algorithmis implementedasa C++ programand

demonstratedontheISCAS’89suiteof benchmarkcircuitsaswell asonanumberof industrialcircuits.The

work describedhereyieldsadditionalinsightsinto thecorrelationbetweencircuit structureandcircuit timing

by characterizingthedegreeto which specificsignalpathslimit theoverall performanceandreliability of a

circuit. This informationis directly applicableto logic andarchitecturalsynthesis.

L. B. KovacsandM. Kotel. Indefinitequadraticprogrammingby gradientprojectionmethodand its applicationto optimal control of a chemicalplant. IFAC symposiumon multi-variablecontrol systemsVDI/VDEFachgruppeReqelungstechnik,Dusseldorf, WestGer-many, 1968.

Abstract. As a subproblemof the optimal control of a chemicalplant an arbitrary (possibly indefinite)

quadraticfunction is to beminimizedsubjectto linearconstraints.Theproblemis solvedby meansof two

computerprogramsthefirst of whichgivesalocaloptimumstartingfrom any feasiblepoint.Thisprogramis

basedon thewell-known gradientprojectionmethodutilizing thespecialstructureof theobjective function.

The gradientprojectionmethodhasbeenchosenbecauseof its goodconvergenceproperties.The second

programgenerates’good’ startingpointsfor thefirst programandterminatesthealgorithmif certaincriteria

aresatisfiedmakingvery likey thefactthatthepresentbestsolutionis aglobalmaximumof theproblem.

K. Koyama,K. Minato, S. Eiho, and M. Kuwahara. A methodfor multivariablequadratic

programmingpossessingspecialstructures.SystemsandControl, 19(7), 402–404,1975.


M. K. Kozlov, S.P. Tarasov, andL. G. Khachiyan.Polynomialsolvability of convex quadraticprogramming.DokladyAkademiiNaukSSSR, 248(5),1049–1051,1979.Seealso,SovietMathematicsDokladyvolume20,pages1108–1111,1979.

Abstract. An algorithmis constructedfor theexactsolutionof a quadraticprogrammingproblem,thevol-

ume(in binarysymbols)of computationbeinglimited by thelengthof theinput,i.e. it is shown thatquadratic

programmingbelongsto the classP of problemssolvableon deterministicTuring machines,in a time ex-

pressedasapolynomialof thebinaryinput length.

M. K. Kozlov, S. P. Tarasov, andL. G. Khachiyan.Polynomialresolutionof convex quadraticprogramming.ZhurnalVychislitel’noi Matematikii Matematicheskoi Fiziki, 20(5),1319–1323,1980.

Abstract. The input length of the probleminput for a quadraticprogrammingproblemis definedas the

numberof binarysymbolsnecessaryto describetheinputdata.An exactsolutionalgorithmfor thequadratic

programmingproblemis developedfor which thenumberof elementalcalculationsis boundedby apolyno-

mial in theinput length.It is shown thatquadraticprogrammingproblemsbelongto classPproblems,which

canbesolvedby deterministicTuring machines.

S.O. Krumke. Einemodifiziertebarrieremethodefur konvexequadratischeoptimierungsprob-

leme.Diplomathesis,Universityof Wurzburg, Germany, 1994.

S.O. Krumke. On theconvergenceof a modifiedbarrierfunctionmethodfor convex quadratic

andlinear programming. In ‘Proceedingsof the Third InternationalConferenceon In-

dustrialandAppliedMathematics(ICIAM’95), Hamburg, Germany’, 1995.

G. Krynska.Theprimalalgorithmusingconjugatedirectionsfor quadraticprogrammingprob-lemswith simpleupperbounds.Optimization, 18(4), 545–560,1987.

Abstract. An algorithmthatsolvesquadraticprogrammingproblemswith aconvex objective functionanda

feasibleregiondefinedby constraintsAx b, β x α is presented.ThealgorithmusestheSUBmethodto

establisha startingfeasiblepoint andto constructthestartingbasismatrix. A feasibledirectionfor thek-th

iterationis chosenfrom a setof row-vectorsof theinverseof thecurrentbasismatrix.After a finite number

of iterationsan optimal solutionis found.A conjugacy propertyof feasibledirections,convergenceof the

algorithm,thequestionof cycling anda comparisonwith otheralgorithmsfrom thesameclassof methods

of feasibledirectionsarediscussed.

C. N. KumarandK. Elango.Unit graphestimationandstabilizationusingquadraticprogram-

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1995.

H. P. KunziandW. Oettli. Integerquadraticprogramming.In R. GravesandP. Wolfe, eds,‘Re-

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USA, 1963.

A. Kuppurajulaand K. R. Nayar. Optimal operationof distribution networks by quadraticprogrammingmethods.Proceedingsof theIEEE, 58(7), 1172–1174,1970.

Abstract. The optimumtap settingsfor transformersat the feederpoints in a distribution network which

minimize(1) thetotalcopperlosses,or (2) thevoltagedeviationsof theloadpoints,areobtainedby quadratic

programmingmethods.

I. V. Kurdyumov, M. V. Mosolova,andV. E. Nazaikinskii. Computingalgorithmfor quadraticprogrammingproblemsof high dimensionality. Zhurnal Vychislitel’noi Matematiki iMatematicheskoi Fiziki, 18(5), 1119–1128,1978.


Abstract. Thealgorithmpresentedis basedon a modificationof thecoupleddirectionmethod,but requires

somewhatfewercomputingoperationsandlessmemory. Thereis alsoaprocedurefor eliminatingcumulative

machineerror.

P. Lachout. Degenerationin Wolfe’s algorithm for quadraticprogramming. EkonomickoMatematicky Obzor, 24(4), 463–468,1988.

Abstract. Wolfe’s Algorithm is awell-known andwidely usedsolutiontechniquefor convex quadraticpro-

grams.This papershows examplesin which, dueto degeneracy, the long form of Wolfe’s algorithmfails

to find the optimal solution.A new algorithmthat solvesevery problemof convex quadraticprogramsis

suggested.

Y. T. Lai, C. C. Kao, andW. C. Shie. A quadraticprogrammingmethodfor interconnectioncrosstalkminimization.In ‘ISCAS’99. Proceedingsof the1999IEEEInternationalSym-posiumonCircuitsandSystemsVLSI . IEEE,Piscataway, NJ,USA’, Vol. 6,pp.270–273,1999.

Abstract. Adjacentwires in a VLSI chip becomecloserasthefabricationtechnologyrapidly evolves.Ac-

cordingly it becomesimportantto considercrosstalkcausedby thecouplingcapacitancebetweenadjacent

wires in thelayoutdesignfor thefastandsafeVLSI circuits.Thecrosstalkis a functionof couplinglength

anddistance.Thecouplinglengthcanbereducedby segmentrearrangementtechnique.This paperpresents

a crosstalkminimizationtechniqueby adjustingthespacebetweenadjacentwires.An exampleis shown in

thispaperto demonstratetheeffectivenessof thetechnique.

C. M. LamandK. T. Fung.A quadraticprogrammingmodelfor optimaldatadistribution. BIT,21(3), 294–304,1981.

Abstract. A quadraticprogrammingmodelis developedto take into considerationa numberof factorsthat

caninfluencetheprocessof optimalallocationof dataamongthenodesin adistributeddatabase.Thefactors

includecommunicationcosts,translationcosts,congestioncostsandstoragecosts.Beale’s methodis used

to solve theresultingquadraticprogram.Somenumericalexamplesarepresentedandthepotentialsof such

anapproachin thedesignandanalysisof distributeddatabasesarediscussed.

A. H. LandandG. Morton. An inverse-basismethodfor Beale’s quadraticprogrammingalgo-rithm. ManagementScience, 19(5), 510–516,1973.

Abstract. This paperpresentsa versionof Beal’s QuadraticProgrammingAlgorithm (1959)for solving a

problemof maximisingaquadraticfunctionunderlinearconstraints.Themodificationdiscussedheremakes

it possibleto retain the ’inversebasis’ tableauwhich hasto be augmentedby additionalconstraintsto be

called’auxiliary.’ Thealgorithmhasbeensuccessfullytestedonacomputer.

A. H. LandandS.Powell. Fortrancodesfor mathematicalprogramming:linear, quadratic and

discrete. J.Wiley andSons,Chichester, England,1973.

L. Landesa-Porras,J. O. Rubinos-Lopez,andA. Garcia-Pino. QuadraticprogrammingandTikhonov regularizationfor illuminationof reflectorantennaswith low spillover. In ‘Pro-ceedingsICECOM ’97. 14th InternationalConferenceon Applied ElectromageticsandCommunications.KoREMA, Zagreb,Croatia’,pp.63–66,1997.

Abstract. Two new methodto synthesizea feed array for the illumination of reflectorantennasarepre-

sented.Spillover radiationpatternor total spillover power can be consideredto obtainefficient feedsby

usingquadraticprogrammingor Tikhonov regularization,respectively. Apertureor surfacefield optimiza-

tion is consideredto obtaina prescribedaperturedistribution. Optimizationis ensuredin the globalsense,

andthespillover playstherole of aconstraint.

M. C. Lang. Designof nonlinearphaseFIR digital filters usingquadraticprogramming.1997IEEEInternationalConferenceonAcoustics,Speech,andSignalProcessingIEEECom-put.Soc.Press,LosAlamitos,CA,USA, 3, 2169–2172,1997.


Abstract. This paperpresentstwo methodsfor thedesignof FIR filters with arbitrarymagnitudeandphase

responsesaccordingto a weightedmeansquarederrorcriterionwith constraintson theresultingmagnitude

andphaseerrors.This constrainedleastsquarecriterion allows for an arbitrary trade-off betweenpureL2

filtersandChebyshev filters.Theresultingnonlinearoptimizationproblemis eitherconvertedinto astandard

quadraticprogrammingproblem(method1) or exactly solvedby asequenceof quadraticprograms(method

2). Thequadraticprogrammingproblemscanbesolvedefficiently usingstandardsoftware.

D. J.Laughhunn.Quadraticbinaryprogrammingwith applicationsto capitalbudgetingprob-

lems.OperationsResearch, 18, 454–461,1970.

R. Lazimy. Mixed-integerquadraticprogramming.MathematicalProgramming, 22(3), 332–349,1982.

Abstract. Considersmixed-integer quadraticprogramsin which the objective function is quadraticin the

integer and in the continuousvariables,and the constraintsare linear in the variablesof both types.The

generalizedBenders’decompositionis asuitableapproachfor solvingsuchprograms.However, theprogram

doesnot becomemore tractableif this methodis used,sinceBenders’cuts are quadraticin the integer

variables.A new equivalent formulationthat rendersthe programtractableis developed,underwhich the

dual objective function is linear in the integer variablesandthe dual constraintset is independentof these

variables.Benders’cuts that arederived from the new formulationare linear in the integer variables,and

the original problemis decomposedinto a seriesof integer linear masterproblemsandstandardquadratic

subproblems.The new formulationdoesnot introducenew primary variablesor new constraintsinto the

computationalstepsof thedecompositionalgorithm.

R. Lazimy. Improved algorithm for mixed-integer quadraticprogramsand a computational

study. MathematicalProgramming, 32(1), 100–113,1985.

T. H. A. Le, T. PhamDinh, andD. M. Le. A combineddc optimization-ellipsoidalbranch-and-boundalgorithmfor solvingnonconvex quadraticprogrammingproblems.Journalof CombinatorialOptimization, 2(1), 9–28,1998.

Abstract. In this paperwe proposea new branch-and-boundalgorithmby usinganellipsoidalpartition for

minimizing an indefinitequadraticfunctionover a boundedpolyhedralconvex setwhich is not necessarily

givenexplicitly by a systemof linearinequalitiesand/orequalities.It is requiredthatfor this setthereexists

anefficient algorithmto verify whethera point is feasible,andto find a violatedconstraintif this point is

not feasible.The algorithmis baseduponthe fact that the problemof minimizing an indefinitequadratic

form over anellipsoidcanbeefficiently solvedby someavailable(polynomialandnonpolynomialtime) al-

gorithms.In particular, theD.C. (differenceof convex functions)algorithm(DCA) with restartingprocedure

recentlyintroducedby PhamDinh TaoandL.T. HoaiAn is appliedto globallysolvingthisproblem.DCA is

alsousedfor locally solvingthenonconvex quadraticprogram.It is restartedwith currentbestfeasiblepoints

in thebranch-and-boundscheme,andimprovedthemin its turn.ThecombinedDCA-ellipsoidalbranch-and-

boundalgorithmthenenhancestheconvergence:it reducesconsiderablytheupperboundandtherebya lot

of ellipsoidscanbeeliminatedfrom furtherconsideration.Severalnumericalexperimentsaregiven.

C. Y. C. Lee, D. R. Wiff, and V. G. Rodgers. Calculatinga relaxationspectrumfrom ex-perimentaldatavia quadraticprogrammingwith andwithout regularization.Journal ofMacromolecularScience—Physics, B19(2), 211–225,1981.

Abstract. Theregularizationquadraticprogrammingapproachto infer relaxationspectrafrom experimental

mechanicaldatawasstudied.It wasfoundthattheinferior of thesumof thesquareddifferencebetweenthe

inputdataandtheback-calculatedvaluesdid notyield asatisfactoryrelaxationspectrumastheregularization

weightingparameterα wasincreased.A modificationwasmadeto thefunctionto beminimizedsothatall

datapointswereequallyweighted.Quadraticprogrammingwasthenfoundto besufficient to infer a reliable

relaxationspectrumif theinputexperimentaldatahadahighdegreeof accuracy. Whentheexperimentaldata

werenotsufficiently accurate,regularizationover-regularizedthehigh-valueregion of thesolutionspectrum


beforeit could improve the low-valueregion. Decreasingthenumberof sought-afterpointsin thesolution

spectrumcancompensatefor thenoiselevel in theinputexperimentaldataandwill allow inferringareliable

relaxationspectrumfrom theexperimentaldata.

L. J.Lefkoff andS.M. Gorelick. Designandcost-analysisof rapidaquiferrestorationsystems

usingflow simulationandquadraticprogramming.GroundWater, 24(6),777–790,1986.

L. L. Lei andJ. E. Epperson. A regional-analysisof vegetableversusrow crop production

usingquadraticprogramming.AmericanJournalof Agricultural Economics, 69(5),1097,

1987.

C. E. Lemke. A methodof solutionfor quadraticprograms.ManagementScience, 8, 442–453,

1962a.

C. E. Lemke. Orthogonality, duality, andquadratictype problemsin mathematicalprogram-

ming. RPI MathRep56, Departmentof Mathematics,RensselaerPolytechnicInstitute,

Troy, NY, USA, 1962b.

M. L. LenardandM. Minkoff. Randomlygeneratedtestproblemsfor positivedefinitequadraticprogramming.ACM Transactionson MathematicalSoftware, 10(1), 86–96,1984.

Abstract. A procedureis describedfor randomlygeneratingpositive definitequadraticprogrammingtest

problems.The testproblemsareconstructedin the form of linear least-squaresproblemssubjectto linear

constraints.Theprobabilitymeasurefor theproblemssogeneratedis invariantunderorthogonaltransforma-

tions.The procedureallows the userto specifythe sizeof the least-squaresproblem(numberof unknown

parameters,numberof observations,andnumberof constraints),therelative magnitudeof theresiduals,the

conditionnumberof the Hessianmatrix of the objective function, and the structureof the feasibleregion

(thenumberof equalityconstraintsandof inequalitieswhich will beactive at thefeasiblestartingpoint and

at the optimal solution).An exampleis given illustrating how theseproblemscanbe usedto evaluatethe

performanceof asoftwarepackage.

A. LentandY. Censor. Extensionsof Hildreth’srow-actionmethodfor quadraticprogramming.SIAMJournalon Control andOptimization, 18(4), 444–454,1980.

Abstract. An extendedversionof Hildreth’s iterative quadraticprogrammingalgorithmis presented,geo-

metrically interpreted,andproved to producea sequenceof iteratesthat (i) convergesto the solution,and

(ii) hasan importantintermediateoptimality property. This extendedHildreth algorithmis castinto a new

form which morepronouncedlybringsout its primal-dualnature.Theapplicationof thealgorithmmaybe

governedby an index sequencewhich is moregeneralthana cyclic sequence,namely, by analmostcyclic

control,andasequenceof relaxationparametersis incorporatedwithout ruiningconvergence.Thealgorithm

is a row-actionmethodwhich is particularlysuitablefor handlinglarge(or huge)andsparsesystems.

D. J.Leo andD. J. Inman. A quadraticprogrammingapproachto thedesignof active-passivevibrationisolationsystems.Journalof SoundandVibration, 220(5), 807–825,1999.

Abstract. A quadraticprogrammingalgorithmispresentedfor studyingthedesigntradeoffs of active-passive

vibration isolationsystems.The novelty of the techniqueis that the optimal control problemis posedasa

quadraticoptimizationwith linear constraints.The quadraticcost function representsthe meansquarere-

sponseof thepayloadaccelerationandisolatorstroke,andthelinearconstraintsrepresentasymptotictrack-

ing requirementsandpeakresponseconstraints.Posingtheproblemasa quadraticoptimizationguarantees

thataglobaloptimalsolutioncanbefoundif oneexists,andtheexistenceof anoptimalsolutionguarantees

that the vibration isolationsystemsatisfiesthe specifieddesignconstraints.The utility of the techniqueis

demonstratedon a comparisonof passive vibration isolationandactive-passive vibration isolationutilizing

relative displacementfeedback.


B. H. Leung.Decimationfiltersfor oversampledanalogto digital convertersbasedonquadraticprogramming.1990IEEE InternationalSymposiumon CircuitsandSystemsIEEE,NewYork, NY, USA, 2, 899–901,1990.

Abstract. A descriptionis givenof a quadratic-programming-basedmethodologyfor thedesignof decima-

tion filtersin CMOStechnology. Themethodologypresentedtacklesthefilter designproblemby formulating

a quadraticprogrammingproblemthat minimizesthe integral of the aliasednoisesubjectto the passband

andstopbandconstraints.Thismethodologyofferstheflexibility of incorporatingnonsinusoidalquantization

noisespectraldensityin thedesignprocess.Moreover thefilter responsecanbeoptimizedto meetarbitrary

antialiasrequirements.Becausethe projectedHessianmatrix of the objective function is positive definite,

thequadraticfunctionhasauniqueminimum.

B. H. Leung.Designmethodologyof decimationfiltersfor oversampledadcbasedonquadraticprogramming.IEEETransactionson CircuitsandSystems, 38(10),1121–1132,1991.

Abstract. A designmethodologyfor oversampledanalog-to-digitalconverterdecimationfilters is presented.

Themethodologytacklesthefinite-impulse-response(FIR) filter designproblemby formulatingaquadratic

programmingproblemthatminimizestheintegral of thealiasednoisesubjectto thepassbandandstopband

constraints.Theapproachoffersa designwhoseresponseis optimizedto meetarbitraryquantizationnoise

power spectraldensityandanti-aliasrequirements.Becausethe projectedHessianmatrix of the objective

function is positive definite,the quadraticfunctionhasa uniqueminimum.Themethodologyis appliedto

designfilters for differentrequirementsandtheperformanceis comparedto conventionalapproaches.

A. J. Levy. A fast quadraticprogrammingalgorithm for positive signal restoration. IEEETransactionson Acoustics,Speech andSignalProcessing, 31(6), 1337–1341,1983.

Abstract. Whena signalor pictureis processedby deconvolution, any additionala priori informationis of

prime interestsinceit canpotentiallyleadto an improvementin resultsandto superresolutionby reducing

ambiguity. The authorproposesa deconvolution methodthatassumes,first, that the unknown signalhasa

known finite extent, andsecond,that this signal is positive. The problemis statedin termsof a quadratic

programmingproblemis statedin termsof aquadraticprogrammingproblemwith positivity constraintsand

proposedis a new algorithmdeliveredfrom a conjugategradientmethod,especiallysuitedto this particular

situation,which leadsto a low costsolution.Experimentalresultson two-dimensionalsignals,emphasizing

relevantsuperresolutionarethenpresented.

J. T. Lewis, R. Murphy, andD. W. Tufts. Designof minimum noisedigital filters subjecttoinequalityconstraintsusingquadraticprogramming. IEEE Transactionson Acoustics,Speech andSignalProcessing, ASSP-24(5), 434–436,1976.

Abstract. A numericalmethodis presentedfor designingdigital filters.Themethodallows oneto minimise

themean-squareerroror noisepower over someintervals of frequency, while simultaneouslyconstraining

themaximumerror in otherintervals of frequency. Thus,for example,onecanminimisenoisepower from

a stopbandof frequencieswhile constrainingsignal fidelity in a passbandof frequenciesby limiting the

maximumpassbanddeviation.

W. Li. Error boundsfor piecewiseconvex quadraticprogramsandapplications.SIAMJournalon Control andOptimization, 33(5), 1510–1529,1995.

Abstract. In this paper, we establisha local error estimatefor feasiblesolutionsof a piecewise convex

quadraticprogramandaglobalerrorestimatefor feasiblesolutionsof aconvex piecewisequadraticprogram.

Theseerror estimatesprovide a unified approachfor deriving many old andnew error estimatesfor linear

programs,linear complementarityproblems,convex quadraticprograms,andaffine variationalinequality

problems.Theapproachrevealsthefact thateacherrorestimateis a consequenceof somereformulationof

theoriginalproblemasapiecewiseconvex quadraticprogramor aconvex piecewisequadraticprogram.In a

sense,evenRobinson’s resulton theupperLipschitzcontinuityof a polyhedralmappingcanbeconsidered

asa specialcaseof errorestimatesfor approximatesolutionsof a piecewise convex quadraticprogram.As


anapplication,wederive new (global)errorestimatesfor iteratesof theproximalpointalgorithmfor solving

aconvex piecewisequadraticprogram.

W. Li. Differentiablepiecewisequadraticexactpenaltyfunctionsfor quadraticprogramswith

simpleboundconstraints.SIAMJournalonOptimization, 6(2), 299–315,1996.

W. Li andJ.J.deNijs. An implementationof QSPLINEmethodfor solvingconvex quadraticprogrammingproblemswith simpleboundconstraints.Technicalreport,DepartmentofMathematicsandStatistics,Old DominionUniversity, Norfolk, Virginia,U.S.A.,2000.

Abstract. A convex quadraticprogrammingproblemwith simpleboundconstraintscanbereformulatedas

anunconstrainedminimizationproblemwith a convex quadraticspline(i.e., a differentiableconvex piece-

wise quadraticfunction) as the objective function. This leadsto a new paradigmfor solving the original

quadraticprogrammingproblem,in which variousunconstrainedminimizationalgorithmscanbe usedto

find a stationarypoint of the convex quadraticspline.In this paper, we give an implementationof a reg-

ularizedNewton method(calledQSPLINEMethod)for finding a stationarypoint of the convex quadratic

spline.QSPLINEMethodcanalsobeconsideredasan implicit active-setmethodwith two novel features:

(i) a mixedprimal-dualapproachfor identifying active indices,and(ii) a line searchstrategy for a dynamic

balancebetweentheneedof minimizingtheoriginalobjective functionandthatof forcingtheiteratesto stay

in thefeasibleregion.Theimplementedversionof QSPLINEMethodusesa matrix up datingtechniquefor

computingNewton directionsanda correctionstrategy for robust reductionof the convex quadraticspline

whenline searchin aNewtondirectionfails.Wehavetestedthecodeonavariationof thetestproblemsused

by More andToraldooraldo(1989).For our generatedtestproblem,the Hessianof the objective function

couldbeann by n positivesemidefinitematrixwith rank2n 3 , andonethird of theLagrangemultiplierscor-

respondingto active constraintsat ageneratedoptimalsolutioncouldbezero.Thatis, our testproblemsare

degenerateandhave highly singularHessians.Thecodefindsveryaccuratenumericalsolutionsfor all 2800

randomlygeneratedtestproblems(with n 300and500)eventhoughall theseproblemsaredegenerateand

about60%of theseproblemshave infinitely many solutions.

W. Li andJ.Swetits.A Newton methodfor convex regression,datasmoothing,andquadratic

programmingwith boundedconstraints.SIAMJournal on Optimization, 3(3), 466–488,

1993.

W. Li andJ. Swetits. A new algorithmfor solvingstrictly convex quadraticprograms.SIAMJournalon Optimization, 7(3), 595–619,1997.

Abstract. We reformulateconvex quadraticprogramswith simple boundconstraintsand strictly convex

quadraticprogramsasproblemsof unconstrainedminimizationof convex quadraticsplines.Therefore,any

algorithmfor finding a minimizer of a convex quadraticsplinecanbe usedto solve thesequadraticpro-

grammingproblems.In this paper, we proposea Newton methodto find a minimizerof a convex quadratic

splinederived from the unconstrainedreformulationof a strictly convex quadraticprogrammingproblem.

TheNewton methodis a ”naturalmixture” of a descentmethodandanactive-setmethod.Moreover, it is an

iterative method,yet it terminatesin finite operations(in exactarithmetic).

X. Li andZ. Xuan. An interior-point QP algorithm for structuraloptimization. StructuralOptimization, 15(3–4),172–179,1998.

Abstract. A new algorithmfor convex quadraticprogramming(QP)is presented.Firstly, thesurrogateprob-

lem for QPis developed,andtheKarush-Kuhn-Tucker conditionsof the surrogateproblemhold if theun-

constrainedminimumof theobjective function doesnot satisfyany constraints.Then,Karmarkar’s (1984)

algorithmfor linear programming(LP) is introducedto solve the surrogatedual problem.In addition, the

caseof generalconstraintsis alsodiscussed,andsomeexamplesof optimumtrusssizingproblemsshow that

theproposedalgorithmis robustandefficient.


R.H. LiangandY. Y. Hsu.Hydroelectricgenerationschedulingusingaquadraticprogrammingbasedneuralnetwork. In ‘IPEC ’95. Proceedingsof theInternationalPowerEngineeringConference.NanyangTechnol.Univ, Singapore’,Vol. 2, pp.684–689,1995.

Abstract. A new approachbasedon neuralnetworks is proposedfor hydroelectricgenerationscheduling.

Thepurposeof hydroelectricgenerationschedulingis to determinetheoptimalamountsof generatedpowers

for the hydro units in the systemfor the next N (N 24 in this paper)hoursin the future.The proposed

approachis basicallya two-stagesolutionmethod.In thefirst stage,a quadraticprogrammingbasedneural

network is developedin orderto reacha preliminarygenerationschedulefor the hydro units.Sincesome

practicalconstraintsmaybeviolatedin thepreliminaryschedule,a heuristicrule basedsearchalgorithmis

developedin thesecondstageto reachafeasiblesuboptimalschedulewhichsatisfiesall practicalconstraints.

The proposedapproachis applied to hydroelectricgenerationschedulingof Taiwan power system.It is

concludedfrom theresultsthat theproposedapproachis very effective in reachingproperhydrogeneration

schedules.

N. Limic andA. Mikelic. ConstrainedKriging usingquadraticprogramming.Journal of the

InternationalAssociationfor MathematicalGeology, 16(4), 423–429,1984.

Y. Lin andJ. Pang. Iterative methodsfor large convex quadraticprograms:a survey. SIAM

Journalon Control andOptimization, 25, 383–411,1987.

G. Ling. An optimal neuronevolution algorithm for constrainedquadraticprogramminginimagerestoration.IEEETransactionsonSystems,ManandCybernetics,Part A (SystemsandHumans), 26(4), 513–518,1996.

Abstract. An optimalneuronevolutionalgorithmfor therestorationof linearlydistortedimagesis presented

in thispaper. Theproposedalgorithmis motivatedby thesymmetricpositive-definitequadraticprogramming

structureinherentin restoration.Theoreticalanalysisandexperimentalresultsshow that the algorithmnot

only significantlyincreasesthe convergencerateof processing,but alsoproducesgoodrestorationresults.

In addition,the algorithmprovidesa genuineparallelprocessingstructurewhich ensurescomputationally

feasiblespatialdomainimagerestoration.

C. T. Liu, G. C. Temes,andH. Samueli.FIR filter designfor sigma-deltaa/dconvertersusingquadraticprogramming.IEEE Pacific RimConferenceon Communications,ComputersandSignalProcessingIEEE,New York, NY, USA, 2, 760–763,1991a.

Abstract. Thedesignof overallfinite impulseresponse(FIR) filter for sigma-deltaanalog-to-digitalconvert-

ersis formulatedasa quadraticprogramwith thetotal outputnoiseastheobjective function,subjectto the

passbandandstopbandconstraints.The formulationis soflexible that it cansuit differentrequirementsfor

suppressingthequantizationnoiseaswell asrejectingtheout-of-bandspurioussignal.Bothone-stagesingle

rateandmultistagemultiratefilter structuresareconsidered.For thepurposeof suppressingthequantization

noise,themultistagemultirateFIR filter with combfiltersasprefiltershasmuchlower complexity with little

performancedegradationrelative to theone-stageoptimalfilter.

C. T. Liu, G. C. Temes,and H. Samueli. FIR filter designusing quadraticprogramming.1991IEEEInternationalSymposiumonCircuitsandSystemsIEEE,New York, NY, USA,1, 148–151,1991b.

Abstract. Theoptimumdesignof FIR (finite impulseresponse)filters is formulatedasaquadraticprogram.

An algorithmis describedfor theefficient useof computermemoryandfor guaranteedconvergenceto the

solutionof theoriginal problem.Examplesillustrating thedesignof decimationfilters for sigma-deltaA/D

convertersusingthenew techniquearealsopresented.Thequadraticprogrammingapproachis moreflexible

andoftenyieldsbetterresultsthanthepreviously reportedleast-squaresdesignapproaches.


J. Q. Liu, T. T. Song,andD. Z. Du. On the necessaryandsufficient conditionof the local

optimal solutionof quadraticprogramming. ChineseAnnalsof MathematicsSeriesB,

3(5), 625–630,1982.

X. Liu, Y. Sun,andW. Wang.Stabilizingcontrolof robustnessfor systemswith maximumun-certainparameters-aquadraticprogrammingapproach.Control TheoryandApplications,16(5), 729–732,1999.

Abstract. Basedon a sufficient conditionfor stabilization,a way to stabilizelinear discrete-timesystems

with uncertainparametersis developed.By quadraticprogramming,theoptimalcontrollerof fixedorderwe

haveobtainedcanstabilizethesystemwith maximuminfinite normof theuncertainsystemparametervector.

Theconclusionis confirmedby simulation.

K. L. Lo and S. P. Zhu. A decoupledquadraticprogrammingapproachfor optimal powerdispatch.Electric PowerSystemsResearch, 22(1), 47–60,1991.

Abstract. Theauthorspresenta decoupledapproachfor optimalactive andreactive power dispatchfor the

economicoperationof an electricutility. The proposediterative schemereporteddecomposesthe optimal

loadflow probleminto a P-subproblemanda Q-subproblem,wherethecontrolvariablesaretheactive and

reactivepoweroutputsof generators.At eachiterationthesubproblemsaresolvedby quadraticprogramming

using the flexible toleranceoptimizationmethod(FTOM). In this approach,constraintsare linearizedby

usingsensitivity analysisandtheperturbationtechnique,in orderto obtaina wider near-feasibleregion for

this quadraticprogrammingmethod.This approachhasa goodcomputationspeed.The resultsof two test

systemsandthreecasesarepresentedandcomparedwith theresultsof othermethods.

W. LochrieandD. Isaacs.Theapplicationof quadraticprogrammingin adaptive filtering. In‘Proceedingsof the8th annualAllerton conferenceon circuit andsystemtheory. IEEE,New York, NY, USA’, pp.101–110,1970.

Abstract. The problemof designingan adaptive filter for a nonlineardynamicsystemis formulatedasa

constrainedestimationproblemin which themeasurementnoisevariancesareestimated.An adaptive filter

mechanizationwith animbeddedconstrainedestimationalgorithmis thendeveloped.A quadraticprogram-

mingalgorithmis incorporatedin themechanizationdevelopedfor learningthesevariancevalues.

E. Loehman,D. Pingry, andA. B. Whinston.Quadraticprogrammingandestimationproblems.Technicalreport,PurdueUniv , Lafayette,IN, USA, 1970.

Abstract. Thepaperpresentsvariousproblemsrelatingto multivariateregressionin a unifiedfashionusing

thequadraticprogrammingtableau.Thetopicsconsideredaremulticollinearityandthegeneralizedinverse,

picking the bestregressionmodel,andconstrainedestimatesandhypothesistestingfor regressioncoeffi-

cients.

H. Lofgren. LiberalisingEgypt’s agriculture:a quadraticprogramminganalysis. Journal of

African Economies, 2(2), 238–261,1993.

F. A. LootsmaandJ.D. Pearson.An indefinite-quadratic-programmingmodelfor acontinuous-productionproblem.PhilipsResearch Reports, 25(4), 244–254,1970.

Abstract. A model is presentedfor a problemof schedulingthe lengthsof N productionperiodson one

machine,whichwill manufacture1 products.Theproblemis to chooseproductionperiodssoasto minimize

thesumof inventorycostsfor theq productsin thepresenceof givendemands.Mathematically, theproblem

is one of minimizing an indefinite quadraticfunction subjectto linear constraints.A numericalexample

concludesthepaper.

S.L. Louwes,J.C. G. Boot,andS.Wage.A quadraticprogrammingapproachto theproblem

of theoptimaluseof milk in thenetherlands.Journal of Farm Economics, 45, 309–317,

1963.


Z. Lu andZ. Wei. Decompositionmethodfor quadraticprogrammingproblemwith box con-straints.MathematicaNumericaSinica, 21(4), 475–482,1999.

Abstract. A decompositionmethodfor solving quadraticprogramming(QP) with box constraintsis pre-

sentedin this paper. It is similar to the iterative methodfor solving linear systemof equations.The main

ideasof thealgorithmto split theHessianmatrix Q of theQPprobleminto thesumof two matricesN and

H suchthatQ=N+H and(N-H) is symmetricpositive definitematrix ((N,H) is calleda regular splitting of

Q). A new quadraticprogrammingproblemwith Hessianmatrix N to replacethe original Q is easierto

solve thantheoriginal problemin eachiteration.Theconvergenceof thealgorithmis provedundercertain

assumptions,andthesequencegeneratedby thealgorithmconvergesto optimalsolutionandhasalinearrate

of R-convergenceif thematrixQ is positive definite,or astationarypoint for thegeneralindefinitematrixQ,

andthenumericalresultsarealsogiven.

A. Lucia andJ. Xu. ChemicalprocessoptimizationusingNewton-like methods.ComputersandChemicalEngineering, 14(2), 119–138,1990.

Abstract. Variousinterrelatedissuesthat effect the reliability andefficiency of Newton-like methodsfor

chemicalprocessoptimizationarestudied.An algorithmfor solving large, sparsequadraticprogramming

(QP)problemsthat is basedon anactive setstrategy anda symmetric,indefinitefactorizationis presented.

TheQPalgorithmis fastandreliable.A simpleasymmetrictrust region methodis proposedfor improving

the reliability of successive QP methods.Ill-defined QP subproblemsareavoidedby adjustingthe sizeof

thetrustregion in anautomaticway. Finally, it is shown thatreliableinitial valuesof theunknown variables

andmultiplierscanbegeneratedautomaticallyusinggenericprobleminformation,short-cuttechniquesand

simulationtools.Many relevantnumericalresultsandillustrationsarepresented.

A. Lucia, J. Xu, and G. C. D’Couto. Sparsequadraticprogrammingin chemicalprocessoptimization.Annalsof OperationsResearch, 42(1–4),55–83,1993.

Abstract. The quadraticprogrammingaspectsof a full spacesuccessive quadraticprogramming(SQP)

methodaredescribed.In particular, fill-in, matrix factor andactive set updating,numericalstability, and

indefinitenessof the Hessianmatrix arediscussedin conjunctionwith a sparsemodificationof Bunchand

Parlett factorizationof symmetricindefinite(Kuhn-Tucker) matricesof the typeoften encounteredin opti-

mization.A new pivoting strategy, calledconstrainedpivoting, is proposedto reducefill-in andcompared

with complete,partial and thresholdpivoting. It is shown that constrainedpivoting often significantly re-

ducesfill-in andthusthe iterative computationalburdensassociatedwith the factorizationandsolutionof

Kuhn-Tucker conditionswithin theQPsubproblem.Several chemicalprocessoptimizationproblems,with

smallandlargedegreesof freedom,areusedastestproblems.Theseincludeminimumwork calculationsfor

multistageisothermalcompression,minimumareatargetingfor heatexchangernetworksanddistillationop-

timizationinvolving someazeotropicandextractive distillations.Numericalresultsshow uniformly thatboth

theproposedQPandSQPalgorithms,particularlythefull spaceNewton method,arereliableandefficient.

No failureswereexperiencedateitherlevel.

F. T. Luk andM. Pagano.Quadraticprogrammingwith M-matrices. Linear Algebra and ItsApplications, 33, 15–40,1980.

Abstract. Studiestheproblemof quadraticprogrammingwith M-matrices.Theauthorsdescribeaneffective

algorithmfor thecasewherethevariablesaresubjectto alower-boundconstraint,andananalogousalgorithm

for thecasewherethevariablesaresubjectto lower-and-upper-boundconstraints.Theauthorsdemonstrate

thespecialmonotonebehavior of the iterateandgradientvectors.Theresulton thegradientvectoris new.

It leadsusto considera simpleupdatingprocedurewhich preservesthemonotonicityof bothvectors.The

procedureusesthefactthatanM-matrix hasanonnegative inverse.Two new algorithmsarethenconstructed

by incorporatingthisupdatingprocedureinto thetwo givenalgorithms.Theauthorsgivenumericalexamples

whichshow thatthenew methodscanbemoreefficient thantheoriginalones.

L. Luksan.Dualmethodfor solvingaspecialproblemof quadraticprogrammingasasubprob-lem at linearly constrainednonlinearminimaxapproximation.Kybernetika, 20(6), 445–457,1984.


Abstract. Describesthedualmethodfor solvinga specialproblemof quadraticprogrammingasa subprob-

lem at linearly constrainednonlinearminimax approximation.Completealgorithmof the dual methodis

presentedandits convergenceafterafinite numberof stepsis proved.

L. Luksan. Dual methodfor solving a specialproblemof quadraticprogrammingasa sub-problemto nonlinearminimax approximation. AplikaceMatematiky, 31(5), 379–395,1986.

Abstract. The dual methodfor solving a specialproblemof quadraticprogrammingas a subproblemto

nonlinearminimax approximationis described.Two casesareanalyzedin detail; they differ in the linear

dependenceof thegradientsof theactive functions.Thecompletealgorithmof thedualmethodis presented

andits finite stepconvergenceis proved.

A. D. Lutzenko andA. V. Martynov. Minimax solutionsof problemsin linear andquadraticprogramming.SovietJournalof ComputerandSystemsSciences, 2, 22–27,1968.

Abstract. A methodisgivenfor obtainingtheoptimumsolutionof monimaxproblemsin linearandquadratic

programming.It is shown thatfor minimaxproblemsin linearprogrammingthesaddle-pointconditionis not

satisfied.

C. Y. MaaandM. A. Shanblatt.Linearandquadraticprogrammingneuralnetwork analysis.IEEE Transactionson Neural Networks, 3(4), 580–594,1992.

Abstract. Neuralnetworksfor linearandquadraticprogrammingareanalyzed.Thenetwork proposedby M.

P. KennedyandL. O. Chua(IEEE Trans.Circuits Syst.,vol. 35, pp.554–562,May 1988)is justified from

the viewpoint of optimizationtheoryandthe techniqueis extendedto solve optimizationproblems,such

asthe least-squaresproblem.For quadraticprogramming,the network convergeseitherto an equilibrium

or to anexact solution,dependingon whetherthe problemhasconstraintsor not. The resultsalsosuggest

ananalyticalapproachto solve the linearsystemBx b without calculatingthematrix inverse.Theresults

aredirectly applicableto optimizationproblemswith C2 convex objective functionsandlinear constraints.

Thedynamicsandapplicabilityof thenetworks aredemonstratedby simulation.Thedistancebetweenthe

equilibriaof thenetworksandtheproblemsolutionscanbecontrolledby theappropriatechoiceof anetwork

parameter.

J. H. Maddocks. Restrictedquadratic-forms,inertia theorems,and the Schurcomplement.

LinearAlgebra andIts Applications, 108, 1–36,1988.

K. MadsenandH. Schjær-Jacobsen.Minimax optimizationusingquadraticprogramming.In‘Proceedingsof theIEEEInternationalConferenceonCircuitsandComputersICCC80.IEEE,New York, NY, USA’, pp.1135–1137,1980.

Abstract. A new methodfor solving thenon-linearminimaxproblemis presented.The problemis solved

througha sequenceof quadraticprogramsandno line searchis used.Themethodmaybeusefulin sparse

problemsandanalgorithmfor this caseis briefly outlined.

K. Madsen,H. B. Nielsen,andM. C. Pinar. A finite continuationalgorithmfor boundcon-strainedquadraticprogramming.SIAMJournalon Optimization, 9(1), 62–83,1998.

Abstract. Thedualof thestrictly convex quadraticprogrammingproblemwith unit boundsis posedasa lin-

ear * 1 minimizationproblemwith quadraticterms.A smoothapproximationto thelinear * 1 functionis used

to obtainaparametricfamily of piecewise-quadraticapproximationproblems.Theuniquepathgeneratedby

theminimizersof theseproblemsyieldsthesolutionto theoriginal problemfor finite valuesof theapprox-

imationparameter. Thus,a finite continuationalgorithmis designed.Theresultsof extensive computational

experimentsarereported.

K. Madsen,H. B. Nielsen,andM. C. Pinar. Boundconstrainedquadraticprogrammingviapiecewisequadraticfunctions.MathematicalProgramming, 85(1), 135–156,1999.


Abstract. We considerthestrictly convex quadraticprogrammingproblemwith boundedvariables.A dual

problemis derivedusingLagrangeduality. Thedualproblemis theminimizationof anunconstrained,piece-

wisequadraticfunction.It involvesa lower boundof λ 1 , thesmallesteigenvalueof a symmetric,positive

definitematrix,andis solvedby Newton iterationwith line search.Thepaperdescribesthealgorithmandits

implementationincludingestimationof λ 1 , how to geta goodstartingpoint for theiteration,andup- and

downdatingof Cholesky factorization.Resultsof extensive testingandcomparisonwith othermethodsfor

constrainedQParegiven.

R.A. Maggio.An explorationof theconvex quadratic-programmingblendingmodel.Modeling

andSimulation:RobotsandGeneral Modeling, 18(5), 1617–1621,1987.

M. MahfoufandD. A. Linkens.Constrainedmultivariablegeneralizedpredictivecontrol(GPC)for anaesthesia:the quadratic-programmingapproach(QP). International Journal ofControl, 67(4), 507–527,1997.

Abstract. Thispaperconsiderstheextensionof thestandardGPCalgorithmto includeinputrate,magnitude

andoutputconstraintsusingthequadraticprogramming(QP)approachonaderivednonlinearmultivariable

anaesthesiamodel comprisingsimultaneouscontrol of musclerelaxation(paralysis)andunconsciousness

(in termsof bloodpressuremeasurements).Simulationresults,whicharepresented,analysedanddiscussed,

demonstratethesuperiorityof theextendedversionin thedeterministicandstochasticcasesevenwhenlow

outputpredictionhorizonsarechosen,andalsothegreatflexibility with respectto choosingthelimits on the

manipulatedaswell asthe outputvariables.The studyalsorevealsthat whenheavy externaldisturbances

occur, the algorithm,which combinesinput andoutputconstraints,performsbetterthaneitherthe uncon-

strainedoneor theonethat includesonly input constraints.Underextremeconditions,thesamealgorithm

reducesto an algorithm with only input constraintswhen the phenomenonof constraintsincompatibility

occurs.

G. Maier. A quadraticprogrammingapproachfor certainclassesof nonlinearstructuralprob-

lems.Meccanica, 3, 121–130,1968.

A. Majthay. Optimality conditionsfor quadraticprogramming.MathematicalProgramming,

1(3), 359–365,1971.

A. Majthay, A. B. Whinston,and J. Coffman. Local optimisationfor nonconvex quadraticprogramming.NavalResearch LogisticsQuarterly, 21(3), 465–490,1974.

Abstract. Presentsan algorithmfor determininga local minimum to the following generalquadraticpro-

grammingproblemMinimize q x> pT x 1 2xT Qxsubjectto Ax b x 0,whereQ is ann by n symmetric

matrixA is anmby n matrix p is ann by 1 vectorb is anmby 1 vector. No additionalrestrictionsareplaced

onthematrixQ. And so,theabove problemis theminimizationof apossiblynonconvex quadraticobjective

functionsubjectto asetof linearinequalityconstraints.

K. Malanowski. Onapplicationof aquadraticprogrammingprocedureto optimalcontrolprob-lemsin systemsdescribedby parabolicequations.Control andCybernetics, 1(1–2),43–56,1972.

Abstract. An optimal control problem of a systemdescribedby linear partial differential equationof

parabolictypeis considered.Bothboundaryanddistributedcontrolsareinvestigated.Theperformanceindex

is aquadraticone.It is shown thatthisproblemcanbereducedto somequadraticprogrammingproblemin a

Hilbert space.An iterative procedureof solvingthis problemis proposed.Onedimensionalheatconduction

systemis consideredasanexample.Somenumericalresultsobtainedusinga digital computeraregiven.

K. Malanowski. A quadraticprogrammingmethodin Hilbert spaceandits applicationto opti-mal controlof systemsdescribedby parabolicequations.5th IFIP Conferenceon Opti-mizationTechniques(Abstractsonly received).Univ Rome, Rome, Italy, p. 37,1973.


Abstract. Abstractonly givensubstantiallyasfollows. Theproblemof finding theminimumof a quadratic

functionalJ y onaclosed,boundedandconvex setGammain aHilbert spaceis considered.ThesetGamma

maynot begiven in anexplicit form. An iterative procedureis proposed,basedon successive minimization

of the functionalJ y on somesubsetsΓi of the original setΓ . Threemethodsof constructingthesubsets

Γi areproposed.The useof the above procedurefor solving optimal control problemsfor linear systems

describedby parabolicequationswith quadraticperformanceindex is proposed.Somenumericalexamples

of optimalboundarycontrolof onedimensionalheat-transferequationarepresented.On thebasisof these

examplesthethreemethodsof constructingthesubsetsΓi arecompared.Thespeedof convergenceandthe

simplicity of theobtainedcontrolaretakeninto accountin thiscomparison.

K. Man’chakandY. A. Tomashevski. Solvingquadraticprogrammingproblemsonananaloguecomputer. ArchiwumAutomatykii Telemechanika, 16(2), 185–204,1971.

Abstract. In thepapera questionof solving quadraticprogrammingproblemsis exemplifiedby a specific

taskof waterresourceallocation.Gradientmethodsof quadraticprogrammingareexamined.Theprinciple

of theapplicationof ananaloguecomputerfor solvingsuchproblemsis discussed.Resultsof simulationand

experimentalinvestigationsaregiven.Thelatteraredividedinto two series.Thefirst oneis concernedwith

thestaticalcase,thesecondonewith thedynamicalcase.

D. W. Manchala,A. B. Palazzolo,A. F. Kascak,G. T. Montague,andG. V. Brown. Constrainedquadraticprogramming,activecontrolof rotatingmassimbalance.Journalof SoundandVibration, 205(5), 561–580,1997.

Abstract. Jetenginesmayexperienceseverevibrationdueto thesuddenimbalancecausedby bladefailure.

Thecurrentresearchinvestigatesemploymentof piezoelectricactuatorsto suppressthis usingactive vibra-

tion control.This requiresidentificationof thesourceof thevibrationsvia anexpertsystem,determination

of therequiredphaseanglesandamplitudesfor thecorrectionforces,andapplicationof thedesiredcontrol

signalsto thepiezoelectricactuators.Correctionforcesmayexceedthephysicallimitationsof theactuators;

henceresultsof ”constrainedforce” quadraticprogramming,leastsquaresandmulti-point correctionalgo-

rithmswill becompared.It is demonstratedthatsimply scalingdown theleastsquarespredictedcorrection

forcesto satisfythe actuatorsaturationconstraintsdoesnot necessarilyyield optimal reductionsin vibra-

tion. In thispapertestresultsareshown for suddenimbalance,andthecomputationaltimerequirementsand

balancingeffectivenessfor thevariousapproachesarecompared.

O. L. Mangasarian.Duality in quadraticprogramming.In ‘NonlinearProgramming’,chapter

8-2,pp.123–126.McGraw-Hill, New York,USA,1969.ReprintedasClassicsin Applied

Mathematics10, SIAM, Philadelphia,USA, 1994.

O. L. Mangasarian. Locally uniquesolutionsof quadraticprograms,linear and non-linear

complementarityproblems.MathematicalProgramming, 19(2), 200–212,1980.

O. L. Mangasarian.Sparsity-preservingSOR algorithmsfor separablequadraticand linear

programming.ComputersandOperationsResearch, 11, 105–112,1984.

O. L. MangasarianandH. Stone.Two-personnonzero-sumgamesandquadraticprogramming.

Journalof MathematicalAnalysisandApplications, 9, 348–355,1963.

H. B. Mann. Quadraticformswith linear constraints.TheAmericanMathematicalMonthly,

50, 430–433,1943.

B. D. Manos.Leastsquarefit andquadraticprogramming.Diastase, 2, 3–14,1986. In Greek.

B. D. ManosandG. I. Kitsopanidis. A quadraticprogrammingmodelfor farmplanningof aregion in centralmacedonia.Interfaces, 16(4), 2–12,1986.


Abstract. Quadraticprogrammingmodelsareusedin farm planningbecauserisk anduncertaintyarein-

volvedin thetechnicalandeconomiccoefficientsusedandthequantitiesandpricesof resources.A special

quadraticprogrammingmodel(the E-V model)wasusedto plan a Greekfarm region, the formerLake of

Giannitsa.The resultingplan is preferredby farmersto thoseresultingfrom the linear andmixed-integer

programmingmodelsandto thepreviously usedplanbecauseit includescropsexpectedto give thehighest

minimumtotalgrossmargin with thesametotalfixedcosts.Thefarmerswantplansthatachieve notonly the

highestbut alsothemoststableeconomicresults.

M. A. Marino andH. A. Loaiciga. Multireservoir operationplanningvia quadraticprogram-

ming. In ‘Scientific Basisfor WaterResourcesManagement’,Vol. 153, pp. 231–239,

1985.

H. M. Markowitz. The optimizationof a quadraticfunction subjectto constraints. Naval

Research LogisticsQuarterly, 3, 111–133,1956.

I. MarosandC.Meszaros.A repositoryof convex quadraticprogrammingproblems.Optimiza-tion MethodsandSoftware, 11-12, 671–681,1999.

Abstract. The introduction of a standard set of linear programming problems, to be found in

NETLIB/LP/DATA, hadanimportantimpactonmeasuring,comparingandreportingtheperformanceof LP

solversUntil recentlytheefficiency of new algorithmicdevelopmentshasbeenmeasuredusingthis impor-

tantreferencesetPresently, wearewitnessinganevergrowing interestin theareaof quadraticprogramming

Theresearchcommunityis somewhat troubledby the lack of a standardformat for defininga QPproblem

andalsoby the lack of a standardreferencesetof problemsfor purposessimilar to thatof LP In thepaper

weproposeastandardformatandannouncetheavailability of a testsetof 138collectedQPproblems.

J.E. Marowitz andK. A. Fegley. Systemdesignusingquadraticprogramming.IEEE Transac-tionson AutomaticControl, 16(3), 241–247,1971.

Abstract. Quadraticprogrammingis appliedto theconstraineddesignandcompensationof stochasticand

deterministicsystems.The developedconstraintscontrol systemstability, plant compensatorrealizability,

systemrise time, andbandwidth.The techniqueseemsattractive to caseswherethe designinformationis

givenonly in numericalform.

A. D. Martin. Mathematicalprogrammingof portfolio selections. ManagementScience,1(2), 152–166,1955.

Abstract. PresentsMarkowitz modelasa mathematicalprogramanda solutionprocedurefor thequadratic

program

B. Martos. Quadraticprogrammingwith a quasiconvex objective function. OperationsRe-search, 19(1), 87–97,1971.

Abstract. Givesbothnecessaryandsufficient conditionsfor a quadraticfunction to bequasiconvex in the

nonnegative orthant.Methodsof pseudoconvex programming(suchasthoseof FrankandWolfe) cansolve

linearlyconstrainedquadraticprogrammingproblemswith suchanobjective function.

Y. K. L. MatitskasandG.S.Palubetskis.Graphcuttingand0–1quadraticprogramming.SovietJournalof ComputerandSystemsSciences, 25(1), 100–107,1987.

Abstract. A problemof 0–1 programmingwith a quadraticobjective function andtwo constraintsin the

form of inequalities,whichcanbeinterpretedasaproblemof cuttingagraphinto two parts,is examined.An

effective algorithmfor cuttinga treeandabranch-and-boundalgorithmfor anarbitrarygrapharepresented.

Theresultsof a computationalexperimentaregiven.The feasibility of usinga reduciblelower estimateof

theoptimumfor thepurposeof testingheuristicalgorithmsis indicated.


R. McBride andJ. Yormark. Finding all solutionsfor a classof parametricquadraticinteger

programmingproblems.ManagementScience, 26, 784–795,1980.

B. A. McCarl andT. Tice. Shouldquadratic-programmingproblemsbeapproximated.Ameri-

canJournalof Agricultural Economics, 64(3), 585–589,1982.

B. A. McCarl, H. Moskowitz, andH. Furtan. Quadraticprogrammingapplications.Omega,5(1), 43–55,1977.

Abstract. This paperreviews andextendssomeof the methodologicalareaswhereQP is applicable,dis-

cussingandillustratingthecharacteristicsandaspectsof theaccompanying solutions.Someof themethod-

ologicalareasamenableincluderegressionanalysis,decisionanalysis,andquadraticapproximationsto gen-

erally complex functions.In the functionalmanagementareas,QP is applicableto problemsin economics

suchasdemand-supplyresponseandenterpriseselection.In finance,it is usedin portfolio analysis;in agri-

culture,in cropselection.

M. P. Mckenna,J. P. Mesirov, and S. A. Zenios. Data-parallelquadraticprogrammingonbox-constrainedproblems.SIAMJournalon Optimization, 5(3), 570–589,1995.

Abstract. We develop designsfor thedataparallelsolutionof quadraticprogrammingproblemssubjectto

box constraints.In particular, we considertheclassof algorithmsthat iteratebetweenprojectionstepsthat

identify candidateactive setsandconjugategradientstepsthat explore the working space.Using the al-

gorithm of More andToraldo[ReportMCS-p77–0589, ArgonneNationalLaboratory, Illinois, 1989] asa

specificinstanceof thisclassof algorithmsweshow how its componentscanbeimplementedefficiently ona

data-parallelSIMD computerarchitecture.Alternative designsaredevelopedfor botharbitrary, unstructured

Hessianmatricesandfor structuredproblems.Implementationsarecarriedout on a ConnectionMachine

CM-2. They areshown to bevery efficient, achieving a peakcomputingrateover 2 Chops.Problemswith

severalhundredthousandvariablesaresolvedwithin oneminuteof solutiontimeonthe8K CM-2.Extremely

largetestproblems,with up to 2.89million variables,arealsosolvedefficiently. Thedataparallelimplemen-

tationoutperformsa benchmarkimplementationof interior point algorithmson an IBM 3090–6009vector

supercomputerandasuccessive overrelaxationalgorithmonanIntel iPSC/860hypercube.

P. Mcleanmeyinsse.Interregionalflowsof corn—aquadratic-programmingapproach.In ‘Mod-

eling andSimulation’,Vol. 19,pp.419–424,1988.

W. G. Medlin and J. F. Kaiser. Bandpassdigital differentiatordesignusing quadraticpro-gramming.ICASSP91.1991InternationalConferenceon Acoustics,Speech andSignalProcessingIEEE,New York, NY, USA, 3, 1977–1980,1991.

Abstract. A novel techniqueutilizing quadraticprogrammingfor thedesignof bandpassFIR (finite impulse

response)digital differentiatorsof arbitraryorderis presented.Thenew differentiatorshave linearphaseand

aremaximally accurateat the centerof the differentiationband.Their designis basedon a minimization

procedurefor the integratedsquareerror of the frequency responseover designatedapproximationbands.

Theclosed-formsolutionfor the filter coefficients is obtainedby the methodof Lagrangemultipliers.The

inclusionof stopbandin the designprocessis alsodiscussed.This techniquehasbeensuccessfullyused

by the authorsfor thedesignof optimal low passdifferentiators.The new differentiatorsareimportantfor

applicationswherethefirst-, second-,or higher-orderderivative of a digital signalis requiredto beaccurate

atmidrangefrequencies.

K. Meer. On thecomplexity of quadraticprogrammingin realnumbermodelsof computation.TheoreticalComputerScience, 133(1), 85–94,1994.

Abstract. The complexity of linearly constrained(nonconvex) quadraticprogrammingis analyzedwithin

the framework of real numbermodels,namelythe oneof L. Blum, M. Shub,andS. Smale(1989)andits

modificationrecently introducedby Koiran (1993) (”weak BSS-model”).In particularwe show that this

problemis not NP-completein theKoiransetting.Applicationsto the(full) BSS-modelarediscussed.


N. MegiddoandA. Tamir. Linear time algorithmsfor someseparablequadraticprogrammingproblems.OperationsResearch Letters, 13(4), 203–211,1993.

Abstract. A large classof separablequadraticprogrammingproblemsis presented.The problemsin the

classcanbesolvedin lineartime.Theclassincludestheseparableconvex quadratictransportationproblem

with afixednumberof sourcesandseparableconvex quadraticprogrammingwith nonnegativity constraints

andafixednumberof linearequalityconstraints.

S.MehrotraandJ.Sun.An algorithmfor convex quadraticprogrammingthatrequiresOn3 ? 5L

arithmeticoperations.Mathematicsof OperationsResearch, 15(2), 342–363,1990.

Abstract. A new interior point methodfor minimizing a convex quadraticfunction over a polytopeis de-

veloped.The authorsshow that the methodrequiresO n3 ( 5L arithmeticoperations.In the algorithmthey

constructa sequencePz0 Pz1 . . . Pzk , . . . of nestedconvex setsthatshrink towardsthesetof optimalsolu-

tion(s).During iterationk they take a partial Newton stepto move from anapproximateanalyticcenterof

Pzk @ 1 to an approximateanalyticcenterof Pzk . A systemof linear equationsis solved at eachiteration to

find the stepdirection.The solutionthat is availableafter O mL iterationscanbe convertedto an opti-

mal solution.Theanalysisindicatesthat inexactsolutionsto thelinearsystemof equationscouldbeusedin

implementingthis algorithm.

A. MelmanandR. Polyak. TheNewton modifiedbarriermethodfor QPproblems.AnnalsofOperationsResearch, 62, 465–519,1996.

Abstract. Themodifiedbarrierfunctions(MBF) haveelementsof bothclassicalLagrangians(CL) andclas-

sical barrierfunctions(CBF). The MBF methodsfind anunconstrainedminimizer of somesmoothbarrier

functionin primalspaceandthenupdatetheLagrangemultipliers,while thebarrierparametereitherremains

fixedor canbeupdatedateachstep.Thenumericalrealizationof theMBF methodleadsto theNewtonMBF

method,wheretheprimalminimizeris foundby usingNewton‘smethod.Thisminimizeris thenusedto up-

datetheLagrangemultipliers.In this paper, theauthorsexaminetheNewton MBF methodfor thequadratic

programming(QP)problem.It is shown thatunderstandardsecond-orderoptimality conditions,thereis a

ball aroundtheprimalsolutionandacut conein thedualspacesuchthatfor asetof Lagrangemultipliersin

this cut cone,themethodconvergesquadraticallyto theprimal minimizer from any point in theaforemen-

tionedball, andcontinuesto do soaftereachLagrangemultiplier update.TheLagrangemultipliers remain

within thecut coneandconvergelinearly to their optimalvalues.Any point in this ball will becalleda ”hot

start”. Startingat such”hot start”, at mostO loglogε ' 1 Newton stepsaresufficient to performtheprimal

minimizationwhich is necessaryfor theLagrangemultiplier update.Here,ε 3 0 is thedesiredaccuracy. Be-

causeof the linearconvergenceof theLagrangemultipliers, this meansthatonly O logε ' 1 O loglogε ' 1 Newtonstepsarerequiredto reachanε-approximationto thesolutionfrom any ”hot start”. In orderto reach

the”hot start”,onehasto performO mlogC Newtonsteps,wherem characterizesthesizeof theproblem

andC 3 0 is the conditionnumberof the QP problem.This conditionnumberis characterizedexplicitly

in termsof key parametersof the QP problem,which in turn dependon the input dataandthe sizeof the

problem.

C. Y. Meng,S.Frimpong,andM. J.Zuo. A quadraticprogrammingmodelfor blastscheduling.Journalof Universityof ScienceandTechnologyBeijing, 6(3), 165–167,1999.

Abstract. A quadraticprogrammingmodelis establishedto choosetheblocksto beblastedin a givenpe-

riod. The lengthof this perioddependson theproductionplanningrequirements.During the given period,

theblocks’ parametersareavailablefrom thegeologicaldatabaseof themine.Theobjective is to minimize

the deviation of the averageore gradeof blastedblocksfrom the standardore graderequiredby the mill.

Transportationability constraint,productionquantitydemandconstraint,minimumsafetybenchconstraint,

block sizeconstraintandblock, benchprecedenceconstraintsareconsideredin forming the programming

model.This modelhasmorepracticalobjective functionandreasonableconstraintscomparedwith theex-

istingmodelfor thiskind of problems.

H. M. Merrill. Failurediagnosisusingquadraticprogramming.IEEE Transactionson Relia-bility, R-22(4), 207–213,1973.


Abstract. This paperdiscussestheproblemof determiningwhich of a largesetof possiblebut improbable

malfunctionsgave riseto agivensetof measurements.Theclassesof systemsunderconsiderationgenerally

leadto underdeterminedsetsof equations.Threemethodsof formulatingandsolvingthis classof problems

arepresented:1) thepseudoinversemethod:this leadsto aneasily-solved computationalproblembut it is

not physicallyrealisticandit tendsto give poor results;2) a patternrecognitionapproachbasedon a more

realisticproblemformulation: unfortunately, the computationalproblemsassociatedwith this formulation

maybeformidable;and3) aquadraticprogrammingapproach:this is basedonminimizationof aphysically

realisticobjective function.

P. MerzandB. Freisleben.Geneticalgorithmsfor binaryquadraticprogramming.In ‘GECCO-99.Proceedingsof theGeneticandEvolutionaryComputationConference.JointMeetingof theEighthInternationalConferenceonGeneticAlgorithms(ICGA-99)andtheFourthAnnualGeneticProgrammingConference(GP-99).MorganKaufmannPublishers,SanFrancisco,CA, USA’, Vol. 1, pp.417–424,1999.

Abstract. Geneticalgorithmsfor theunconstrainedbinaryquadraticprogrammingproblem(BQP)arepre-

sented.It is shown that for smallproblems,a simplegeneticalgorithmwith uniform crossover is sufficient

to find optimumor best-known solutionsin ashorttime,while for problemswith ahighnumberof variables

(n¿or=200),it is essentialto incorporatelocal searchto arrive at high quality solutions.A hybrid genetic

algorithmincorporatinglocal searchis testedon 40 probleminstancesof sizescontainingbetweenn=200

andn=2500.The resultsof the computerexperimentsshow that the approachis comparableto alternative

heuristicssuchastabu searchfor small instancesandsuperiorto tabu searchandsimulatedannealingfor

largeinstances.New bestsolutionscouldbefoundfor 14 largeprobleminstances.

C. Meszaros. On the sparsityissuesof interior point methodsfor quadraticprogramming.TechnicalReportWP 98-4, Laboratoryof OperationsResearchandDecisionSystems,HungarianAcademyof Sciences,1998a.

Abstract. In this paperwe will investigatehow thesparsityof non-separablequadraticprogrammingprob-

lemsbehavesin interior point methods.We will show that for thenormalequationapproach,two orderings

canbeperformedin independentstepsto reducethefill-in duringinteriorpoint iterations.Oneof thepermu-

tationshasto beperformedonthecolumns,whereastheotheron therowsof theproblem.Weshow thatone

caneasilyattribute the sparsityissuesof non-separablequadraticprogrammingproblemsto that of linear

programmingfor which well–developedtechniquesareavailable.We describehow the fundamentalstruc-

tural propertiesof non–separablequadraticprogrammingproblemscanbe representedby a singlematrix

whosesparsitypatterncanserve to determinetherow permutationandto useheuristicsdevelopedfor linear

programmingfor determiningwhich of the augmentedsystemandnormalequationapproachis moread-

vantageous.Numericalresultsaregivenon a wide varietyof non–separableconvex quadraticprogramming

problems.

C. Meszaros.Theseparableandnon-separableformulationsof convex quadraticproblemsin

interior point methods.TechnicalReportWP 98-3,Laboratoryof OperationsResearch

andDecisionSystems,HungarianAcademyof Sciences,1998b.

C. Meszaros.Steplengthsin interior-point algorithmsof quadraticprogramming.OperationsResearch Letters, 25(1), 39–45,1999.

Abstract. An approachto determineprimal anddual stepsizesin the infeasible-interior-point primal-dual

methodfor convex quadraticproblemsis presented.Theapproachreducestheprimalanddualinfeasibilities

in eachstepand allows different stepsizes.The methodis derived by investigatingthe efficient set of a

multiobjective optimizationproblem.Computationalresultsarealsogiven.

M. C. Meyer. An algorithmfor projectionsonto convex coneswith applicationsto nonpara-

metric regressionandquadraticprogramming.TechnicalreportSta99-12,Department

of Statistics,Universityof Georgia,Athens,Georgia,USA, 1999.


M. Minoux. A polynomialalgorithmfor minimum quadraticcostflow problems. European

Journalof OperationsResearch, 18, 377–387,1984.

B. K. MishraandC. Das.Maximin problemandaduality theoremfor mixed-integerquadratic

programming.Zeitschrift fur AngewandteMathematikund Mechanik, 65(7), 310–312,

1985.

G. D. Mistriotis. Quadraticprogramming:anefficientalternativeto micro-simulation.Bulletinof theOperationsResearch Societyof America, 20, B328,1972.

Abstract. This papersuggestsan alternative to micro-simulationasa meansof updatinga given sample

representingsomebackgroundpopulation.Theproblemis translatedto a quadraticprogrammingproblem

with linear equality constraintsby the adjustmentof the individual recordweightsof the sample.In this

way thesamesamplewith thenew revisedrecordweightswould adequatelyrepresenttheagedpopulation.

The philosophyof the problemformulationandits advantagesarediscussed.Even for large lengthrecord

problemstheenormouslyrelative low costandthelow corememoryrequirementsareillustrated.Finally, the

comparative limitationsareoutlined.

A. A. Mitsel andA. N. Khvashchevsky. New algorithmfor solvingthequadraticprogrammingproblem.Optoelectronics,InstrumentationandData ProcessingAbstracts, 3, 1999.

Abstract. An algorithmfor solving the quadraticprogrammingproblemis considered.It differs from the

known algorithmsby theabsenceof artificial variables.In someparticularcasesthealgorithmis reducedto

solvingasystemof linearalgebraicequations.

E. Mitsopoulou. Unilateralcontact,dynamicanalysisof beamsby a time-stepping,quadraticprogrammingprocedure.Meccanica, 18(4), 254–265,1983.

Abstract. Discussesa methodfor thedynamicanalysisof aslenderbeam,undergoing’small’ deformations

andcontrastedwithout friction by a supportingprofile. Therelevant contact-impact(unilateralconstrained)

problemis studied,with referenceto adiscretemodelin spaceandtime,after its formulationasa quadratic

programmingproblem(QPP)with signconstraintsonly. Thecontact-impactproblemfor a rigid supporting

profileis first solved,andsubsequentlythecontactproblemfor anelasticprofile.Amongavailablealgorithms

for the numericalsolution,the implicit unconditionallystablealgorithmof Zienkiewicz, WoodandTaylor

(1980)provedto bethemostefficient for thetimediscretization,while amodificationof Hildreth-d’Esopo’s

algorithmwith theintroductionof avariableoverrelaxationfactorwasusedfor thesolutionof theQPP.

E. Mitsopoulou and I. Doudoumis. Unilateral contactof elastic bodiesby finite-element

methodandquadratic-programmingtechnique.Zeitschrift fur AngewandteMathematik

undMechanik, 66(5), T421–T423,1986.

S. Mizuno andK. G. Murty. Computationof theexactsolutionfrom a nearoptimumsolutionfor convex QP. TechnicalReport10/92,Departmentof OR andIE, Cornell,New York,USA, 1992.

Abstract. Considera convex quadraticprogramof sizeL involving m inequalityconstraintsandpossibly

someequality constraintsin n variables.Given a feasiblesolution whoseobjective value is within 2' 2L

of the optimum objective value, we discussa procedurethat finds a true optimum solution in at mostm

iterations,whereeachiterationinvolvesminimizing thequadraticobjective functiononanaffine space(this

takesat mostn conjugategradientsteps).

Y. S.Mo, W. S.Lu, andA. Antoniou.An iterativequadraticprogrammingmethodfor multiratefilter design.In ‘ISCAS ’98. Proceedingsof the1998IEEE InternationalSymposiumonCircuitsandSystems.IEEE,New York, NY, USA’, Vol. 5, pp.41–44,1998.


Abstract. An iterative quadraticprogrammingmethodfor multiratefilter designis described.The design

problemis formulatedas a 4th-ordernonlinearoptimizationproblemin which the objective function is

a weightedsum of a reconstructionterm andan aliasing-errorterm. Constraintson the filter’s frequency

responsein thepassbandandstopbandareimposedasa setof linearinequalities.Theoptimizationproblem

issolvedby iteratively minimizingaquadraticfunctionsubjecttoasetof linearconstraints.Explicit formulas

for evaluatingtheminimumsof thesequadraticfunctionsaredescribed,which leadto anefficient andfast

algorithm.An exampleis includedto illustratethedesignmethod.

N. Moal andJ. J. Fuchs. Sinusoidsin white noise: a quadraticprogrammingapproach. In‘Proceedingsof the1998IEEEInternationalConferenceonAcoustics,SpeechandSignalProcessing,ICASSP’98. IEEE,New York, NY, USA’, Vol. 4, pp.2221–2224,1998.

Abstract. We addressthe problemof the estimationandidentificationof real sinusoidsin white Gaussian

noiseusinga correlation-basedmethod.We estimatea partial covariancesequencefrom the dataandseek

a representationof thesenew observationsasa super-positionof a small numberof cosineschosenfrom

a redundantbasisandthe white noisecontribution. We proposeto minimize a quadraticprogramin order

to choosea parsimoniousdecompositionamongthe many that allow the reconstruction.We develop op-

timality conditionsfor the criterion that canbe geometricallyinterpretedandpresenta dual criterion that

hasanappealingphysicalinterpretation.Somesimulatedexamplesarealsopresentedto show theexcellent

performancein resolutionof theapproach.

I. B. Mohd andY. Dasril. Constraintexplorationmethodfor quadraticprogrammingproblem.AppliedMathematicsandComputation, 112(2–3),161–170,2000.

Abstract. In this paper, we representa methodwhich is basedon the violatedconstraintsby the uncon-

strainedminimum of the objective function of the quadraticprogrammingproblemfor exploring, locating

andcomputingtheoptimalsolutionof theproblemwithout usingadditionalinformationashave beendone

in mostof thefavouriteestablishedmethods.

A. Mohri. A computationalmethodfor optimal control of a linear systemby quadraticpro-gramming.InternationalJournalof Control, 11(6), 1021–1039,1970.

Abstract. The numericalmethodderived is applicableto the problem,wherethe control u is considered

to be held at a constantvalueduring the constanttime interval, and the criterion function is given by the

following integral type:J # x; Qx u; Ru dt. But themethodintroducedin thispaperrequiresnonumerical

integrationexceptto obtainthetransitionmatrix.Thereforethecomputationaleffort is considerablysaved.

J.A. Momoh,R. E. L. Adapa,andM. E. Hawary. A review of selectedoptimalpowerflow lit-eratureto 1993.I. Nonlinearandquadraticprogrammingapproaches.IEEETransactionson PowerSystems, 14(1), 96–104,1999.

Abstract. Thepaperpresentsa review of literatureon optimalpower flow tracingprogressin this areaover

from 1962–93.Part 1 dealswith theapplicationof nonlinearandquadraticprogramming.

R.D. C.Monteiro,I. Adler, andM. G.C.Resende.A polynomial-timeprimal-dualaffinescalingalgorithmfor linear andconvex quadraticprogrammingandits power seriesextension.Mathematicsof OperationsResearch, 15(2), 191–214,1990.

Abstract. Describesan algorithmfor linear andconvex quadraticprogrammingproblemsthat usespower

seriesapproximationof the weightedbarrier path that passesthroughthe current iteratein order to find

thenext iterate.If r 1 is theorderof approximationused,theauthorsshow that their algorithmhastime

complexity O n1A 2 B 1C 1r ! L 1 9 1r ! iterationsandO n3 n2r arithmeticoperationsper iteration,wheren is

thedimensionof theproblemandL is thesizeof the input data.Whenr 1, they show that thealgorithm

canbeinterpretedasanaffine scalingalgorithmin theprimal-dualsetup.

R. D. C. Monteiro and I. Adler. Interior path following primal-dualalgorithms.2. Convex

quadraticprogramming.MathematicalProgramming, 44(1), 43–66,1989.


R. D. C. Monteiro andT. Tsuchiya. Global convergenceof the affine scalingalgorithmforconvex quadraticprogramming.SIAMJournalon Optimization, 8(1), 26–58,1998.

Summary. A global convergenceproof of the second-orderaffine scalingalgorithmfor convex quadratic

programmingproblemsis given,wherethenew iterateis thepoint thatminimizestheobjective functionover

the intersectionof the feasibleregion with theellipsoidcenteredat thecurrentpoint andwhoseradiusis a

fixedfractionβ D 0 1E= of theradiusof the largest“scaled”ellipsoid inscribedin thenonnegative orthant.

Theanalysisis basedon the local Karmarkarpotentialfunction introducedby Tsuchiya.For any β D 0 1andwithout makingany nondegeneracy assumptionon the problem,the sequencesof primal iteratesand

dualestimatesconverge to optimalsolutionsof thequadraticprogramandits dual.

J.H. MooreandA. Whinston.Experimentalmethodsin quadraticprogramming.Management

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B. MooresandA. J. Murphy. An integer quadratic-programmingformulationandalgorithm

for the organizationof nursestaffing schedules.In ‘Third InternationalConferenceon

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J.L. Morales-PerezandR. W. H. Sargent.Computationalexperiencewith severalmethodsfor

largesparseconvex quadraticprogramming.AportacionesMatematicas,SerieComuni-

caciones, 14, 141–158,1994.

J.J.MoreandG.Toraldo.Algorithmsfor boundconstrainedquadraticprogrammingproblems.NumerischeMathematik, 55(4), 377–400,1989.

Abstract. Presentsanalgorithmwhich combinesstandardactive setstrategieswith thegradientprojection

methodfor thesolutionof quadraticprogrammingproblemssubjectto bounds.Theauthorsshow, in particu-

lar, thatif thequadraticis boundedbelow onthefeasiblesetthenterminationoccursatastationarypoint in a

finite numberof iterations.Moreover, if all stationarypointsarenondegenerate,terminationoccursata local

minimizer. A numericalcomparisonof thealgorithmbasedon thegradientprojectionalgorithmwith astan-

dardactive setstrategy shows thatonmildly degenerateproblemsthegradientprojectionalgorithmrequires

considerablelessiterationsandtime thantheactive setstrategy. On nondegenerateproblemsthenumberof

iterationstypically decreasesby at leasta factorof 10. For stronglydegenerateproblems,theperformance

of thegradientprojectionalgorithmdeteriorates,but it still performsbetterthantheactive setmethod.

J. J. More andG. Toraldo. On the solutionof large quadraticprogrammingproblemswith

boundconstraints.SIAMJournalonOptimization, 1(1), 93–113,1991.

J.J.Moreau.Quadraticprogrammingin mechanics:Dynamicsof one-sidedconstraints.SIAM

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H. Mori and S. Tsuzuki. A robust QP-basedalgorithm for power systemstateestimation.In C. E. Lin, ed., ‘IASTED InternationalConference.High Technologyin the PowerIndustry. PowerHigh Tech’91. ACTA Press,Calgary, Alta , Canada’,pp.130–134,1991.

Abstract. The authorspresenta new robust algorithm for power systemstateestimation.The proposed

methodis basedon the fact that the power flow equationcan be expressedas the exact quadraticform

of nodevoltagesin rectangularcoordinates.As a result, the stateestimationproblemis formulatedas a

quadraticprogrammingproblem.The advantageof the proposedmethodis that the obtainedestimatesare


very robustto grosserrorsof measurementsor baddatadueto theexactobservationequationin a quadratic

form. Themethodhasbeensuccessfullyappliedto asamplesystemandits efficiency is demonstrated.

M. Morse. Subordinatequadraticformsandtheir complementaryforms. RevueRoumainede

Math Pureset Appliques, 16, 558–569,1971.

A. M. Morshedi,C.R.Cutler, andT. A. Skrovanek.Optimalsolutionof dynamicmatrixcontrol

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T. S. Motzkin. Quadraticformspositive for nonnegativevariablesnot all zero. Noticesof the

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Abstract. A modifiedBeale’s algorithmis describedwhich computesthe local minimizerof any quadratic

objective function subjectto linear constraints.Someextensionsare given, first of all the possibility of

movementto the neighbouringlocal minimizer with a reducedobjective function value in somespecial

cases.

J. A. Muckstadt. A dual decompositionalgorithm for solving mixed integer-continuousquadraticprogrammingproblems. Technical report, Air Force SystemsCommand,Wright-PattersonAFB, OH, USA, 1969.

Abstract. The report containsa presentationof an algorithm for solving the following mixed integer-

continuousmathematicalprogrammingproblem:mincTg cT x 1 2xT Dx: Ax By. b, y is anelement

of S, x O) whereg c x y, andb arecolumnvectorsof appropriatedimension;A B, andD arematrices

of appropriatedimension;andS is a closed,boundedset whoseelementshave integer components.The

quadraticform is assumedto bepositive semi-definite.Thealgorithmis a generalizationof onedeveloped

by J. F. Bendersfor solving the above problemfor thespecialcasewhereD O. The primarymotivation

for developingthis algorithmwasa desireto constructa methodfor solving both an aircraft maintenance

schedulingproblemandtheunit commitment-economicdispatchschedulingproblemencounteredin large

scalepower systems.

W. Murray. An algorithm for finding a local minimum of an indefinite quadraticprogram.TechnicalReportNAC 1, NationalPhysicalLaboratory, London,England,1971.

Abstract. An algorithmis describedthatwill find eithertheglobalminimumof a strictly convex quadratic

programor a local minimum of an indefinite quadraticprogram.The algorithm is a generalizationof a

recentlypublishedalgorithmfor linear programming.A featureof the new algorithm is that numerically

stablefactorsof amatrixarerecurredasopposedto its inverseor unstablefactors.

K. G. Murty. On thenumberof solutionsto thecomplementaryquadraticprogrammingprob-lem. Technicalreport, California Univ., OperationsResCenter, Berkeley, CA, USA,1968.

Abstract. Therelationshipbetweenthenumberof solutionsto thecomplementaryproblem,theright-hand

constantvectorq andthe matrix M is explored.Most of the proofsarebasedon mathematicalinduction.

Counterexamplesaregivento show thatthetheoremsfail if any of thehypothesesarenotsatisfied.

K. G. Murty andS. N. Kabadi. SomeNP-completeproblemsin quadraticandnonlinearpro-

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L. D. Muu and W. Oettli. An algorithm for indefinite quadraticprogrammingwith convexconstraints.OperationsResearch Letters, 10(6), 323–327,1991.

Abstract. Proposesabranch-and-boundmethodfor minimizing anindefinitequadraticfunctionover acon-

vex set.Theboundingoperationis basedonacertainrelaxationof theconstraints.Thebranchingis asimplex

bisection.

W. C. Mylander. Nonconvex quadraticprogrammingby a modificationof Lemke’s method.

TechnicalPaperRAC-TP-414,ResearchAnalysisCorporation,Mclean,VA, USA, 1971.

W. C. Mylander. Finite algorithmsfor solving quasiconvex quadraticprograms.Operations

Research, 20, 167–173,1972.

W. C. Mylander. Processingnonconvex quadratic programmingproblems. PhDthesis,Depart-

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N. NabonaandL. L. Freris.Optimumallocationof spinningreserveby quadraticprogramming.Proceedingsof theInstitutionof Electrical Engineers, 122(11),1241–1246,1975.

Abstract. In power systemoptimumdispatchproblems,thespinning-reserve characteristicsof the individ-

ual generatorscannotbeeffectively linearisedthroughthesensitivity coefficients.To solve this problem,a

techniqueknown asquadratic-approximationprogrammingis used.Limits on thetransmission-linecurrent

arealso incorporatedin the algorithm. It is shown that the minimum-costdispatchproblem,subjectto a

specifiedspinningreserve andthemaximum-spinning-reserve problem,canbeeasilysolvedwith moderate

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A. Nagurney. An algorithmfor thesolutionof a quadratic-programmingproblem,with appli-

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Y. NakamuraandS.Yamashiro.Probabilisticoperationof electricpower systemsconsideringenvironmentalconstraint.X. Comparisonwith quadraticprogrammingmethod.Memoirsof theKitami Instituteof Technology, 18(2), 173–179,1987.

Abstract. For pt.IX seeibid., volume18, number1 (1987).Theauthorsdescribea fastschedulingmethod

to considerevery line capacityandNOx emissionconstraintwheneachline andthermalunit fail proba-

bilistically. Themethodis anapproximateonebecauseonly two thermalunitscontrol theoutputpower to

satisfythe linescapacityandonly a healthystatecontrolstheemissionexceptingfaulty states.Theuseful-

nessof the methodis confirmedby comparingit with a strict method.The strict scheduleis estimatedby

thequadraticprogrammingmethod.Thereportdescribestheoutlinesof both theproposedmethodandthe

quadraticprogrammingmethod.Both methodsareappliedto amodelsystemandtheresultsareshown.

J. Nanda,D. P. Kothari, andS. C. Srivastava. New optimal power-dispatchalgorithmusingFletcher’s quadraticprogrammingmethod. IEE ProceedingsC (Generation, Transmis-sionandDistribution), 136(3), 153–161,1989.

Abstract. A freshattempthasbeenmadeto develop a new algorithmfor optimalpower flow (OPF)using

Fletcher’s quadratic-programmingmethod(1971).Thenew algorithmconsiderstwo decoupledsubproblems

needingminimum cost of generationand minimum system-transmissionlosses.Thesehave beensolved

sequentiallyto achieve optimalallocationof realandreactive power generationandtransformertapsettings

with considerationof system-operatingconstraintson generation,busbarvoltageandline-flow limits. The

potentialof the new algorithmfor OPFhasbeendemonstratedthroughsystemstudiesfor two IEEE test

systemsandanIndiansystem.Resultsrevealthattheproposednew algorithmhaspotentialfor onlinesolving

of OPFproblems.


D. L. NashandG. W. Rogers.Risk managementin herdsireportfolio selection:a comparison

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Abstract. Thispaperis concernedwith solvingquadraticprogrammingproblemswith varioustypesof linear

constraints.The typesconsideredarequite generalandinclude(1) equalityconstraints,(2) nonnegativity

constraints,(3) inequalityconstraints,and(4) any combinationof the first threetypes.In eachcase,it is

assumedthatthematrix of thequadraticobjective functionis at leastpositive semidefinite.Theexpressions

for the solutionvectorsaregiven in termsof the matricesof the objective function andconstraints,or of

submatricesof these.Extensive useis madeof thepropertiesof thepseudoinverse(thegeneralizedinverse

of Penrose).

Y. Nesterov. Polynomialmethodsin the linearandquadraticprogramming.Soviet Journal ofComputerandSystemsSciences, 26(5), 98–101,1988.

Abstract. New polynomialalgorithmsfor solving problemsof the linear andquadraticprogrammingare

proposed.Thedependenceof theestimateof thenumberof arithmeticoperationsuponthedimensionof the

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Y. Nesterov. Semidefiniterelaxationand nonconvex quadraticoptimization. Optimization

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Y. Nesterov. Global quadraticoptimization on the sets with simplex structure. CORE

DiscussionPaper9915,Centerfor OperationsResearchandEconometrics,Universite

catholiquedeLouvain,Louvain-la-Neuve,Belgium,1999.

Y. Nesterov andA. Nemirovsky. Accelerationandparallelizationof thepath-following method

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564,1991.

B. P. Ng, M. H. Er, andC.Kot. A flexible arraysynthesismethodusingquadraticprogramming.IEEE Transactionson AntennasandPropagation, 41(11),1541–1550,1993.

Abstract. A highly flexible synthesismethodfor anarbitraryarrayis proposedto bestapproximateadesired

arraypatternin a minimum-mean-square-errorsense.Thebasicideaof thetechniqueis to form a quadratic

programwith its cost function given by the mean-squareerror betweenthe arrayresponseanda properly

selectedpatterndescribedby a known mathematicalfunction.This quadraticprogramcanbea constrained

or unconstrainedoptimizationproblemdependingon the requirementsof the desiredarraypattern.In for-

mulatingthequadraticprogram,noassumptionhasbeenmadeon thegain/phaseresponseor characteristics

of theindividual arrayelements.Therefore,onecansynthesizeanarrayof arbitraryshapeto any appropriate

patternwith thecharacteristicof thearrayelementstaken into considerationaslong asoneis ableto model

the arrayaccurately. The proposedmethodis usedto synthesizearraysof different shapes,linear aswell

asplanararrays(including rectangularandcircular planararrays),usinga Chebyshev polynomialor zero

functionasa designtemplate,to illustratethefeasibilityof theproposedmethod.

H. NicholsonandM. J. H. Sterling. Optimumdispatchof active andreactive generationbyquadraticprogramming. IEEE Transactionson Power Apparatus and Systems, PAS-92(2), 644–654,1973.

Abstract. Optimumschedulingof generationsubjectto constraintson maximumandminimum levels and

ratesof changeof generation,sparecapacityandline flow limits, is studied.A linearprogrammingformu-

lation for systemconstraintsis usedwith aquadraticcostfunctionfor generationandtransmissionline loss.


An optimal solution for real power dispatchis obtainedby quadraticprogramming,andoptimumalloca-

tion of reactive power is basedon a gradienttechniquewhich minimisestransmissionlosssubjectto nodal

voltageandreactive power limits. An ac load flow analysisis incorporatedtogetherwith a load prediction

programbasedonaspectralanalysisof pastloaddata.Thefeasibilityof simulatingtheoverall systemprob-

lem, including load-frequency controlof equivalentgeneration,usinga combinedanalogue-digitalprocess

computersystemlinkedto a largescientificdigital computerby datalink is investigated.

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A. Nilsberth,D. Sjelvgren,andH. Brannlund.Applicationof quadraticprogrammingin proba-bilistic productioncosting.In ‘PSCC.Proceedingsof theTenthPower SystemsCompu-tationConference.Butterworths,London,England’,pp.678–683,1990.

Abstract. Thelongtermproductionplanningof amixedhydro-thermalpowersystemoftenhasto betreated

asan interactionbetweentwo subproblems:Probabilisticproductioncostingfor the thermalunits,andop-

timal schedulingof theavailablehydrodischargeswithin the limits of the river system.In this interaction,

marginal costsregardingthehydroenergy arecalculatedfor usein thehydroscheduling.Theallocationof

thehydrodischarges/energies is thenusedfor anothercalculationof hydromarginal coststo beusedin the

next iteration.In theplanningsystemdevelopedat theSwedishStatePower Boardthehydrosubproblemis

modeledwith atime-incrementof oneweekwhile theloadduringeachweekis describedby threeloaddura-

tion curves-onefor eachof day-time,night-timeandweekends.To calculatethecorrectmarginalcostfor the

hydroenergy, theavailableweeklyenergieshasto beallocatedoptimally within eachweek.This optimiza-

tion problemcanbeformulatedasaquadraticprogrammingproblemwith onelinearequalityconstraint:the

fixed amountof energy to be distributed.Sincethis methodusesecondorderderivatives the solutionwill

convergebetterthanfor themorecommonmethodswhichsolve setsof linearequations.

J.NocedalandS. J.Wright. Quadraticprogramming.In ‘NumericalOptimization’,Seriesin

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Abstract. Within anelectricutility, timely informationon thecontribution of peakdemandfrom customer

classescan be extremely useful in establishingboth load forecastsand pricing policies.Unfortunately, a

utility’s normaldatacollectionproceduresdoesnot provide this informationandobtainingit, even infre-

quently, hasentailedcostlystatisticalsamplingoperations.A statisticalidentificationmodelwith inequality

constraintson the unknown variableshasbeenconstructedasa meansfor providing estimateson the de-

siredquantities.Model inputsincludemonthlycustomerbilling dataandaggregatepeakdemands.Testcase

resultsdemonstratingthequalityof theestimatesarepresented.

S. NordeboandI. Claesson.A well-conditionedquadraticprogramfor uniquedesignof two-dimensionalweightedChebyshev FIR filters. In ‘1996 IEEE InternationalConferenceon Acoustics,Speech,andSignalProcessingConferenceProceedings.IEEE,New York,NY, USA’, Vol. 3, pp.1355–1358,1996.

Abstract. TheweightedChebyshev designof two-dimensionalFIR filters is in generalnot uniquesincethe

Haarconditionis not generallysatisfied.However, for a designbasedon thediscretefrequency domain,the

Haarconditionmight befulfilled. Thequestionof uniquenessis, however, ratherextensive to investigate.It

is thereforedesirableto definesomesimpleadditionalconstraintsto theChebyshev designin orderto obtain

a uniquesolution.The weightedChebyshev solutionof minimum Euclideanfilter weight norm is always

unique,andrepresentsa sensibleadditionalconstraintsinceit impliesminimumwhite noiseamplification.


It is shown that this uniqueChebyshev solution can always be obtainedby using an efficient quadratic

programmingformulationwith astrictly convex objective functionandlinearconstraints.An examplewhere

aconventionalChebyshev solutionis non-uniqueis discussed.

S. Nordebo,I. Claesson,andS. Nordholm. WeightedChebyshev approximationfor the de-signof broadbandbeamformersusingquadraticprogramming.IEEE SignalProcessingLetters, 1(7), 103–105,1994.

Abstract. A methodto solve a generalbroadbandbeamformerdesignproblemis formulatedasa quadratic

program.As a specialcase,theminimaxnear-field designproblemof a broadbandbeamformeris solvedas

a quadraticprogrammingformulationof theweightedChebyshev approximationproblem.Themethodcan

alsobeappliedto thedesignof multidimensionaldigital FIRfilterswith anarbitrarilyspecifiedamplitudeand

phase.For linearphasemultidimensionaldigital FIR filters,thequadraticprogrambecomesalinearprogram.

Examplesaregiventhatdemonstratetheminimaxnear-field behavior of thebeamformersdesigned.

S. Nordebo,I. Claesson,andS. Nordholm. Quadraticprogrammingfor the designof two-

dimensionalweightedChebyshev FIR filters with incompletespecifications.Nonlinear

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S. Nordebo,I. Claesson,andZ. Zang. Optimumwindow designby semi-infinitequadraticprogramming.IEEESignalProcessingLetters, 6(10),262–265,1999.

Abstract. This letterpresentsa new extendedactive setstrategy for optimumfinite impulseresponse(FIR)

window designby semi-infinitequadraticprogramming.Thewindows maybeasymmetriccorrespondingto

frequency responseswith generalnonlinearphase.Theoptimalitycriterionis to minimizethesidelobeenergy

(L2-norm)subjectto a peaksidelobemagnitude-constraint(L∞-constraint).Additional linearconstraintsare

usedto form the mainlobe(unity DC gain).Numericalexamplesinvolving groupdelayspecificationsare

usedto illustratetheusefulnessof thealgorithm.

A. Nou. An algorithmfor a singly constrainedquadraticprogramsubjectto lower bounds.

TechnicalReportTRITA/MAT-97-OS2,Departmentof Mathematics,KTH, Stockholm,

Sweden,1997.

B. Novak andI. Novak. Quadraticprogrammingappliedto shorttermhydrothermalschedul-ing. In H. B. SiguerdidjaneandP. Bernhard,eds,‘Control Applicationsof NonlinearProgrammingandOptimization1989.Proceedingsof the 8th IFAC Workshop.Perga-mon,Oxford,England’,pp.57–62,1990.

Abstract. Thepurposeof optimalshorttermschedulingis to maximizeelectricalenergy productiondepend-

ing onnaturalwaterinflowsof ariver, accordingto daily loaddurationcurveandoperationof thermalpower

plantswith minimaloscillatingload.Thermalpowerplantshavealongstart-uptimeandcannotfollow rapid

changesin loaddemand,sothey areusedto cover theconstantpartof theload,while peaksin loaddemand

arecoveredby hydrounits.A mathematicalmodelof a hydroplantchainis developedwhich usesonly two

insteadof threevariables.TheoptimizationalgorithmusesNewton’s method,which is appliedon theexact* 1 penaltyfunction,allowing a nonfeasiblestartingpoint sothatno phaseI simplex or projectionmethodis

neededat thebeginningof optimization.Theprogramwasappliedto thehydro-chainon theriver Drava.

I. Nowak.Someheuristicsandtestproblemsfor nonconvex quadraticprogrammingoverasim-plex. Preprintseries,Institut fur Mathematik,Humboldt-UniversitatzuBerlin, Germany,1998.

Abstract. In thispaperwecomparetwo methodsfor estimatingaglobalminimizerof anindefinitequadratic

form overasimplex. Thefirst methodis basedon theenumerationof localminimizersof aso-calledcontrol

polytope.Thesecondmethodis basedon anapproximationof theconvex envelopeusingsemidefinitepro-

gramming.In orderto testthealgorithmsa methodfor generatingrandomtestproblemsis presentedwhere


theoptimal solutionis known andthe numberof bindingconstraintsis prescribed.Moreover, it is investi-

gatedif somemodificationsof theobjective functioninfluencetheperformanceof thealgorithms.Numerical

experimentsarereported.

I. Nowak. A global optimality criterion for nonconvex quadraticprogrammingover a sim-plex. Preprintseries,Institut fur Mathematik,Humboldt-UniversitatzuBerlin, Germany,1998b.

Abstract. In this paperwe proposea global optimality criterion for globally minimizing a quadraticform

over the standardsimplex, which in additionprovidesa sharplower boundfor the optimal value.The ap-

proachis basedonthesolutionof asemidefiniteprogram(SDP)andaconvex quadraticprogram(QP).Since

thereexist fast(polynomialtime) algorithmsfor solvingSDP’s andQP’s thecomputationaltime for check-

ing theglobaloptimality criterionandfor computingthelower boundis reasonable.Numericalexperiments

on randomtestexamplesup to 30variablesindicatethattheoptimality criterionverifiesaglobalsolutionin

almostall instances.

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Abstract. Theestimationsaregivenby thesolutionof alinearsystemof equationbymeansof themethodsof

interval arithmetic.Therefore,it isquitecertainthattheroundoff errorsarealsoincludedin theseestimations.

J. H. Oh andE. Feron. Primal-dualquadraticprogrammingapproachto multiple conflict res-olution. In ‘Proceedingsof the 1998AmericanControl Conference.AmericanAutom.ControlCouncil,Evanston,IL, USA’, Vol. 5, pp.2802–2806,1998.

Abstract. This paperconsidersa multiple conflict resolutionproblemfor air traffic control systems.The

timerequiredto optimallysolveaircraftconflictsis known to grow exponentiallywith thenumberof aircraft

involvedandmaybecomeprohibitive whenlargenumbersof aircraftareinvolved.As anattemptto circum-

ventthis issue,aheuristicpolynomial-timeconflict resolutionalgorithmis proposedonthebasisof analysis

resultson therelationshipbetweenprimalanddualquadraticprograms.

Y. Ohashi,H. Kando,H. Ukai, andT. Iwazumi.Optimaldesignof FIR linearphasedigital filtersvia convex quadraticprogrammingmethod. InternationalJournal of SystemsScience,19(8), 1469–1482,1988.

Abstract. Many methodsexist for designingFIR linear phasedigital filters by usingvariousoptimization

techniques.An optimizationdesignmethodvia the convex quadraticprogrammingmethodis investigated

in the paper. The aim of this methodis to designdigital filters with certaindesiredfrequency responses

that keepthe passbandripple to a minimum while maintaininga balancebetweenoppositecharacteristics

of frequency responses,i.e. thenarrow transitionbandandthegreatstopbandattenuation.In orderto verify

the effectivenessof this method,low-passfilters are designedandcomparisonstudieswith other typical

optimizationmethodsaregiven.

S. Ohtaki andK. I. Hata. Planeelastic-plasticstressanalysisusingquadraticprogramming.adoptionof theyield functioncontainingthethird invariantof thedeviatoricstresstensor.Memoirsof theHokkaidoInstituteof Technology, 15, 35–41,1987.

Abstract. Thefinite elementelastic-plasticanalysisbasedon thetheoryof plasticitytakinginto accountthe

secondordereffect usingthe quadraticprogrammingmethodis appliedto the planestressproblem.This

secondordereffect is causedfrom theyield functionderived from Prager’s theory, which is assumedto be

expressedin termsof not only thesecondinvariantof thedeviatoric stresstensorJ;2 but alsoof thethird one

J;3 Matricesconcerningwith thismethodareformulatedundertheplanestressstate.A numericalcalculation

is presentedfor thestressconcentrationproblemof a thin rectangularplatewith semi-circularsidenotches

subjectedto auni-axialtension.Resultsobtainedby usingthesecondorderplastictheoryarecomparedwith

thoseof theordinaryJ2 theory.


S.OhtakiandA. Kurimura.An analysisof two-dimensionalelastic-plasticstressproblemusingthequadraticprogramming.(Thecaseof adoptingPrandtl-ReussequationandvonMisesyield function). Bulletin of theJapanSocietyof MechanicalEngineers, 28(243),1864–1867,1985.

Abstract. G. Maier (1971)investigatedproblemsin whichnormalityconditionswith regardto plasticstrain

incrementspertainingto the yield surfacein stressspacearenot satisfiedandwork-softeningeffects are

consideredwith theaidof quadraticprogrammingconcepts.Theauthorsappliedthismethodfor solvingthe

elastic-plasticplanestressproblemasa quadraticoptimizationin termsof displacementratesandplastic

multiplier rates.The matricesareformulatedin explicit forms for the problemof a perforatedstrip under

tension.A comparisonis madebetweenthe resultsobtainedby the presentmethodand by the ordinary

matrixmethod.Theelastic-plasticproblemis fairly tractableto thepresentmethod.

A. OhuchiandI. Kaji. Algorithmsfor optimalallocationproblemshaving quadraticobjectivefunction. Journalof theOperationsResearch Societyof Japan, 23(1), 64–79,1980.

Abstract. An optimal allocationproblem(APQ) with a quadraticobjective function,a total resourcecon-

straintandanupperandlowerboundconstraintis considered.TheAPQis averybasicandsimplemodelbut

it canserve asa subproblemin thesolutionof thegeneralizedallocationproblem.Applying the Lagrange

relaxationmethodanexplicit expressionof thedualfunctionassociatedwith theAPQandanequationwhich

theoptimaldualvariablemustsatisfyandderivedfirst. Then,somepropertiesof theequationarediscussed.

Finally, threealgorithmsfor solvingtheequationareproposed,andsomecomputationalresultsfor theAPQ

aregiven.Theseresultsrevealtheeffectivenessof thealgorithm.

A. OhuchiandI. Kaji. An algorithmfor theHitchcocktransportationproblemswith quadraticcost functions. Journal of the OperationsResearch Societyof Japan, 24(2), 170–181,1981.

Abstract. In thispaperanalgorithmisgivenfor theHitchcocktransportationproblems(HTPQ).A procedure

hasbeenproposedwhich involvessuccessively minimising the Lagrangianwith respectto eachof its dual

coordinates.Thefactthatthealgorithmsfor optimalallocationproblemshaving quadraticobjective functions

(APQ) canbeusedfor the maximisingprocedurein network transportationproblemswith quadraticcosts

functionswasinvestigated,andAPQstudiedin detail.Theapplicationof algorithmsproposedby theauthors

to solvingHTPQis considered.In particularseveral examplesaregiven of relatively large-scaleproblems,

with computationaltimes.

M. E. O’Kelly. A quadraticinteger-programfor thelocationof interactinghubfacilities.Euro-

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D. P. O’Leary. A generalizedconjugategradientalgorithm for solving a classof quadraticprogrammingproblems.Linear Algebra andIts Applications, 34, 371–399,1980.

Abstract. Applies matrix splitting techniquesanda conjugategradientalgorithmto the problemof mini-

mizing a convex quadraticform subjectto upperandlower boundson the variables.This methodexploits

sparsitystructurein the matrix of the quadraticform. Choicesof the splitting operatorarediscussed,and

convergenceresultsareestablished.

D. P. O’Leary. Sparsequadraticprogrammingwithout matrix updating.TechnicalReportTR-

1200,ComputerScienceDepartment,Universityof Maryland,USA, 1982.

D. P. O’Leary andW. H. Yang. Elasto-plastictorsionby quadraticprogramming.ComputerMethodsin AppliedMechanicsandEngineering, 16(3), 361–368,1978.

Abstract. A finite elementscheme(togetherwith a conjugategradientalgorithm)is demonstratedto be a

very effective methodfor analyzingelastoplastictorsionof prismaticbarsposedasquadraticprogramming

problems.Solutionsfor barswith elliptical andSokolovsky’s oval cross-sectionsarepresented.Thesolutions


for the elliptical barsagreewith the existing elasticand limit plasticsolutionsat the two extremesof the

elastic-plasticrange.ThealgorithmalsoreproducesaccuratelytheSokovsky solutionandextendsit beyond

its limitations.

T. OnodaandH. Sekimoto. A neutronspectrometryunfolding codebasedon quadraticpro-gramming.NuclearInstrumentsandMethodsin PhysicsResearch, SectionA (Accelera-tors,Spectrometers,DetectorsandAssociatedEquipment), 272(3), 844–846,1988.

Abstract. A new unfoldingcodebasedon quadraticprogramminghasbeendevelopedfor precisetreatment

of the nonnegativity constraintof the neutronspectrum.This codedoesnot requireany initial guessand

enablesa globaloptimumsolutionto bederived.

A. Orden.Stationarypointsof quadraticfunctionsunderlinearconstraints.ComputerJournal,

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S. Ou andM. Pedram.Timing-drivenbipartitioningwith replicationusingiterative quadraticprogramming.In ‘Proceedingsof theASP-DAC ’99 Asia andSouthPacific DesignAu-tomationConference1999.IEEE,Piscatatway, NJ,USA’, Vol. 1, pp.105–108,1999.

Abstract. We presentanalgorithmfor solving a generalmin-cut, two-way partitioningproblemsubjectto

timing constraints.The problemis formulatedasa constrainedprogrammingproblemandsolved in two

phases:cut-setminimizationandtiming satisfaction.A mathematicalprogrammingtechniquebasedon it-

erative quadraticprogramming(TPIQ) is usedto find anapproximatesolutionto the constrainedproblem.

Whenthetiming constraintsaretoo strict to have a feasiblesolution,nodereplicationis usedto satisfythe

constraints.Experimentalresultson the ISCAS89benchmarksuiteshow that TPIQ cansolve the timing-

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K. A. Palaniswamy, J.K. Sharma,andK. B. Misra. Minimization of loadcurtailmentin powersystemusing quadraticprogramming. Journal of the Institution of Engineers (India)ElectricalEngineeringDivision, 65, 213–218,1985.

Abstract. The authorsdescribethe problemsof steadystateoptimal load sheddingin power systemsand

presenta methodfor minimizing load curtailmentundera given set of emergency operatingconditions.

Electric power consumersare inconveniencedif there is an inadequatelevel of supply voltageand load

curtailment.Theinconvenienceis to beminimizedsubjectto thesystempower-flow equationsandthelimits

on realandreactive power generationalongwith the limits on the line flows. This optimizationproblemis

solvedusingquadraticprogrammingwhichhasbeenfoundeffective in theassignmentof distributionof load

curtailmentover theentiresystem.Themethodis illustratedon a samplesystemfor generation-outageand

line-outageconditions.

V. N. Palyaeva. A linearization method with quadraticprogrammingfor linearly con-

strainedapproximation-problems. VestnikLeningradskogo UniversitetaSeriyaMatem-

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J.S.Pang.An equivalencebetweentwo algorithmsfor quadraticprogramming.MathematicalProgramming, 20(2), 152–165,1981.

Abstract. ThispaperdemonstratesthattheVandePanne-Whinstonsymmetricsimplex methodwhenapplied

to a certainimplicit formulation of a quadraticprogramgeneratesthe samesequenceof primal feasible

vectorsasdoestheVon Hohenbalken simplicial decompositionalgorithmspecializedto thesameprogram.

Suchanequivalenceof thetwo algorithmsextendsearlierresultsfor a least-distanceprogramdueto Cottle-

Djang.

J.S.Pang.Methodsfor quadraticprogramming:asurvey. ComputersandChemicalEngineer-ing, 7(5), 583–594,1983.


Abstract. Thispaperpresentsasurvey onmethodsfor solvingthegeneralquadraticprogrammingproblem.

The discussionis centeredon (i) the unificationof several classesof finite methods,(ii) recentdevelop-

mentsof iterative methodsand(iii) thenumericalimplementationof severalfinite anditerative methodsfor

largescaleapplications.

P. M. Pardalos.Generationof large-scalequadraticprogramsfor useasglobaloptimizationtest

problems.ACM Transactionson MathematicalSoftware, 13(2), 133–137,1987.

P. M. Pardalos.Quadraticproblemsdefinedonaconvex hull of points.BIT, 28, 323–328,1988.

P. M. Pardalos. Polynomial time algorithms for some classesof constrainednonconvex

quadraticproblems.Optimization, 21(6), 843–853,1990.

P. M. Pardalos.Constructionof testproblemsin quadraticbivalentprogramming.ACM Trans-

actionsonMathematicalSoftware, 17(1), 74–87,1991a.

P. M. Pardalos. Global optimizationalgorithmsfor linearly constrainedindefinitequadratic

problems.ComputationalMathematicsandApplications, 21(6/7),87–97,1991b.

P. M. PardalosandN. Kovoor. An algorithmfor asinglyconstrainedclassof quadraticprograms

subjectto upperandlowerbounds.MathematicalProgramming, 46(3), 321–328,1990.

P. M. PardalosandG. P. Rodgers. Parallel branchandboundalgorithmsfor unconstrained

quadratic0–1programming.In R.Sharda,ed.,‘Impactof RecentComputerAdvanceson

OperationsResearch’,pp.131–143,North Holland,Amsterdam,theNetherlands,1989.

P. M. PardalosandG. P. Rodgers.Computationalaspectsof a branchandboundalgorithmfor

quadratic0–1programming.Computing, 45, 131–144,1990a.

P. M. PardalosandG. P. Rodgers. Parallel branchandboundalgorithmsfor quadratic0–1

programson the hypercubearchitecture.Annalsof OperationsResearch, 22, 271–292,

1990b.

P. M. PardalosandG. Schnitger. Checkinglocal optimality in constrainedquadraticprogram-ming is NP-hard.OperationsResearch Letters, 7(1), 33–35,1988.

Abstract. Proves that the problemof checkinglocal optimality for a feasiblepoint and the problemof

checkingif a local minimum is strict, areNP-hardproblems.As a consequencethe problemof checking

whethera functionis locally strictly convex is alsoNP-hard.

P. M. PardalosandS. A. Vavasis. Quadraticprogrammingwith onenegative eigenvalue is

NP-hard.Journalof Global Optimization, 1, 15–23,1991.

P. M. Pardalos,M. V. Ramana,andY. Yajima. Cutsandsemidefiniterelaxationsfor nonconvex

quadraticproblems.SIAMJournalon Optimization, (submitted), 1998.

P. M. Pardalos,Y. Ye, andC. G. Han. An interior point algorithmfor large-scalequadratic

problemswith boxconstraints.In A. BensoussanandJ.L. Lions,eds,‘AnalysisandOp-

timizationof Systems: Proceedingsof the9thInternationalConference,Antibes,France,

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K. C. Park, P. H. Chang,andS. H. Kim. The enhancedcompactQP methodfor redundantmanipulatorsusingpracticalinequalityconstraints. In ‘Proceedings.1998IEEE Inter-nationalConferenceon RoboticsandAutomation.IEEE, New York, NY, USA’, Vol. 1,pp.107–114,1998.

Abstract. For resolvingthe manipulatorredundancy underinequalityconstraints,the compactQPmethod

(includingthe improvedCompactQPmethod)is very effective andefficient. Froma viewpoint of practical

applications,however, it turnsout, we found, to have someperformancelimitationssuchasunrealistically

high torquedueto joint anglelimit constraintsandtrackingerrorsdueto joint torquelimit constraintsunder

parametervariationanddisturbance.Remedyingthe limitations, the enhancedcompactQP methodis de-

velopedby usingthepracticalinequalityconstraintswith p-step-aheadpredictorandtime delayestimation.

Throughdynamicsimulationresults,it is verified that the enhancedcompactQP methodsignificantly im-

provesthecompactQPmethodin termsof efficiency andeffectivenessfor thereal-timecontrolof redundant

manipulatorsunderphysicallimits.

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review. SpecialStudiesPaper5, Boardof Governorsof the FederalReserve System,

USA, 1970.

T. D. Parsons.A combinatorialapproachto convexquadraticprogramming. PhDthesis,Prince-

ton University, 1966.

T. D. Parsons.A combinatorialapproachto convex quadraticprogramming.Linear AlgebraandIts Applications, 3(3), 359–378,1970.

Abstract. Theproblemof minimizing a convex quadraticfunctionof many variablesconstrainedby linear

inequalitieshasbeenstudiedextensively, anda numberof algorithmshave beendeveloped.Most of them

aresimilarandarecloselyrelatedto thesimplex methodfor linearprogramming.Thispaperis essentiallyan

extensionof A. W. Tucker’sCombinatorialTheoryunderlyinglinearprogramsto convex quadraticprograms.

As such,it sharesthepurposeof thatpaper’ to discover theunderlyingtheoreticalstructure’of theproblem,

andto unitethiswith theactualstepsof computationto make clearthelimits of generality.

N. Pelillo andA. Jagota.Feasibleandinfeasiblemaximain a quadraticprogramfor maximumclique. Journalof Artificial Neural Networks, 2(4), 411–420,1995.

Abstract. T. S.Motzkin andE. G. Straus(1965)relatedglobalmaximaof acertainquadraticprogramto the

maximumcliquesizein a certaingraph.We extendthis resultto relatestrict local maximaof this program

to certainmaximalcliques,andcertainmaximato noncliques.Our resultsareusefulto a companionpaper

whichemploys thisQPin aneuralnetmodelto find largecliquesin graphs.

A. F. Perold. An alternative methodfor a global analysisof quadraticprogramsin a finite

numberof steps.MathematicalProgramming, 15(1), 105–109,1978.

P. Pfaff. Portfolio selectionusingquadraticprogrammingon a microcomputer. In ‘Modeling

andSimulation: Proceedingsof the Twelfth Annual Pittsburgh Conference’,Vol. 1–4,

pp.993–997,1981.

T. PhamDinh, T. Q. Phong,R. Horaud,andL. Quan.Stability of Lagrangianduality for non-convex quadraticprogramming.solutionmethodsandapplicationsin computervision.RAIRO-MathematicalModellingandNumericalAnalysis—ModelisationMathematiqueet AnalyseNumerique, 31(1), 57–90,1997a.

Abstract. Theproblemof minimizing a quadraticform over a ball centeredat theorigin is considered.The

stabilityof Lagrangianduality is establishedandcompletecharacterizationsof aglobaloptimalsolutionare


given. On the basisof this theoreticalstudy, two principal solutionmethodsarepresented.An important

applicationof nonconvex quadraticprogrammingis thecomputationof rite stepto anew iteratein theTrust

Region (TR) approachmethodswhich areknown to beefficient for nonlinearoptimizationproblems.Also,

wediscussthemathematicalmodelsof someimportantproblemsencounteredin ComputerVision.Most of

themcanbeformulatedasaminimizationof asumof squaresof nonlinearfunctions.A practicalTR-based

algorithmis proposedfor nonlinearleastsquaresproblemwhichseemsto bewell suitedfor ourapplications.

T. PhamDinh, P. ThaiQuynh,R. Horaud,andL. Quan.Stabilityof Lagrangianduality for non-convex quadraticprogramming.Solutionmethodsandapplicationsin computervision.MathematicalModellingandNumericalAnalysis, 31(1), 57–90,1997b.

Abstract. Theproblemof minimizing a quadraticform over a ball centeredat theorigin is considered.The

stability of Lagrangianduality is establishedand completecharacterizationsof a global optimal solution

aregiven.On thebasisof this theoreticalstudy, two principalsolutionmethodsarepresentedAn important

applicationof nonconvex quadraticprogrammingis thecomputationof thestepto a new iteratein theTrust

Region (TR) approachmethodswhich areknown to beefficient for nonlinearoptimizationproblems.Also,

wediscussthemathematicalmodelsof someimportantproblemsencounteredin ComputerVision.Most of

themcanbeformulatedasaminimizationof asumof squaresof nonlinearfunctions.A practicalTR-based

algorithmis proposedfor nonlinearleastsquaresproblemwhichseemsto bewell suitedfor ourapplications.

J. Philip. Restorationof picturesby quadraticprogrammingandby Fouriertransformationinthecomplex domain.Proceedingsof theIEEE, 61(4), 468–469,1973.

Abstract. Two methodsfor digital restorationof picturesblurredby imperfectionsof theopticalsystemand

by randomnoisearesuggested.Methodoneis basedontheapriori informationthatapicturehasnonnegative

intensityandleadsto quadraticprogramming.In methodtwo, thefinite extentof thepictureis shown to make

Fouriertransformationin thecomplex domainuseful.

J. Philip. Digital imageandspectrumrestorationby quadraticprogrammingandby modifiedFourier transformation. IEEE Transactionson Pattern Analysisand Machine Intelli-gence, PAMI-1 (4), 385–399,1979.

Abstract. Considerstheconvolution equationf & h e d, where f is sought,h is a known ’point spread

function’, e representsrandomerrorsandd is the measureddata.All thesefunctionsare definedon the

integersmod N . A mathematical-statistical formulationof the problemleadsto minf F f & h d F A, where

theA-normis derivedfrom thestatisticaldistributionof e. If f is known to benonnegative, this is aquadratic

programmingproblem.UsingthediscreteFourier transforms(DFTs)F H, andD of f h, andd, theauthor

arrivesat a minimization in anothernorm: minF F F H D F α. A solutionwould be F D H, but H has

zeros.He considersthe theoreticalandpracticaldifficulties that arisefrom thesezerosanddescribestwo

methodsfor calculatingF numericallyalsowhenH haszeros.Numericaltestsof methodsarepresented,in

particulartestswith oneof themethods,called’thederivative method’,whered is ablurredimage.

A. T. Phillips andJ. B. Rosen.A parallelalgorithmfor constrainedconcave quadraticglobal

minimization.MathematicalProgramming, SeriesB, 42(2), 421–448,1988.

A. T. Phillips andJ. B. Rosen. Guaranteedffl-approximatesolution for indefinitequadratic

globalminimization.NavalResearch LogisticsQuarterly, 37, 499–514,1990.

H. X. PhuandN. D. Yen. On the stability of solutionsto quadraticprogrammingproblems.MathematicalProgramming, 89(3), 385–394,2001.

Abstract. We considerthe parametricprogrammingproblem(Qp) of minimizing the quadraticfunction

f x p : xT Ax bTx subjectto theconstraintCx d, wherex IRn, A IRn G n, b IRn, C IRm G n, d IRm,

andp : H A b C d is theparameter. Here,thematrix A is not assumedto bepositive semidefinite.Theset

of the global minimizersand the set of the local minimizersto (Qp) are denotedby M p andMl oc p ,respectively. It is provedthat if thepoint-to-setmappingMl oc + is lower semicontinuousat p thenMl oc p


is anonemptysetwhichconsistsof atmostI m4 n points,whereI m4 n Jm

min6K<m 2=L n8NM is themaximal

cardinalityof theantichainsof distinctsubsetsof 6 1 2 O m8 which have at mostn elements.It is proved

also that the lower semicontinuityof M + at p implies that M p is a singleton.Under someregularity

assumption,thesenecessaryconditionsbecomethesufficient ones.

J. W. Pitton. Positive time-frequency distributionsvia quadraticprogramming. Multidimen-sionalSystemsandSignalProcessing, 9(4), 439–445,1998.

Abstract. A new methodfor computingpositive time-frequency distributions(TFD) for nonstationarysig-

nalsis presented.This work extendstheearlierwork of theauthorandhis colleaguesin computingpositive

TFD (1994).This paperdescribesa generalquadraticprogrammingapproachto theproblemof computing

thesesignal-dependentdistributions.Themethodis basedon anevolutionaryspectrumformulationof posi-

tive TFD. Theminimizationproblemreducesto a linearly-constrainedquadraticprogrammingproblem,for

whichstandardsolutionsarewidely available.

S.PoljakandH. Wolkowicz. Convex relaxationsof (0,1)-quadraticprogramming.Mathematicsof OperationsResearch, 20(3), 550–561,1995.

Abstract. We considerthreeparametricrelaxationsof the (0,1)-quadraticprogrammingproblem.These

relaxationsare to: quadraticmaximizationover simple box constraints,quadraticmaximizationover the

sphere,andthemaximumeigenvalueof aborderedmatrix.Whenminimizedover theparameter, eachof the

relaxationsprovidesanupperboundontheoriginaldiscreteproblem.Moreover, theseboundsareefficiently

computable.Our mainresultis that,surprisingly, all threeboundsareequal.

S.Poljak,F. Rendl,andH. Wolkowicz. A recipefor semidefiniterelaxationfor (0,1)-quadraticprogramming.Journalof Global Optimization, 7(1), 51–73,1995.

Abstract. Wereview variousrelaxationsof (0,1)-quadraticprogrammingproblems.Theseincludesemidef-

inite programs,parametrictrust region problemsandconcave quadraticmaximization.All relaxationsthat

weconsiderleadto efficiently solvableproblems.Themaincontributionsof thepaperarethefollowing. Us-

ing Lagrangianduality, we prove equivalenceof therelaxationsin a unifiedandsimpleway. Someof these

equivalenceshave beenknown previously, but our approachleadsto shortandtransparentproofs.Moreover

weextendtheapproachto thecaseof equalityconstrainedproblemsby takingthesquaredlinearconstraints

into theobjective function.We show how this techniquecanbeappliedto theQuadraticAssignmentProb-

lem, theGraphPartition ProblemandtheMax- CliqueProblem.Finally we show our relaxationto bebest

possibleamongall quadraticmajorantswith zerotrace.

D. B. Ponceleon. Barrier methodsfor large-scalequadratic programming. PhDthesis,Depart-

mentof ComputerScience,StanfordUniversity, Stanford,California,USA, 1990.

M. Pontil, S. Rogai,andA. Verri. Supportvectormachines:a very largeQP. In R. D. Leone,

A. Murli, P. M. PardalosandG. Toraldo,eds,‘High PerformanceAlgorithmsandSoft-

warein NonlinearOptimization’,pp.315–336,Kluwer AcademicPublishers,Dordrecht,

TheNetherlands,1998.

D. N. Pope,G. L. Curry, andD. T. Phillips. Closedform solutionsto non-serial,nonconvex

quadratic-programmingproblemsusingdynamic-programming. Journal of Mathemati-

cal AnalysisandApplications, 86(2), 628–647,1982.

M. J.D. Powell. An uppertriangularmatrixmethodfor quadraticprogramming.In O. L. Man-

gasarian,R. R. Meyer andS.M. Robinson,eds,‘NonlinearProgramming,2’, Academic

Press,London,1981.


M. J.D. Powell. ZQPCVX,aFortransubroutinefor convex quadraticprogramming.Technical

ReportDAMTP/NA17, Departmentof Applied Mathematicsand TheoreticalPhysics,

CambridgeUniversity, Cambridge,England,1983.

M. J.D. Powell. On thequadraticprogrammingalgorithmof GoldfarbandIdnani. Mathemat-ical ProgrammingStudies, 25, 46–61,1985.

Abstract. Two implementationsof the algorithmof Goldfarb andIdnani (1983)for convex quadraticpro-

grammingareconsidered.A pathologicalexampleshows that thefasteronecanbeunstable,but numerical

testingon somedifficult problemsindicatesthat both implementationsgive excellentaccuracy. Therefore

theauthor(1983)hasprovided for generalusea FORTRAN subroutinethatappliesthefasterimplementa-

tion.Thissubroutineis comparedwith two widely availablequadraticprogrammingsubroutinesthatemploy

feasiblepoint methods,namelyQPSOL(Gill, Murray, Saunders,andWright, 1983)andVEO2A (Fletcher,

1970).Theauthorconcludesthatthealgorithmof GoldfarbandIdnaniis verysuitablein practicefor convex

quadraticprogrammingcalculations.

J.C. Preisig.Copositivity andtheminimizationof quadraticfunctionswith nonnegativity and

quadraticequalityconstraints.SIAMJournalon Control andOptimization, 34(4), 1135–

1150,1996.

L. G. Proll. A computerroutine for quadraticand linear programmingproblems(algorithm

r431). CollectedAlgorithmsfromACM, 2589, 1974.

M. L. Psiaki andK. Park. Parallel orthogonalfactorizationnull-spacemethodfor dynamicquadraticprogramming.Journal of OptimizationTheoryandApplications, 85(2), 409–434,1995.

Abstract. An algorithmhasbeendevelopedto solve quadraticprogramsthathave a dynamicprogramming

structure.It hasbeendevelopedfor useaspart of a paralleltrajectoryoptimizationalgorithmandaimsto

achieve significantspeedwithout sacrificingnumericalstability. The algorithmmakesuseof the dynamic

programmingproblemstructureandthedomaindecompositionapproach.It parallelizestheorthogonalfac-

torizationnull-spacemethodof quadraticprogrammingby developingaparallelorthogonalfactorizationand

aparallelCholesky factorization.Testsof thealgorithmona32-nodeINTEL iPSC/2hypercubedemonstrate

speedupfactorsaslargeas10 in comparisonto thefastestknown equivalentserialalgorithm.

R. Pytlak. A range-spacemethodfor piecewise-linearquadraticprogramming:anapplicationto optimalcontrolalgorithms.In ‘Proceedingsof the33rdIEEEConferenceonDecisionandControl.IEEE,New York, NY, USA’, Vol. 2, pp.1462–1463,1994.

Abstract. A new methodfor solvinga convex optimizationproblemwith box constraintsis presented.The

objective functionhasa positive-definitequadratictermanda piecewise-linearterm.Themethodis derived

from a rangespacemethodfor QP problems.The approachis particularlyefficient if the piecewise-linear

termhasfew breakpoints.Numericalcomparisonswith anefficient implementationof anull-spaceactive-set

algorithm(LSSOL)arealsopresented.

A. J. Quist,E. Deklerk,C. Roos,andT. Terlaky. Copositive relaxationfor generalquadraticprogramming.OptimizationMethodsandSoftware, 9(1–3),185–208,1998.

Abstract. We considergeneral,typically nonconvex, QuadraticProgrammingProblems.TheSemi-definite

relaxationproposedby Shorprovidesboundson the optical solution.but it doesnot alwaysprovide suffi-

cientlystrongboundsif linearconstraintsarealsoinvolved.To getrid of thelinearside-constraints.another,

strongerconvex relaxationis derived.This relaxationusescopositive matrices.Specialcasesarediscussed

for whichbothrelaxationsareequal.At theendof thepaper, thecomplexity andsolvability of therelaxations

arediscussed.


S.R.Radhakrishnan.Capitalbudgetingandmixed0–1integerquadraticprogramming.Bulletinof theOperationsResearch Societyof America, 20, B331,1972.

Abstract. Thecapitalbudgetingproblemis treatedasapartof thegeneraltheoryof choice,whereutility is

to bemaximizedsubjectto theopportunitiesandconstraints.Theutility functionis assumedto bequadratic

to takecareof any risk-aversebehaviour of theshareholdersandaquadraticmixed0–1integerprogramming

modelis developed.Theduality conceptsin discreteprogrammingareusedto derive propertiesof optimal

solutionsto themodel.Specialsolutiontechniquesfor themixed0–1integerquadraticprogrammingmodel

arediscussed.

M. R. Rakhimov. On somemethodsof solvinga problemof linearly-quadraticprogrammingfor systemswith distributedparameters.Zhurnal Vychislitel’noi Matematikii Matem-aticheskoi Fiziki, 26(12),1797–1812,1986.

Abstract. The authorformulatesa theory of the sufficient conditionsof optimality of a certainclassof

nonlinearequationsof thesecondorder. By theuseof spectralexpansionof anon-self-conjugateoperator, a

solutionof theproblemof linearly-quadraticprogrammingis constructedin closedform.A proof is obtained

of aseriesof theoremsof thepropertiesof thesolutionof aRikkati equationby substantiationof themethod

of dynamicprogramming.Theresultsobtainedarebasedonthepropertiesof aRissbasisformedby theeigen

andadjointelementsof a stationaryoperatorforming theprincipalpartof thedynamicequationexamined.

It is shown thattheapproximatesolutionof theproblemconvergesto anexactone.

M. R. Rakhimov. The use of methodsof spectrumseriesand dynamic-programmingfor

studyingtheproblemsof linear-quadraticprogramming.DokladyAkademiinaukSSSR,

292(4), 818–821,1987.

R. RamabhadranandJ. K. Antonio. Fastsolutiontechniquesfor a classof optimal trajectoryplanningproblemswith applicationsto automatedspraycoating. TechnicalReportTR-EE95–9,PerdueDepartmentof ECE,PerdueUniversity, ???,USA, 1995.

Abstract. Optimal trajectoryplanningproblemsareoften formulatedasconstrainedvariationalproblems.

In general,solutionsto variationalproblemsare determinedby appropriatelydiscretizingthe underlying

objective functionalandsolving the resultingnonlineardifferentialequation(s)and/ornonlinearprogram-

ming problem(s)numerically. Thesegeneralsolutiontechniquesoftenrequirea significantamountof time

to becomputed,andthereforeareof limited valuewhenoptimaltrajectoriesneedto befrequentlycomputed

and/orre-computed.In this paper, a realisticclassof optimal trajectoryplanningproblemsis definedfor

which theexistenceof fastnumericalsolutiontechniquesaredemonstrated.To illustratethepracticalityof

thisclassof trajectoryplanningproblemsandtheproposedsolutiontechniques,threeoptimaltrajectoryplan-

ningproblemsfor spraycoatingapplicationsareformulatedandsolved.Basedontheproposeddiscretization

technique,it is shown thattheseproblemscanbereducedto eithera linearprogramor aquadraticprogram,

which arereadilysolved.In contrast,usingthestandarddiscretizationof theseproblemsgenerallyleadsto

nonconvex nonlinearprogrammingproblemsthatrequirea significantamountof computationto arrive at a

(possibly)locally optimalsolution.

H. V. Ramakrishnan.Largescaleenergy systemplanningusingquadraticprogramming.Jour-nal of theInstitutionof Engineers (India) Electrical EngineeringDivision, 68, 170–176,1988.

Abstract. A quadraticprogrammingapproachto the problemof planninga large scaleenergy systemis

presented.Theapproachdeterminesoptimalplantlocation,typeandcapacityof generatingunitsto beadded

to any existingsystembyminimizingtheobjective functionconsistingof annualamortizedcostof generation

andtransmissionof thesystem,simultaneouslysatisfyingtheconstraintsof peakandaveragedemands.An

illustrative examplefrom publishedliteratureis workedout.Theeffectof additionalconstraintsfor minimum

generationisalsostudied.Highlightsof theresults,acritical comparisonwith previousstudies,andimportant

conclusionsaregiven.


J. R. J. Rao and R. Bradlaw. A unified characterizationof nonunique responseinelastic/perfectly-plasticand elastic/locking structuresusing constraint qualification.TechnicalReport 9, Departmentof MechanicalEngineering,University of Houston,Texas,USA, 1995.

Abstract. Mathematicalprogrammingmethodshave traditionallybeenusedextensively in theanalysisand

designof elastic/plasticstructures.However, someof therecentresultsfrom parametricnonlinearprogram-

minghavenotbeenfully exploitedin mechanicalapplications,particularlythoserelatingto thecausesof sin-

gularitiesdueto parametricvariations.In this paper, we reexaminethephenomenonof nonuniquedisplace-

mentsin elastic/perfectly-plastic structuresunderproportionalloading.Oncetheanalysismodelasderived

from a minimumenergy principleis formulatedasaquadraticprogram,it turnsout thattheresponsecanbe

characterizedasbeinguniqueor nonuniquedependingon thesatisfactionof thelinear independenceor the

Mangasarian-Fromovitz constraintqualification.A completelyanalogousresultis thenalsoderivedfor elas-

tic/locking structures,thusindicatingtheusefulnessof constraintqualificationsin analyzingthenonunique

behavior in otherapplicationsaswell.

A. Ravindran. A computerroutinefor quadraticandlinearprogrammingproblems(algorithm

a431).CollectedAlgorithmsfromACM, 2285, 1972.

A. RavindranandH. K. Lee. Computerexperimentson quadraticprogrammingalgorithms.EuropeanJournalof OperationalResearch, 8(2), 166–174,1981.

Abstract. Comparesthe computationalperformanceof five quadraticprogrammingalgorithms.Thesein-

cludeWolfe’ssimplex method,Lemke’scomplementarypivot method,convex simplex methodandquadratic

differentialalgorithm.Executiontimeanditerationcountareusedasthemajorcriteriafor comparison.Since

Lemke’salgorithmout-performedall othermethodsin thestudy, adetailedstatisticalanalysiswasperformed

to determinetherelative importanceof problemparameterson theefficiency of Lemke’salgorithm.An anal-

ysis of varianceshowed that the numberof variables,the percentof positive linear termsin the objective,

thenumberof constraints,andtheir interactionswerethesignificantfactorsfor bothiterationcountandexe-

cutiontime.Finally, regressionequationsfor iterationcountandexecutiontime arederivedasa functionof

fifteenproblemparameters.

B. S. Razumikhin. Method of physicalmodelling in mathematicalprogrammingand eco-nomics. III. Solution algorithms for problemsof linear and quadraticprogrammingand for equilibrium problemsof exchangemodels. Automationand RemoteControl,33(6), 992–1001,1972.

Abstract. For pt. II seeAutom. andRemoteControl (USA), 33(4) pp. 628–39,(1972).Recurrencealgo-

rithms for numericalsolutionof linear andquadraticprogrammingproblemsandequilibrium problemsof

linearexchangemodelsaregiven.Thealgorithmsfollow from thephysicalpropertiesof thecorresponding

problemsandthemethodof redundantrelations.

B. D. ReddyandG. P. Mitchell. Theanalysisof elastic-plasticplates:aquadraticprogrammingproblemandits solutionby finite elements.ComputerMethodsin AppliedMechanicsandEngineering, 41(2), 237–248,1983.

Abstract. An extendedkinematicminimumprinciplein classicalplasticityis usedasthebasisfor thefinite

elementformulationof the rateproblemfor elastic-plasticplates.A simplealgorithmis usedto solve the

resultingquadraticprogrammingproblem.Thenumericalsolutionof theproblemis carriedout in two ways:

onemethodinvolvesloadstepsizeswhich arescaledsothatoneor moregausspointsjust becomeplastic,

andthe othermethodinvolves load stepsizeswhich arefixed onceandfor all at the outset.Examplesare

givenanddiscussed.

K. P. Reddy, A. K. Mittal, andS. K. Gupta. Bivalentquadraticprogrammingproblem-acom-putationalstudy. Opsearch, 21(3), 153–166,1984.


Abstract. Theproblemconsideredis theminimisationof a quadraticfunctionof n 0–1variables.Effective

boundingstrategiesaredevelopedandincorporatedin abranch-and-boundalgorithm.Thealgorithmis used

to identify the relatively difficult problems.A heuristicalgorithmis alsoproposedto solve difficult and/or

large sizeproblems.Computationalresultspertainingto the effectivenessof the proposedalgorithmsare

reported.

A. T. Redpath,B. L. Vickery, andD. H. Wright. A new techniquefor radiotherapy planningusingquadraticprogramming.Physicsin MedicineandBiology, N1(5), 781–791,1976.

Abstract. Previousattemptsat optimisationof radiotherapy planningaredescribedandcriticised,andcon-

siderationof theseattemptshasresultedin thedevelopmentof anew techniqueusingquadraticprogramming.

Uniformity of tumourdoseis selectedasthemostimportantfeatureof any plan,andthis is achievedby min-

imising thevarianceof thedoseto preselectedpointswithin thetumour. Thedoseto vulnerableregionscan

beconstrainednot to exceeda givenpercentageof themeantumourdose.Optimisationof field weightand

field typeis possible.Theoperationof thesystemis describedandsometypical resultsaregiven.

G. F. ReidandL. Hasdorff. Economicdispatchusingquadraticprogramming.IEEE Transac-tionson PowerApparatusandSystems, PAS-92(6), 2015–2023,1973.

Abstract. The economicdispatchproblemis formulatedasa quadraticprogrammingproblemandsolved

usingWolfe’s algorithm.The methodis capableof handlingboth equalityandinequalityconstraintson p,

q, andv andcansolve theloadflow aswell astheeconomicdispatchproblem.Thequadraticprogramming

algorithmdoesnot requirethe useof penaltyfactorsor the determinationof gradientstepsizewhich can

causeconvergencedifficulties.Convergencewasobtainedin threeiterationsfor all testsystemsconsidered

andsolutiontime is small enoughto allow the methodto be usedfor on-line dispatchingat practicaltime

intervals.Resultsarepresentedfor 5, 14,30,57,and118bustestsystems.

R. RhodeandR. Weber. Multiple objectivequadratic-linearprogramming.In J.P. Brans,ed.,‘OperationalResearch’81. Proceedingsof the Ninth IFORSInternationalConference.North-Holland,Amsterdam,Netherlands’,pp.405–420,1981.

Abstract. A vector optimizationproblemis formulatedand discussedwhich containsone quadraticand

several linear objective functions.It is shown that suchproblemsmay be solved easily in differentways.

Algorithmsfor thestrictly concave casearepresented.

F. Riciniello. A survey on themethodsof linearandquadraticprogramming.NoteRecensionie Notizie, 22(2–3),270–315,1973.

Abstract. Somemethodsof linear andquadraticprogrammingare reviewed. Attention is focusedon the

algorithmswhich have beenrecognizedasthemostsuitablefor computerimplementationbecauseof their

limited memoryoccupancy andlow sensitivity to roundoff errors.

N. L. Ricker. Useof quadraticprogrammingfor constrainedinternalmodelcontrol. IndustrialandEngineeringChemistry, ProcessDesignandDevelopment, 24(4), 925–936,1985.

Abstract. Absoluteboundson control actionandothercontrol systemconstraintscanbe handledconve-

niently througha combinationof quadratic-programmingandmultivariableInternalModel Control (IMC).

Threealternative simplex-type quadraticprogramming(QP) algorithmsare considered,and the resulting

IMC algorithmsareappliedto thecontrolof asimulatedmultiple-effect evaporationprocess.Overallcontrol

qualityis excellent;it degradesverylittle whentheprocessis operatednearaconstraintonthecontrolaction.

Calculationsrequiredfor anadjustmentof the controlactionareessentiallyinstantaneouson a PDP11/60

minicomputer. The resultsof the QPcomparisonsuggestthatcertainQPalgorithmscantake advantageof

thespecialstructureof theIMC QPproblem.

K. Rim, R. Chand,and E. J. Haug. Analysis of unbondedcontactproblemsby meansofquadraticprogramming.Journal of OptimizationTheoryandApplications, 20(2), 171–189,1976.


Abstract. Two-body, elastic,unbondedcontactproblemsare formulatedasquadraticprogrammingprob-

lems.Uniquenesstheoremsof quadraticprogrammingtheoryareappliedto show thatthesolutionof a con-

tactproblem,if oneexists,is uniqueandcanbereadilyfoundby themodifiedsimplex methodof quadratic

programming.A solution techniquethat is compatiblewith finite-elementmethodsis developed,so that

contactproblemswith complex boundaryconfigurationscanberoutinelysolved.A numberof classicaland

nonclassicalproblemsaresolved.Goodagreementis foundfor problemswith previously known solutions.

K. Ritter. Uber dasmaximum-problemfur nichtkonkavequadratischefuncktionen. Doctoral

dissertation,Albert-LudwigsUniversity, Frieberg, Germany, 1964.

K. Ritter. Stationarypointsof quadraticmaximumproblems.Zeitschrift fur Wahrscheinlichkeit-

stheorieundverwandteGebiete, 4, 149–158,1965.

K. Ritter. A methodfor solving maximumproblemswith a nonconcave quadraticobjective

function. Zeitschrift fur WahrscheinlichkeitstheorieundverwandteGebiete, 4, 340–351,

1966.

K. Ritter. A decompositionmethodfor structuredquadraticprogrammingproblems.Journal

of ComputerandSystemSciences, 1(3), 241–260,1967.

K. Ritter. On parametriclinearandquadraticprogrammingproblems.In R. W. Cottle,M. L.Kelmansonand B. Korte, eds,‘MathematicalProgramming.Proceedingsof the Inter-nationalCongresson MathematicalProgramming.North-Holland,Amsterdam,Nether-lands’,pp.307–335,1984.

Abstract. An algorithmis describedfor determiningtheoptimalsolutionof parametriclinearandquadratic

programmingproblemsas an explicit piecewise linear function of the parameter. Eachlinear function is

uniquelydeterminedby anappropriatesubsetof active constraints.For every critical valueof theparameter

anew subsethasto bedetermined.A simpleruleis givenfor addinganddeletingconstraintsfrom thissubset.

A. G. Robinson,N. Jiang,andC. S. Lerme. On thecontinuousquadraticknapsack-problem.

MathematicalProgramming, 55(1), 99–108,1992.

R.T. Rockafellar. Linear-quadraticprogrammingandoptimalcontrol.SIAMJournalonControlandOptimization, 25(3), 781–814,1987.

Abstract. A generalizedapproachis taken to linear andquadraticprogrammingin which dual aswell as

primal variablesmaybesubjectedto bounds,andconstraintsmayberepresentedthroughpenalties.Corre-

spondingproblemmodelsin optimal control relatedto continuous-timeprogrammingarethensetup and

theoremson duality and the existenceof solutionsarederived. Optimality conditionsareobtainedin the

form of aglobalsaddlepointpropertywhichdecomposesinto aninstantaneoussaddlepointconditiononthe

primalanddualcontrolvectorsat eachtime,alongwith anendpointcondition.

R.T. Rockafellar. Computationalschemesfor large-scaleproblemsin extendedlinear-quadraticprogramming.MathematicalProgramming, SeriesB, 48(3), 447–474,1990.

Abstract. Numericalapproachesaredevelopedfor solvinglarge-scaleproblemsof extendedlinear-quadratic

programmingthat exhibit Lagrangianseparabilityin both primal anddual variablessimultaneously. Such

problemsarekin to large-scalelinear complementaritymodelsasderived from applicationsof variational

inequalities,and they arisefrom generalmodelsin multistagestochasticprogramminganddiscrete-time

optimalcontrol.Becausetheirobjective functionsaremerelypiecewiselinear-quadratic,dueto thepresence

of penaltyterms,they do not fit a conventionalquadraticprogrammingframework. They have potentially

advantageousfeatures,however, whichsofarhavenotbeenexploitedin solutionprocedures.Thesefeatures

arelaid out andanalyzedfor their computationalpotential.In particular, a new classof algorithms,called


finite-envelopemethods,is describedthat doestake advantageof the structure.Suchmethodsreducethe

solutionof a high-dimensionalextendedlinear-quadraticprogramto thatof a sequenceof low-dimensional

ordinaryquadraticprograms.

R. T. Rockafellar. Large-scaleextendedlinear-quadraticprogrammingand multistageopti-

mization. In ‘Advancesin NumericalPartial Dif ferentialEquationsandOptimization:

Proceedingsof theFifth Mexico-UnitedStatesWorkshop’,Vol. 2, pp.247–261,1991.

R. T. Rockafellar. A simplex-active-setalgorithmfor piecewise quadraticprogramming. In

D.-Z. Du andJ.Sun,eds,‘Advancesin OptimizationandApproximation’,pp.275–292,

Kluwer AcademicPublishers,Dordrecht,TheNetherlands,1994.

R.T. RockafellarandR.J.-B.Wets.A dualsolutionprocedurefor quadraticstochasticprograms

with simplerecourse.TechnicalReportCP-83-017,InternationalInstitute for Applied

SystemsAnalysis,Vienna,Austria,1983.

R. T. RockafellarandR. J. B. Wets. Linear-quadraticprogramming-problemswith stochastic

penalties—thefinite generationalgorithm. Lecture Notesin Control and Information

Sciences, 81, 545–560,1986.

J.E.Rogers,P. T. Boggs,andP. D. Domich.A predictor-corrector-o3dformulationfor quadratic

programming.TechnicalReportNISTIR 5020,NationalInstituteof StandardsandTech-

nology, USA, 1994.

J. S. Rogersand M. Rolko. A quadraticprogrammingmodel for planninggenerationandinter-utility transmission.InternationalJournalof ElectricalPowerandEnergySystems,14(1), 18–22,1992.

Abstract. A new quadraticprogrammingapproachto planningthecoordinatedexpansionof generationand

transmissionbetweentwo electricutilities is described.Earliermethodsusea stepfunctionto approximate

theloaddurationcurve.Theresultinglinearprogrammingmodelswork well if theshapeof theloadcurve is

not changedby the installationof equipment.However, expansionof inter-utility transmissionchangesthe

shapeof theloadcurve whichmustbesuppliedby thegenerationof eachutility, andsoamoreflexible type

of approximationis needed.Theauthorsusea piecewise linearapproximationto theloadcurve to construct

aquadraticprogrammingmodel.Realsystemdataareusedto comparetheresultsfrom thenew modelwith

thosefrom theearliertypeandwith resultsfrom ananalyticalmodelthatgivesanexactsolution.

J. B. Rosenand P. M. Pardalos. Global minimization of large-scaleconstrainedconcave

quadraticproblemsby separableprogramming.MathematicalProgramming, 34(2),163–

174,1986.

V. RuggieroandL. Zanni. A modifiedprojectionalgorithmfor largestrictly-convex quadraticprograms.Journalof OptimizationTheoryandApplications, 104(2), 281–299,2000.

Abstract. In this paper, we proposeamodifiedprojection-typemethodfor solvingstrictly-convex quadratic

programs.This iterative schemerequiresessentiallythe solutionof an easyquadraticprogrammingsub-

problemanda matrix-vector multiplication at eachiteration.The main featureof the methodconsistsin

updatingtheHessianmatrix of thesubproblemsby a convenientscalingparameter. Theconvergenceof the

schemeis obtainedby introducinga correctionformulafor thesolutionof thesubproblemsandvery weak

conditionson the scalingparameter. A practicalnonexpensive updatingrule for the scalingparameteris

suggested.Theresultsof numericalexperimentationenablethisapproachto becomparedwith someclassical

projection-typemethodsandits effectivenessasa solver of largeandvery sparsequadraticprogramsto be

evaluated.


M. H. Rusin. A revisedSimplex methodfor quadraticprogramming. SIAM Journal on Ap-plied Mathematics, 20(2), 143–160,1971.Seealso,Bulletin of theOperationsResearchSocietyof America,volume18,pageB17,1970.

Abstract. A computationalmethodfor solvingquadraticprogrammingproblemsis presentedwhichreduces

to therevisedsimplex methodfor linearprogrammingwhentheobjective functionis linear. Thebasismatrix

which is maintainedis symmetricand may vary in size from iteration to iteration.Comparisonson test

problemsindicatethatthemethodis moreefficient thanotheravailablemethods.

R. S.Sacher. A decompositionalgorithmfor quadraticprogramming.MathematicalProgram-ming, 18(1), 16–30,1980.

Abstract. A decompositionalgorithmusing,Lemke’s methodis proposedfor thesolutionof quadraticpro-

grammingproblemshaving possiblyunboundedfeasibleregions.The feasibleregion for eachmasterpro-

gramis ageneralizedsimplex of minimal size.TThispropertyis maintainedby adroppingprocedurewhich

doesnot affect thefinitenessof the convergence.Thedetailsof thematrix transformationsassociatedwith

anefficient implementationof thealgorithmaregiven.Encouragingpreliminarycomputationalexperience

is presented.

H. D. Sahinoglou,P. M. Pardalos,andI. M. Roussos.Characterizationsof globalminima in

nonconvex quadraticprogramming.In ‘Proceedingsof the19thAnnualPittsburh Con-

ferenceon Modelling andSimulationModelingandSimulation’,pp.1823–1828,1988.

T. Satoh,T. Okita,andM. Itoh. Fastcapacitymethodfor solvingconvex quadraticprogrammingproblem. Transactionsof theSocietyof InstrumentandControl Engineers, 32(4), 587–595,1996.

Abstract. The quadraticprogramming(QP) problemis a naturalgeneralizationof well-known linear pro-

gramming(LP) problem.TheQPproblemis formulatedasaproblemto minimizea linearobjective function

subjectto a systemof linear equalitiesand inequalities.The continuationmethodis a powerful tool for

solvinga systemof nonlinearequations.In thecontinuationmethod,thecontinuationparameterbetais in-

troducedinto the original problemP anda parameterizedproblemP( beta) is constructed.Assumingthat

the optimal solutionto P(P),for beta=0, canbe trivially or easily found, andthat the optimal solutionto

P β , for β 1, will bethedesiredsolutionto theoriginalproblemP, a trajectoryof thesolutionto P β is

followedby increasingthevalueof betafrom 0 to 1. In this paper, basedon thecontinuationmethod,a path

following methodcalledthecapacitymethodis constructedfor solvingtheQPproblem.Thebasicideaof the

capacitymethodis to follow a trajectoryof theKuhn-Tucker pointwhenthevalueof betais increasedfrom

0 to 1. It is shown thatthetrajectoryis expressedasapiecewise-linearfunctionof beta. Thereforetheopti-

mal solutioncanbefoundvery efficiently by following thepiecewise-lineartrajectory. In this paper, firstly,

theassumptionthat the right-hand-sidevectorof the constraintequationsmustbe positive, which restricts

theapplicabilityof thecapacitymethod,is droppedanda methodfor transformingany QPproblemto the

standardform of thecapacitymethodis presented.Secondly, thebasismatrix of QP is factorizedby using

anLDLT factorization,andthefactorizationis efficiently updatedby usingmatrix modificationtechniques.

Thentheproposedmethodis appliedto sometestproblems,andnumericalresultsindicatetheeffectiveness

of themethod.

M. A. Saunders.Stablereductionto KKT systemsin barriermethodsfor linearandquadratic

programming.TechnicalReportSOL96-3,Departmentof OperationsResearch,Stanford

University, California,USA, 1996.

C. Schmidand L. T. Biegler. Quadraticprogrammingmethodsfor reducedHessianSQP.ComputersandChemicalEngineering, 18(9), 817–832,1994.

Abstract. ReducedHessiansuccessive quadraticprogramming(SQP)is well suitedfor thesolutionof large-

scaleprocessoptimizationproblemswith many variablesandconstraintsbut only few degreesof freedom.


Thepreprocessingphasedeterminesaninitial consistentpoint, to selecta nonsingularsetof basisvariables

andto identify linear dependency amongthe equality constraints.Fourer’s (1985,1989)piecewise-linear

simplex techniquesallow us to solve a smallerinitialization problemmoreefficiently thanis possiblewith

standardsimplex techniques.Wepresentanew QPsolver, QPKWIK, basedona dualalgorithmwhichonly

requirestheinverseCholesky factorof theHessianmatrixto besupplied.Theresultingsolutiontechniquefor

theQPsubproblemis O n2 with respectto thedegreesof freedomof theproblem.Further, theunconstrained

optimumis dualfeasible,whichprecludestheneedfor phaseI calculations,andmakesthismethodsuperior

even for problemswith few degreesof freedom.QPKWIK hasbeenimplementedso as to enhancethe

efficiency of theactive setidentificationandis alsoableto determinea searchdirectionwheninfeasibleQP

subproblemsareencounteredby relaxingtheequalityconstraintswithoutviolating thesimpleboundsonthe

variables.Finally, numericalresultsareincludedto illustratetheadvantagesof theproposedtechniquesand

to assesstheoverall performanceof thereducedHessianmethod.Thisapproachis especiallywell-suitedfor

processandreal-timeoptimizationproblems.We demonstratethis on several distillation andfractionation

problems.

U. Schmitz,A. Donati,T. L. James,N. B. Ulyanov, andL. J.Yao. Smallstructuralensembles

for a 17-nucleotidemimic of the tRNA TYC-loop via fitting of dipolar relaxationrates

with thequadraticprogrammingalgorithm.Biopolymers, 46, 329–342,1998.

I. E. SchochetmanandR. L. Smith. Solutionexistencein infinite quadraticprogramming.InA. V. Fiacco,ed., ‘Lecture Notesin PureandApplied Mathematics’,Vol. 195.MarcelDekker, 1998.

Abstract. We consideran infinite quadraticprogrammingproblemwith positive semi-definitequadratic

costs,equality constraintsand unboundedvariables.Sufficient conditionsare given for thereto exist an

optimalsolution.Specifically, werequirethat(1) thecostoperatorbestrictly positivedefinitewhenrestricted

to theorthogonalcomplementof its kernel,and(2) theconstraintoperatorhaveclosedrangewhenrestricted

to thekernelof thecostoperator. Condition(1) is shown to beequivalentto thespectrumof therestrictedcost

operatorbeingboundedaway from zero.Similarly, condition(2) is equivalent to theminimummodulusof

therestrictedconstraintoperatorbeingpositive. In thepresenceof separability, wegiveasufficientcondition

for (2) to hold in termsof finite dimensionaltruncationsof the restrictedconstraintoperator. We apply

our resultsto a broadclassof infinite horizonoptimizationproblems.In this setting,thefinite dimensional

truncationscan be consideredto be finite dimensionalapproximationsto our problemwhoselimit, in a

somewhatformalsense,is our infinite dimensionalproblem.Eachof theseapproximationshasproperties(1)

and(2) by virtue of their finite-dimensionality, i.e., eachadmitsanoptimalsolution.However, our infinite

dimensionalproblemmaynot.Thus,we give sufficient conditionsfor our problemto alsoadmitanoptimal

solution.Finally, we illustrate this applicationin the caseof an infinite horizonLQ regulator problem(a

productionplanningproblem).

I. E. Schochetman,R. L. Smith,andS. K. Tsui. Solutionexistencefor time-varying infinite-horizon quadraticprogramming. Journal of MathematicalAnalysisand Applications,195(1), 135–147,1995.

Abstract. Weconsidera general,time-varying, infinite horizon,purequadraticprogrammingproblemwith

positive-definitecostmatricesandunboundeddecisionvariables.Sufficientconditionsareprovidedfor there

to exist anoptimalsolution.Specifically, we show that if theeigenvaluesof the costmatricesarebounded

away from zero, thena (unique)optimal solutionexists. We apply our resultsto the infinite horizonLQ

tracker problemin optimalcontroltheory.

M. Schocm.Uberdie aquivalenzderallgeminenquadratischeroptimierungsaufgabezu einer

parametrischenkomplimentaritarenoptimierungsaufgabe.Mathematische Operations-

forschungundStatistik,SerieOptimization, 15, 211–216,1984.


L. Schrage.Integer, andQuadratic Programmingwith LINDO. TheScientificPress,third edn,

1986.

P. Scobey andD. G.Kabe.Directsolutionsto somelinearandquadraticprogrammingproblems.IndustrialMathematics, 29(2), 59–75,1979.

Abstract. Someresultsof univariatenormallinear regressiontheoryareusedto presentdirectsolutionsto

somelinearandquadraticprogrammingproblems.Thelinearprogrammingproblemsincludetheknapsack

problem,theextremepointlinearprogrammingproblem,andtheinterval linearprogrammingproblem.Some

inaccuraciesin interval linearprogrammingproblemsandquadraticprogrammingproblemsarepointedout.

P. D. Scott.A quadraticprogrammingdualalgorithmfor minimaxcontrol. IEEE Transactionson AutomaticControl, AC-20(3), 434–435,1975.

Abstract. Theoptimalcontrolproblemwith peakweightingon trajectoryerror is relatedto quadraticpro-

grammingthroughaduality transformation.A seriesof finite dimensionalquadraticprogramsyieldsfinitely

convergentsolutionsfrom whichtheoptimalcontrolmayberecovered.Eachprogramyieldsupperandlower

boundson theoptimalcost.

J. Semple. Infinite positive-definitequadraticprogrammingin a Hilbert space. Journal ofOptimizationTheoryandApplications, 88(3), 743–749,1996.

Abstract. This note generalizesthe resultsof Benson,Smith, Schochetman,and Bean(1995) regarding

the minimizationof a positive-definitefunctionalover the countableintersectionof closedconvex setsin

a Hilbert space.A finite approximatingsubproblemfor the generalcaseis shown to have the samestrong

convergencepropertiesof the earlier work without any of the specializedstructuresimposedtherein.In

particular, the currentdevelopmentdoesnot rely on any propertiesof L2 anddoesnot requirethe Hilbert

spaceto beseparable.

M. SernaandF. Xhafa. On theparallelapproximabilityof someclassesof quadraticprogram-ming. TechnicalReportR95-58,Departamentde Llenguatgesi SistemesInformatics,UniversitatPolitecnicadeCatalunya,Spain,1995.

Abstract. In thispaperweanalyzetheparallelapproximabilityof two specialclassesof QuadraticProgram-

ming. First, we considerConvex QuadraticProgramming. We show that the problemof Approximating

Convex QuadraticProgrammingis P-complete.We alsoconsidertwo approximationproblemsrelatedto

it, SolutionApproximationandValueApproximationandshow bothof thesecannotbesolved in NC, un-

lessP NC. Secondly, on thepositive side,we show thatwe have anNC ApproximationSchemefor those

instancesof QuadraticProgrammingthatare”smooth” and”positive.” Thenwe show how to extendthere-

sult for positive instancesof boundeddegreeSmoothIntegerProgrammingproblems.Finally, we formulate

severalcombinatorialproblemsaspositive QP(or positive IntegerPrograms)in packing/covering form and

show thatthepresentedtechniquescanbeusedto obtainNC ApproximationSchemesfor ”dense”instances

of suchproblems.

M. SernaandF. Xhafa.Theparallelapproximabilityof asubclassof quadraticprogramming.In‘Proceedings.1997InternationalConferenceon ParallelandDistributedSystems.IEEEComput.Soc,Los Alamitos,CA, USA’, pp.474–481,1997.

Abstract. In this paperwe dealwith theparallelapproximabilityof a specialclassof QuadraticProgram-

ming (QP),calledSmoothPositive QuadraticProgramming.This subclassof QP is obtainedby imposing

restrictionson thecoefficientsof theQPinstance.TheSmoothnessconditionrestrictsthemagnitudesof the

coefficientswhile thepositivenessrequiresthatall thecoefficientsbenon-negative. Interestingly, evenwith

theserestrictionsseveral combinatorialproblemscanbe modeledby SmoothQP. We show NC Approxi-

mationSchemesfor the instancesof SmoothPositive QP. This is doneby reducingthe instanceof QP to

an instanceof Positive Linear Programming,finding in NC an approximatefractionalsolution to the ob-

tainedprogram,andthenroundingthefractionalsolutionto anintegerapproximatesolutionfor theoriginal


problem.Thenwe show how to extendthe result for positive instancesof boundeddegreeto SmoothIn-

teger Programmingproblems.Finally, we formulateseveral importantcombinatorialproblemsasPositive

QuadraticPrograms(or Positive Integer Programs)in packing/covering form andshow that the techniques

presentedcanbeusedto obtainNC ApproximationSchemesfor ”dense”instancesof suchproblems.

D. G. Shankland.Quadraticprogrammingusinggeneralizedinverses.Technicalreport,AirForceInst.Tech, Wright PattersonAFB, OH, USA, 1975.

Abstract. A methodfor computingtheoptimumvalueof aquadraticfunctionalsubjectto linearinequalities

rapidly ascertainswhich, if any, of theinequalitiesarebindingat theoptimumpoint.Themethodresembles

thatof H. Theil andC. VandePanne,but nocombinatorialanalysisneedbeperformedto isolatethebinding

constraints.All violatedconstraintsareimposedasequalities,andthosewith positiveLagrangianmultipliers

areretained.Contradictoryequalitiesareautomaticallyresolved by theuseof thegeneralizedinverse.The

methodappearsmostusefulin systemswith largenumbersof variablesandconstraints.

D. G. Shankland. A numericallyefficient procedurefor the Theil-Van de Pannequadraticprogrammingmethod. Journal of OptimizationTheoryand Applications, 31(1), 117–123,1980.

Abstract. A procedureis given for implementingthe Theil-Van de Pannealgorithm, which utilizes the

Cholesky decompositionto reducethecomputationsinvolvedin matrix inversions.

J. K. SharmaandS. Kanti. Indefinitequadraticprogrammingand transportationtechnique.Indian Journalof PureandAppliedMathematics, 8(9), 1029–1031,1977.

Abstract. A technique,similar to the transportationtechniquein linearprogramming,is describedto min-

imize a locally indefinitequadraticfunction.Theproblemis attacked directly startingfrom a basicfeasible

solutionandthe conditionsunderwhich the solutioncanbe improved have beenindicated.Conditionsfor

localoptimality have beenobtained.

J.F. SharpandK. M. Suk. A quadraticprogrammingplanningmodel.EngineeringEconomist,20(1), 71–77,1974.

Abstract. Variousauthorshave suggestedusinga quadraticprogrammingmodelof the firm. A quadratic

programmingapproachhasbeenusedto study the optimal useof milk in the Netherlands.Thereseems

to be a lack of otheractualapplicationsreportedin the literature.The authorshave successfullyapplieda

quadraticprogrammingmarketingmodelto anindustrialfirm. It hasseveralfeaturesnot includedin themilk

application.Theseinclude:morethanonemarket for thesameproduct,limits on therangeof applicability,

andlegal restrictions.A simplifiedversionof thismodelis givenbelow.

R. Shen.Reactive power optimizationin power systemquadraticprogrammingmethod.Pro-ceedingsof theChineseSocietyof ElectricalEngineering., 6(5), 40–48,1986.

Abstract. A quadraticprogrammingmethodof comprehensive reactivepoweroptimizationin powersystems

is proposed.With realpower lossastheobjective functionandoperatingvariablesasconstraints,a mathe-

maticalmodelfor comprehensive reactive power optimizationis setupby sensitivity relationsbetweencon-

trol andstatevariablesin thepower system.Theoptimal locationsandcapabilitiesof compensatedreactive

power, optimal terminalvoltagesof generatorsandoptimal ratiosof on-loadtransformertap-changersare

determinedby optimization.Basedoneconomicanalysis,amathematicalmodelof multiobjective program-

ming for comprehensive reactive power optimizationis setup with realpower lossandtotal compensation

capabilityasobjective functions.Threesystemsareoptimizedby thesemethodsandresultsaregiven and

comparedwith thoseof linearprogrammingmethods.

H. D. SheraliandA. Alameddine.Reformulation-linearizationtechniquefor bilinearprogram-

ming problems.Journalof GlobalOptimization, 2(4), 379–410,1992.


H. D. SheraliandC. H. Tuncbilek.A reformulation-convexificationapproachfor solvingnon-convex quadraticprogrammingproblems. Journal of Global Optimization, 7(1), 1–31,1995.

Abstract. In this paper, we considerthe classof linearly constrainednonconvex quadraticprogramming

problems,andpresentanew approachbasedonanovel Reformulation-Linearization/Convexification Tech-

nique.In this approach,a tight linear (or convex) programmingrelaxation,or outer- approximationto the

convex envelopeof theobjective functionover theconstrainedregion,is constructedfor theproblemby gen-

eratingnew constraintsthroughtheprocessof employing suitableproductsof constraintsandusingvariable

redefinitions.Varioussuchrelaxationsareconsideredandanalyzed,includingonesthat retainsomeuseful

nonlinearrelationships.Efficientsolutiontechniquesarethenexploredfor solvingtheserelaxationsin order

to derive lower andupperboundson theproblem,andappropriatebranching/partitioning strategiesareused

in concertwith theseboundingtechniquesto derive a convergentalgorithm.Computationalresultsarepre-

sentedon a setof testproblemsfrom the literatureto demonstratethe efficiency of the approach.(Oneof

thesetestproblemshadnot previously beensolved to optimality.) It is shown that for many problems,the

initial relaxationitself producesanoptimalsolution.

W. M. Shi. Comparisonof optimizationwith linearandquadraticprogrammingin 2nd-order

design.In ‘Collectionof 8thInternationalSymposiumonGeodeticComputation’,Vol. 9,

pp.157–163,1991.

J.K. Shim. A survey of quadraticprogrammingapplicationsto businessandeconomics.Inter-nationalJournalof SystemsScience, 14(1), 105–115,1983.

Abstract. Oneof theshortcomingsof linearprogramminglies in its linearity assumption,primarily in the

objective function.This compelsoneto work with a constantmarginal rateof substitutionandconstantre-

turn to scale.However, this assumptionis at variancewith theeconomists’preferencepostulates.Thepaper

presentsa survey of quadraticprogrammingpracticesin corporateandeconomicplanningformulations.It

examinesa variety of applicationsof quadraticprogramming-portfolioselection,monopolists’profit max-

imization, inequalityconstrainedleast-squaresestimation,spatialequilibrium analysis,goal programming

with quadraticpreferences,andoptimaldecisionrules.

Y. Shimazu,M. Fukushima,andT. Ibaraki. A successive over-relaxationmethodfor quadraticprogrammingproblemswith interval constraints. Journal of the OperationsResearchSocietyof Japan, 36(2), 73–89,1993.

Abstract. Hildreth’salgorithm(1957)isaclassicaliterativemethodfor solvingstrictly convex quadraticpro-

grammingproblems,whichusestherowsof constraintmatrixjustoneatatime.Thisalgorithmis particularly

suitedto largeandsparseproblems,becauseit actsuponthegivenproblemdatadirectly andthecoefficient

matrix is never modifiedin thecourseof the iterations.Theoriginal Hildreth’s algorithmis mathematically

equivalentto Gauss-Seidelmethodappliedto thedualof thegivenquadraticprogrammingproblem.In this

paper, we proposea successive overrelaxationmodificationof Hildreth’s algorithmfor solvinginterval con-

strainedquadraticprogrammingproblems.We prove globalconvergenceof thealgorithmandshow thatthe

rateof convergenceis linear. Computationalresultsarealsopresentedto demonstratetheeffectivenessof the

algorithm.

R. I. Shrager. Quadraticprogrammingfor nonlinearregression.Communicationsof theACM,15(1), 41–45,1972.Seealso,CollectedAlgorithmsfromACM, 2395.

Abstract. A quadraticprogrammingalgorithmis describedfor usewith themagnifieddiagonalmethodof

nonlinearregressionwith linearconstraints.Theregressionmethodis publishedin JACM, July 1970.

V. Sima.Algorithm podqp—Solvingpositivedefinitequadraticprogrammingproblems.Stud-iesin InformaticsandControl, 1(1), 1992.


Abstract. An efficientandnumericallystabledualalgorithm-PODQP-for solvingpositivedefinitequadratic

programmingproblemsis briefly described.Both inequalityandconstraintsaredealtwiyh. The algorithm

employs anactive setstrategy andcanexploit a priori informationaboutan initial active set.In particular,

the algorithm is strongly recommendedfor solving nonlinearlyconstrainedoptimizationproblemsusing

projectedLagrangiantechniques.Orthogonaltransformationsareusedfor updatingthematricesandmatrix

factorizationsaftereachchangeperformedin theactive set.

S. Simunovic and S. Saigal. Frictionlesscontactwith BEM using quadraticprogramming.Journalof EngineeringMechanics-ASCE, 118(9), 1876–1891,1992.

Abstract. The contactsurface,with its accompanying load transfer, may well constitutethe critical factor

in a structuralmember. Thus it is essentialto be able to performcontactstressanalysisof a component

accuratelyandefficiently. Considerableresearcheffort is representedin the literaturefor contactanalysis

usingfinite elements.To obtain reliable resultsin the contactzone,it is necessaryto provide a very fine

discretizationin thatzone.Many distinctcontactzonesmayexist. which may force theentiredomain,and

not just the contactzones,to be discretizedfinely. This generallyleadsto anexcessive numberof degrees

of freedom(dof), resultingin anuneconomical,andsometimesintractable,analysis.Theboundaryelement

method(BEM), whichdealswith thediscretizationof only theboundaryof thestructurebeinganalyzed,may

beusedto circumventthesedifficultiesandprovide accurate,economicalresults.While afinediscretization

of the contactzoneis still unavoidable,the BEM leadsto a smallernumberof dof’s becausethe rest of

the boundaryneednot have a fine mesh.The problemof frictionlesscontactbetweenan elasticbody and

a rigid surfaceis formulatedasan optimizationproblem.Threedistinct functionsaredefinedin termsof

theunknown variables(displacementsandtractions)correspondingto thecontactsurface,andexpressedas

quadraticobjective functionsthatareto beminimized.Thesolutionis obtainedusingthestandardquadratic

programmingtechniquesof optimization.A numberof exampleproblemswith straightandcurvedcontact

boundariesweresolved.Thepresentformulationswerevalidatedthroughcomparisonof the testproblems

with existing alternative solutions.

S. Simunovic andS. Saigal. Frictionlesscontactwith BEM usingquadraticprogramming—

closure.Journalof EngineeringMechanics-ASCE, 119(12),2540,1993.

S.Simunovic andS.Saigal.Frictionalcontactformulationusingquadraticprogramming.Com-putationalMechanics, 15(2), 173–187,1994.

Abstract. A new solutionprocedurefor contactproblemsin elasticitywith prescribednormaltractionson

contactsurfacehasbeenproposedin thispaper. Theprocedureis basedontheboundaryelementmethodand

quadraticprogramming.It is next usedin a two stepsolutionalgorithmfor theanalysisof contactproblems

with friction. Severalnumericalexamplesarepresentedandcomparedwith resultsobtainedusingalternative

solutionmethods.

S. Simunovic andS. Saigal. Quadratic-programmingcontactformulation for elasticbodiesusingboundary-elementmethod.AIAA Journal, 33(2), 325–331,1995.

Abstract. A methodfor theanalysisof contactof deformablebodiesbasedontheboundaryelementmethod

(BEM) hasbeenpresentedin this paper. The contactproblemis statedin the form of a convex quadratic

programming(QP) problemwritten in termsof the contacttractionson the contactsurface.A strategy for

theincorporationof theBEM contactanalysisinto modelswhosedomainmaybediscretizedusingthefinite

elementmethod(FEM) hasbeeninvestigated.A discussionconcerningthemeritsof theproposedapproach

is providedandseveralexamplesarepresentedto illustratethevalidity of themethod.

F. W. Sinden.A geometricrepresentationfor pairsof dualquadraticor linearprograms.Journal

of MathematicalAnalysisandApplications, 5, 378–402,1963.

J. Skorin-Kapov. On stronglypolynomialalgorithmsfor someclassesof quadraticprogram-ming problems.MathematicalCommunications, 2(2), 95–105,1997.


Abstract. In thispaperwesurvey someresultsconcerningpolynomialand/orstronglypolynomialsolvability

of someclassesof quadraticprogrammingproblems.Thediscussiononpolynomialsolvability of continuous

convex quadraticprogrammingis followedby acoupleof modelsfor quadraticintegerprogrammingwhich,

dueto their specialstructure,allow polynomial (or even stronglypolynomial)solvability. The theoretical

merit of thoseresultsstemsfrom the fact that a running time (i.e. the numberof elementaryarithmetic

operations)of astronglypolynomialalgorithmis independentof theinputsizeof theproblem.

M. Skutella.Convex quadraticprogrammingrelaxationsfor network schedulingproblems.In

J.Nesetril,ed.,‘Algorithms—ESA’99’, Vol. 1643of LectureNotesin ComputerScience,

pp.127–138,SpringerVerlag,Heidelberg,Berlin, New York, 1999.

L. H. Smith. A noteon quadraticprogrammingin activationanalysis.OperationsResearch,18(2), 290–299,1970. Seealso,Bulletin de la SocieteRoyaleBelge desElectriciens,Volume16,page62,1968.

Abstract. Variousstatisticaltechniqueshave beenemployed in ’activation analysis’in orderto provide a

bettermeansof estimatingtheamountsof variouspurechemicalelementscontainedin anunknown mixture.

In particular, the methodof leastsquareshasbeenemployed extensively. However, for the mostpart, the

usualleastsquaresapplicationsin activationanalysishaveutilized theordinarymatrixmodelY Xbeta e,

underthe ’error’ assumptions(a) zeromeans,(b) variancesproportionalto Y, and(c) zerocovariances.In

additionto thefact thatassumptions(b) and(c) may leadto erroneousresults,theusualapplicationsallow

only pointestimation,with noprovision for confidenceintervalsandtestsfor modelgoodnessof fit. Further,

theusualapplicationsfail to eliminatethedrawbackthatnegative coefficientsaresometimesobtained.The

presentpapersetsforth aniterative quadraticprogrammingestimationprocedurethatnotonly eliminatesthe

necessityfor assumptions(b) and(c), but alsoalleviatestheotherabove-mentioneddifficulties.

J. E. Sohl. An applicationof quadraticprogrammingto the deregulationof naturalgas. In‘Proceedingsof the16thAnnualMeetingof theAmericanInstitutefor DecisionSciences.AmericanInst.DecisionSci,Atlanta,GA, USA’, pp.686–689,1984.

Abstract. Examinesthe issueof naturalgasderegulationwithin the context of the linear complementarity

programming(LCP) format. The LCP model forecastsUnited Statesoil, naturalgas,andcoal pricesand

quantitiesfor boththedemandandthesupplyside.Forecastsarepresentedonaregionalizedlevel. Validation

of theLCPmodelis achievedby forecastinghistoricalpricesandquantities.In addition,themodelexamines

severalscenariosconcerningthevariouspolicy optionsfor thedecontrolof naturalgas.

V. N. Solov’ev. Algorithmsof solutionof problemsof quadraticprogrammingandof optimalguaranteedestimation.AutomationandRemoteControl, 51(9), 1213–1218,1990.

Abstract. An algorithmof solutionis proposedfor aquadraticprogrammingproblemwith aStieltjesmatrix

that is often encounteredafter inverting a matrix with nonnegative elements.This algorithmis appliedto

optimalguaranteedestimation.

J.Sommerschuh.Propertiesof thegeneralquadraticoptimizationproblemandthecorrespond-ing linearcomplementarityproblem.Optimization, 18(1), 31–39,1987.

Abstract. Thequadraticnonconvex functionψ x. 1 2xTCx rTx (C symmetric,arbitrary)is considered

ontheconvex polyhedralsetG x IRn 2Gx g , andneighbourhoodsof localminimaaredescribed.Using

thecorrespondingKuhn-Tuckerconditionstheauthorfoundacutexcludinganeighbourhoodof apreviously

determinedlocal minimum.

C. B. SomuahandF. C. Schweppe.Minimum frequency constrainedgenerationmargin alloca-tion usingquadraticprogramming.InternationalJournalof ElectricalPowerandEnergySystems, 9(2), 105–112,1987.


Abstract. Theproblemof allocatingthe total systemmargin amongall thegeneratorsin a power systems,

taking into considerationthe maximumpost-disturbancefrequency deviation, is formulatedasa quadratic

programmingproblem.Theoptimizationproblemusesthesensitivity of thefrequency deviation to changes

in eachgenerator’s margin in the allocationof total systemmargin. The quadraticprogrammingproblem

is solved via the Dantzig-Wolfe decompositiontechniqueusing a sequenceof linear programmingsub-

problems.A significant improvementis found in the post-disturbanceperformanceof the power system

in comparisonwith conventionalmargin allocation.Numericalexampleson two testsystemsareincluded.

Resultsof sensitivity studiesrelatingchangesin power plant margin andsizeof postulateddisturbanceto

changesin systemfrequency deviation andgenerationcostarealsodiscussed.

P. Spellucci. Numericalexperimentswith modernmethodsfor large scaleQP problems. In

‘RecentAdvancesin Optimization’,Vol. 452of LectureNotesin EconomicsandMathe-

maticalSystems, 1997.

F. R. Spena. A quadraticprogrammingapproachto the collapsefor pure torsion of beamswith thin-walled multiply-connectedsection. Journal of Informationand OptimizationSciences, 12(3), 419–429,1991.

Abstract. Referringto a thin-walled, homogeneous,isotropic, perfectly, plastic prismaticalbeamwith a

multiply-connectedcross-section,thecollapsevaluefor thetorquefor anequilibriumstressstateof puretor-

sionis determined.It is shown thatthemathematicalformulationleadsto aquadraticprogrammingproblem

for which the objective function is representedby the collapsetorque.It is alsoproved that the objective

function to bemaximizedis strictly concave, suchthat theoptimal solutionof the problemexists andit is

unique.

P. S. J. Spencer, L. Kersley, andS. E. Pryse. A new solution to the problemof ionospheric

tomographyusingquadraticprogramming.RadioScience, 33, 607–616,1998.

G. E. StaatsandJ. P. Dittman. Surrogatequadraticprogramming.Bulletin of theOperationsResearch Societyof America, 20, B–140,1972.

Abstract. In this paper, a highly efficient algorithmfor solving quadraticprogrammingproblemsis pre-

sented.The algorithm is basedon the resultsobtainedwhen the constraintset is replacedby a surrogate

constraint.Kuhn-Tucker Conditionsarethenappliedto obtaineitheranunconstrainedoptimizationproblem

or anequalityconstrainedproblem.Thisresultsin theneedto solveonly asystemof linearequationsin order

to obtainthesolutionto thesurrogatedproblem.An iterative procedureis given for obtaininga surrogated

problemwhichhasthesamesolutionastheoriginalquadraticprogrammingproblem.

A. Stachurski.Monotonesequencesof feasiblesolutionsfor quadratic-programmingproblems

with M-matricesandboxconstraints.LectureNotesin Control andInformationSciences,

84, 896–902,1986.

A. Stachurski. Analysis of the Newton projectionmethodin applicationto quadraticpro-grammingproblemswith M-matricesandbox constraints.ArchiwumAutomatykii Tele-mechanika, 34(1–2),155–164,1989.

Abstract. This paperis devotedto the analysisof the Newton projectionmethod,whenthis is appliedto

solveaspecialQPproblemwith anM-matrixastheHessianandsimpleboxconstraints.Theproblemarises

astheresultof anapproximationof a partialDirichlet problemwith obstacleby meansof thefinite element

method.It is proved that the propertiesof the Hessian(beinganM-matrix) imply that if the upperbounds

upon variablesdoesnot appearin the problemand the startingpoint is equal to the lower boundupon

variablesthenthenumberof iterationsof theNewtonprojectionmethoddoesnotexceedn (wheren denotes

the size of the problem); that if the upperboundsdo not exist, and the startingpoint is feasible,and ε0

(parameterusedin thedefinitionof theworking setof active constraints)is forcedto beequalto 0 in step


1, thenthenumberof iterationsis not greaterthann 1; andthat if thereexist lower andupperboundson

theproblemvariablesandif someadditionalrequirementsareintroducedinto thealgorithmthentheNewton

projectionmethodgeneratesthesamesequenceof pointsasthatof J. S. Pang(1976).Theanalysiscarried

out indicatessomecorollariesallowing a simplificationof theformulationof theNewton projectionmethod

for theconsideredclassof problems.

A. Stachurski.An equivalencebetweentwo algorithmsfor a classof quadraticprogrammingproblemswith M-matrices.Optimization, 21(6), 871–878,1990.

Abstract. Thebehaviour of thegradientprojectionmethodfor thequadraticprogrammingwith M-matrices

and lower boundson variablesis analysed.It is shown that the Chandrasekaranmethodfor solving such

problemsis simply a realizationof theNewton projectionmethodif for the latteronethestartingpoint has

componentsequalto lower boundsonvariables.

V. I. Starostenko. Constantalgorithmsof quadraticprogrammingandsolutionof the inverseproblemsof gravimetry relative to densities.Geofizicheskii-Sbornik., 64, 52–57,1975.

Abstract. Theproblemsof quadraticandlinearprogrammingareformulatedandtheir solutionsby means

of quadraticprogrammingalgorithmsregulatingaccordingto A. N. Tikhonov arepresented.

F. SteinerandL. Zilahysebess.Gravity interpretationwith theaidof quadraticprogramming—

discussion.Geophysics, 48(10),1413–1414,1983.

T. Sugimoto,M. Fukushima,andT. Ibaraki. A parallel relaxationmethodfor quadraticpro-grammingproblemswith interval constraints. Journal of Computationaland AppliedMathematics, 60(1–2),219–236,1995.

Abstract. Optimizationproblemswith interval constraintsareencounteredin variousfieldssuchasnetwork

flowsandcomputertomography. As theseproblemsareusuallyvery large,they arenoteasyto solvewithout

taking their sparsityinto account.Recently”row-actionmethods”,which originatefrom the classicalHil-

dreth’s methodfor quadraticprogrammingproblems,have drawn muchattention,sincethey areparticularly

usefulfor largeandsparseproblems.Variousrow-actionmethodshave alreadybeenproposedfor optimiza-

tion problemswith interval constraints,but they mostlybelongto theclassof sequentialmethodsbasedon

the Gauss-SeidelandSORmethods.In this paper, we proposea highly parallelizablemethodfor solving

thoseproblems,which may be regardedasan applicationof the Jacobimethodto the dual of the original

problems.We prove convergenceof thealgorithmandreportsomecomputationalresultsto demonstrateits

effectiveness.

M. F. Sukhinin. Step-by-stepquadraticpenaltysolutionof a quadratic-programmingproblem.ComputationalMathematicsandMathematicalPhysics, 34(8–9),1125–1131,1994.

Abstract. An numericalalgorithmfor a quadraticprogrammingproblemis considered.Resultsof experi-

mentson thetransportationproblemaregiven.

J. L. Sullivan, J. W. Adams,andR. Roosta. The designof constrainedminimax FIR digitalfilters usingquadraticprogramming.ConferenceRecord of TheTwentyNinth AsilomarConferenceonSignals,SystemsandComputers.IEEEComput.SocPress,LosAlamitos,CA,USA, 2, 826–830,1996.

Abstract. We (seeAdamset al., ibid., pp. 314, 1994)presentedin an earlierarticle a new quadraticpro-

grammingalgorithmthatcandesignconstrainedleast-squaresFIR digital filters. In this articlewe presenta

new quadraticprogrammingalgorithmfor designingconstrainedminimaxFIR filters.Thenew algorithmis

betterthanthe Parks-McClellan(1973)algorithmbecauseit candesigna muchwider variety of filters. In

particular, it candominimaxoptimizationsubjectto arbitraryequalityandinequalityconstraints.

J.Sun.A convergenceproof for anaffine-scalingalgorithmfor convex quadraticprogrammingwithout nondegeneracy assumptions.MathematicalProgramming, 60(1), 69–79,1993.


Abstract. This paperpresentsa theoreticalresult on convergenceof a primal affine-scalingmethodfor

convex quadraticprograms.It is shown that, as long as the stepsizeis lessthana thresholdvalue which

dependson theinput dataonly, Ye andTse’s interiorellipsoidalgorithmfor convex quadraticprogramming

is globally convergentwithout nondegeneracy assumptions.In addition,its local convergencerateis at least

linearandthedualiterateshave anergodicallyconvergentproperty.

J.Sun.OnpiecewisequadraticNewtonandtrust-regionproblems.MathematicalProgramming,76(3), 451–468,1997.

Abstract. Some recent algorithms for nonsmoothoptimization require solutions to certain piecewise

quadraticprogrammingsubproblems.Two typesof subproblemsareconsideredin this paper. Thefirst type

seekstheminimizationof acontinuouslydifferentiableandstrictly convex piecewisequadraticfunctionsub-

ject to linear equalityconstraints.We prove that a nonsmoothversionof Newton’s methodis globally and

finitely convergentin thiscase.Thesecondtypeinvolvestheminimizationof apossiblynonconvex andnon-

differentiablepiecewisequadraticfunctionover a Euclideanball. Characterizationsof theglobalminimizer

arestudiedundervariousconditions.Theresultsextendaclassicalresulton thetrustregion problem.

J.SunandH. Kuo. Applying a Newton methodto strictly convex separablenetwork quadraticprograms.SIAMJournalon Optimization, 8(3), 728–745,1998.

Abstract. By introducingquadraticpenaltyterms,a strictly convex separablenetwork quadraticprogram

canbe reducedto an unconstrainedoptimizationproblemwhoseobjective is a continuouslydifferentiable

piecewisequadraticfunction.A recentlydevelopednonsmoothversionof Newton’s methodis appliedto the

reducedproblem.The generalizedNewton directionis computedby an iterative procedurewhich exploits

the specialnetwork datastructuresthat originatedfrom the network simplex method.New featuresof the

algorithmincludetheuseof min-maxbasesandadynamicstrategy in computationof theNewtondirections.

Somepreliminarycomputationalresultsarepresented.Theresultssuggesttheuseof ”warmstart” insteadof

”cold start”.

J. SunandJ. Zhu. A predictor-correctormethodfor extendedlinear-quadraticprogramming.ComputersandOperationsResearch, 23(8), 755–767,1996.

Abstract. The saddlepoint form of extendedlinear-quadraticprogramscanbesolved by an interior point

path-following methodin polynomialtime.Thealgorithmmaytake advantageof theblockstructuresof cer-

tainproblemsarisingfrom optimalcontrolandstochasticprogramming.In addition,it needsnoline searches

andtreatsfully or not fully quadraticproblemsequally. Preliminarycomputationalresultsapparentlyshow

thatthealgorithmis effective in solvingaclassof two-stagestochasticprogrammingproblems.

S. M. Sun,H. S. Tzou, andM. C. Natori. Parametericquadraticprogrammingmethodfordynamiccontactproblemswith friction. AIAAJournal, 32(2), 371–378,1994.

Abstract. Basedon theparametricvariationalprinciple, a generalbut effective parametericquadraticpro-

grammingtechniquesatisfyingvariouscontactconditionsif establishedfor dynamicanalysisof contact

problemswith friction anddamping.Thediscretizationwith respectto time andspaceleadsto astaticlinear

complementaryproblem(LCP) for eachtime stepwhich is solved by a quadraticoptimizationalgorithm

suchastheLemkealgorithm,etc.Thus,theconvergenceandnumericalstabilityof thesolutioncanbeguar-

anteed.The substructurecondensationtechniqueis implementedto handlethe unknown contactboundary

conditionsothatthecomputationeffect is considerablyreduced.An anapplicationof themethodpresented,

threedynamiccontactexamplesandnumericalresultsaregiven.

S.SunderandV. Ramachandran.Designof recursive differentiatorsusingquadraticprogram-ming. In ‘Proceedingsof the36thMidwestSymposiumon CircuitsandSystems.IEEE,New York, NY, USA’, Vol. 1, pp.774–777,1993.

Abstract. A methodfor thedesignof first andhigher-degreerecursive differentiatorswith constantgroup-

delaycharacteristicsusinga least-squaresapproachis presented.In this method,a mean-squareerrorbased

on the differencebetweenthe desiredand actual frequency responseis formulatedin a quadraticform.


Quadraticprogrammingis employed whereinthe constrainton stability is accommodatedto designstable

differentiators.Our methodis comparedwith the linear programming(LP) approachin termsof the com-

putationalcomplexity and the variation of magnitudeandgroup-delayerrorswith frequency. It is shown

thatthedifferentiatorsdesignedusingourmethodhaveamuchlowercomputationalcomplexity andsmaller

variationof themagnitudeandgroup-delayerrorwith frequency thanthosedesignedusingtheLP approach.

W. R. S.Sutherland,H. Wolkowicz, andV. Zeidan.An explicit linearsolutionfor thequadratic

dynamic- programmingproblem. Journal of OptimizationTheory and Applications,

58(2), 319–330,1988.

K. Swarup. Indefinite quadraticprogramming. Cahiers du Centre d’Etudesde Recherche

Operationalle, 8, 217–222,1966a.

K. Swarup.Programmingwith indefinitequadraticfunctionwith linearconstraints.Cahiersdu

Centre d’EtudesdeRechercheOperationalle, 8, 132–136,1966b.

K. Swarup.Quadraticprogramming.CahiersduCentred’EtudesdeRechercheOperationalle,

8, 223–234,1966c.

B. A. Szaboand T. T. Chung. The quadraticprogrammingapproachto the finite elementmethod. InternationalJournal for NumericalMethodsin Engineering, 5(3), 375–381,1973.

Abstract. A finite elementapproximationtechniqueis describedin which the potentialenergy function

is minimized subjectto linear constraints.The constraintsrequiresatisfaction of all kinematicboundary

conditionsand interelementcontinuity conditions.An example, indicating the efficiency of the proposed

method,is presented.

K. Tabushi,S. Itoh, M. Sakura,Y. Kutsutaninakamura,T. A. Iinuma,andT. Arai. A methodfor calculatingthe optimum irradiation condition for intracavitary radiotherapy usingquadraticprogramming.Physicsin MedicineandBiology, 33(5), 515–527,1988.

Abstract. A methodof calculatingoptimum irradiation conditionsfor intracavitary radiotherapy using

quadraticprogramminghasbeenformulatedandthenmodifiedfor practicalapplication.Theallowablerange

of obtaineddose,which is usuallyfixed in advance,is automaticallycomputedto be assmall aspossible.

Thevarianceof theproductof theactivity andthe irradiationtime of the tandemsourceis alsominimised

to avoid theoccurrenceof cold and/orhot spots.Optimumirradiationconditionsfor conventionalintracav-

itary radiotherapy of carcinomaof the uterinecervix wereobtainedon the basisof isodosecurvespassed

throughthepointsA of theManchestersystem.Thosefor carcinomaof theotherorgansandspecialcasesof

carcinomaof theuterinecervixcanbedeterminedafterconsiderationof thetumourstate.

M. Takamoto,N. Yamada,Y. Kobayashi,H. Nonaka,and S. Okoshi. 0–1 quadraticpro-grammingalgorithmfor resourceleveling of manufacturingprocessschedules.Systemsand Computers in Japan, 26(10), 68–76,1995. Seealso,Transactionsof the Instituteof Electronics,InformationandCommunicationEngineers,volumeJ77D-II(10)pages2075–2082,1994.

Abstract. In industrialplant constructionscheduling,it is necessaryto minimize thefluctuation,themaxi-

mumpeakof thefluctuationandthemaximumpeakvalueof theamountof daily resources,which is calcu-

latedasthesumof daily resourcesfor eachprocess.Minimization of thefluctuationor themaximumpeak

valuecorrespondsto leveling thepile of resources.To performthis resourceleveling, we needto decideon

anobjective functionwhich is a monotonefunctionthatsimply increaseswith thedegreeof resourcelevel-

ing andthensolve theoptimizationproblemby fixing theprocessstartdateswhile minimizing theobjective

function. Industrialplant constructionis, however, in many casesa large-scaleschedulingwith an entire


periodof morethan1000daysandmorethan100processes,so it is very difficult to obtaina global opti-

mizationsolution.Wehavedevelopedanalgorithmwhichsolvesa large-scaleoptimizationproblemto level

necessaryresources.This algorithmcanquickly searchfor a goodsuboptimalsolutioncloseto the global

optimalsolutionof a0–1quadraticprogrammingproblem.Thealgorithmsearchesby repeatingapivot oper-

ationusingvariableselectionrulesfor resourceleveling.Weappliedthisalgorithmto large-scalescheduling

for anactualplantconstructionschedule,andsuccessfullyobtaineda practicalsuboptimalsolutionwithin a

few minutes(CPUpower: 28MIPS).Theresultssuggestthatthealgorithmis practicalfor resourceleveling

of large-scaleconstructionscheduling.

T. Takayamaand N. Uri. A note on spatialand temporalprice and allocationmodeling—

quadraticprogrammingor linearcomplementarityprogramming.Regional Scienceand

Urban Economics, 13(4), 455–470,1983.

A. Takeda,Y. Dai, M. Fukuda,andM. Kojima. Towardstheimplementationof successivecon-vex relaxationmethodfor nonconvex quadraticoptimizationproblems.ResearchReporton InformationSciencesB-347,Departmentof MathematicalandComputingSciences,Tokyo Instituteof Technology, Japan,1999.

Abstract. RecentlyKojima and Tuncel proposednew successive convex relaxationmethodsand their

localized-discretized variantsfor generalnonconvex quadraticprograms.Although an upperboundof the

objective function valuewithin a prior precisioncanbe found theoreticallyby solving a finite numberof

linearprograms,severalimportantimplementationproblemsremainunsolved.In thispaperwediscussthese

issues,presentpracticallyimplementablealgorithmsandreportnumericalresults.

N. N. Tam and N. D. Yen. Continuity propertiesof the Karush-Kuhn-Tucker point set inquadraticprogrammingproblems.MathematicalProgramming, 85(1), 193–206,1999.

Abstract. We obtainnecessaryandsufficient conditionsfor thesetof theKarush-Kuhn-Tucker pointsin a

canonicalquadraticprogrammingproblemto beuppersemicontinuousor lowersemicontinuouswith respect

to theproblemparameters.

K. Tammer. Possibilitiesfor theapplicationof theresultsof parametricprogrammingto solvingindefinitequadraticprogrammingproblems. Mathematische OperationsforschungundStatistik, 7(2), 209–222,1976.

Abstract. Somepossibilitiesfor the applicationof the resultsof the convex and nonconvex parametric

quadraticprogrammingto the computationof nonconvex quadraticprogrammingproblemsaregiven. In

thefirst casetheproblemcanbesolvedby solvinga stronglyconvex parametricquadraticproblemwith in

generalmorethanoneparameterandsomesmallerquadraticproblems,in thesecondcaseby solvinga one

parametricnonconvex quadraticproblem.

H. TamuraandY. Kihara. Multistagequadraticprogrammingfor discrete-timeoptimalcontrolwith stateandcontrolconstraints.SystemsandControl, 21(12),702–709,1977.

Abstract. A multistagequadraticprogrammingtechniqueis developedfor discrete-timelinear-quadratic(L-

Q) optimalcontrolproblemwith stateandcontrolconstraints.Theproblemis convertedto a linearprogram

with bilinear constraints.Then, taking advantageof the staircase-structureof equalityconstraints(system

equation),Dantzig-Wolfe decompositionprinciple is appliedrepeatedlyin eachstage,wherethedecompo-

sition techniquefor the bilinear constraintsis newly developedin this paper. The significantadvantageof

themultistagequadraticprogrammingtechniquein this paperis that it canhandlea largenumberof stages

without increasingthecomputationtime andstoragerequirementsenormously, andthat it canhandlestate

andcontrolconstraintswithout difficulty. Therefore,a substantialreductionof computationalburdenis ob-

tainedfrom theexisting discrete-timeoptimalcontrolalgorithms.Numericalexamplesfor comparisonwith

Wolfe’s quadraticprogrammingmethodareincluded.


Y. TanandC. Deng.Solvingfor aquadraticprogrammingwith aquadraticconstraintbasedona neuralnetwork frame.Neurocomputing, 30(1–4),117–128,2000.

Abstract. In many applications,a classof optimizationproblemscalledquadraticprogrammingwith aspe-

cial quadraticconstraint(QPQC)oftenoccurs,suchasin thefieldsof maximum-entropy spectralestimation,

FIR filter designwith time-frequency constraints,and the designof an FIR filter bankwith a perfectre-

constructionproperty. In order to deal with this kind of optimizationproblemand to be inspiredby the

computationalvirtueof analogor dynamicneuralnetworks,a feedbackneuralnetwork is proposedfor solv-

ing for thisclassof QPQCcomputationproblemsin realtime.Thestability, convergenceandcomputational

performanceof theproposedneuralnetwork have alsobeenanalyzedandprovedin detail,soasto theoreti-

cally guaranteethecomputationaleffectivenessandcapabilityof thenetwork. Fromthetheoreticalanalyses,

it turnsoutthatthesolutionof aQPQCproblemis just thegeneralizedminimumeigenvectorof theobjective

matrixwith respectto theconstrainedmatrix.A numberof simulationexperimentshavebeengivento further

supportour theoreticalanalysisandto illustratethecomputationalperformanceof theproposednetwork.

H. TanakaandH. Lee. Interval regressionanalysisby quadraticprogrammingapproach.IEEETransactionsonFuzzySystems, 6(4), 473–481,1998.

Abstract. Whenwe uselinear programmingin possibilisticregressionanalysis,somecoefficients tendto

becomecrispbecauseof thecharacteristicof linearprogramming.On theotherhand,a quadraticprogram-

mingapproachgivesmorediversespreadcoefficientsthanalinearprogrammingone.Therefore,to overcome

thecrispcharacteristicof linearprogramming,we proposeinterval regressionanalysisbasedon a quadratic

programmingapproach.Another advantageof adoptinga quadraticprogrammingapproachis to be able

to integrateboth the propertyof centraltendency in leastsquaresand the possibilisticpropertyin fuzzy

regression.By changingthe weightsof the quadraticfunction,we cananalyzethe given datafrom differ-

entviewpoints.For datawith crisp inputsandinterval outputs,thepossibilityandnecessitymodelscanbe

considered.Therefore,theunifiedquadraticprogrammingapproachobtainingthepossibilityandnecessity

regressionmodelssimultaneouslyis proposed.Eventhoughtherealwaysexist possibilityestimationmodels,

theexistenceof necessityestimationmodelsis not guaranteedif we fail to assumea properfunctionfitting

to thegivendataasaregressionmodel.Thus,weconsiderpolynomialsasregressionmodelssinceany curve

canberepresentedby thepolynomialapproximation.Usingpolynomials,we discusshow to obtainapprox-

imationmodelswhich fit well to thegivendatawherethemeasureof fitnessis newly definedto gaugethe

similarity betweenthepossibilityandthenecessitymodels.Furthermore,from theobtainedpossibilityand

necessityregressionmodels,a trapezoidalfuzzyoutputcanbeconstructed.

H. Tanaka,K. Koyama,andH. Lee. Interval regressionanalysisbasedon quadraticprogram-ming. In ‘Proceedingsof the Fifth IEEE InternationalConferenceon FuzzySystems.FUZZ-IEEE’96. IEEE,New York, NY, USA’, Vol. 1, pp.325–329,1996.

Abstract. Weproposeanapproachto interval regressionanalysisbasedonquadraticprogramming.Secondly

weproposea unifiedapproachfor thepossibilityandthenecessityregressionmodelsin interval regression.

Fromtheobtainedtwo models,fuzzy outputcanbedefined.Moreover, a performanceindex for theunified

approachto measurethefitnessof thegivendatato theobtainedmodelsis defined.

J. Tang and D. Wang. An interactive approachbasedon a geneticalgorithm for a type ofquadraticprogrammingproblemswith fuzzy objective andresources.Computers andOperationsResearch, 24(5), 413–422,1997.

Abstract. A type of modelof fuzzy quadraticprogrammingproblems(FQP) is proposed.The modelde-

scribesthefuzzy objective andresourceconstraintswith differenttypesof membershipfunctionsaccording

to differenttypesof fuzzyobjective andfuzzy resourceconstraintsin actualproductionproblems.This arti-

cledevelopsaninexactapproachto solve this typeof modelof quadraticprogrammingproblemswith fuzzy

objective andresourceconstraints.Insteadof finding anexactoptimalsolution,we usea geneticalgorithm

(GA) with mutationalongtheweightedgradientdirectionto find afamily of solutionswith acceptablemem-

bershipdegrees.Thenby meansof thehuman-computerinteraction,thesolutionspreferredby theDM under

differentcriteriacanbeachieved.


W. S. Tang. A discrete-timerecurrentneuralnetwork for solving quadraticprogramswithapplicationto fir filter synthesis. In ‘SMC 2000ConferenceProceedings.2000 IEEEInternationalConferenceon Systems,Man andCybernetics.‘CyberneticsEvolving toSystems,Humans,Organizations,andtheirComplex Interactions,IEEE,Piscataway, NJ,USA’, Vol. 4, pp.2491–2496,2000.

Abstract. A discrete-timerecurrentneuralnetwork is presentedfor solving convex quadraticprograms.It

is the discrete-timeversionof its continuous-timecounterpartwhich was developedby J. Wang and H.

Li (1994).Sharingthesamecharacteristicwith its continuous-timecounterpart,theproposeddiscrete-time

neuralnetwork couldcomputetheexactoptimalsolutionto a quadraticprogramwithout usingany penalty

parameter. However, thediscrete-timeversionis moredesirablein practicalrealizationin view of theavail-

ability of digital hardwareand the good compatibility to computer. The condition for the neuralnetwork

globallyconverging to theoptimalsolutionof aquadraticprogramis given.Theneuralnetwork is appliedto

FIR filter synthesisfor illustratingits effectiveness.

W. S.TangandJ.Wang.A discrete-timelagrangiannetwork for solvingconstrainedquadraticprograms.InternationalJournalof Neural Systems, 10(4), 261–265,2000.

Abstract. A discrete-timerecurrentneuralnetwork which is calledthediscrete-timeLagrangiannetwork is

proposedfor solvingconvex quadraticprograms.It is developedbasedontheclassicalLagrangeoptimization

methodandsolvesquadraticprogramswithout usingany penaltyparameter. The condition for the neural

network to globally converge to the optimal solutionof the quadraticprogramis given.Simulationresults

arepresentedto illustrateits performance.

Q. TaoandD. Sun.Thesimplificationof neuralnetwork for quadraticprogrammingproblemsandits applicationsin optimal control. In ‘Proceedingsof the 3rd World CongressonIntelligentControlandAutomation,IEEE,Piscataway, NJ,USA’, Vol. 5, pp.3504–3508,2000.

Abstract. In thispaper, akind of neuralnetwork for quadraticprogrammingproblemsis first simplified.The

simplificationis necessaryfor high accuracy solutionsandlow costimplementation;the simplified model

hashigh performance.Theproposedneuralnetwork is usedto solve quadraticcontrolproblemsfor discrete

linearsystemswith constraints.Thesimulationprovedtherationalityof theresultsobtained.

R. M. Teny and A. K. Kochhar. Solution of the aggregateproductionplanning problemin a multi-stage-multi-productmanufacturing systemusing functional spaceanalysisand quadraticprogrammingapproaches. International Journal of SystemsScience,14(3), 325–342,1983.

Abstract. It is shown thatbothof theseapproachescanbeusedto determinetheproductionplanningstrate-

giesfor the numberof periodsunderconsideration.Statisticalinferencetechniquesareusedto determine

whetherthefunctionalspaceanalysistechniqueresultsin aglobalminimum.Testscarriedouton two differ-

entsetsof datashow that thequadraticprogrammingapproachalwaysresultsin a minimumcostsolution,

althoughit requiresa large amountof computingpower. Althoughcomplicatedmathematicsis requiredto

formulatetheproblemandsolve it, theresultingsolutioncanbeeasilyunderstoodandappliedto apractical

manufacturingsystem.

T. Terlaky. A new algorithmfor quadraticprogramming.EuropeanJournalof OperationalRe-search, 32(2), 294–301,1987. Seealso,AlkalmazottMatematikaiLapok.12(3–4):283–293,1986.

Abstract. Presentsanew finite algorithmfor quadraticprogramming.Thealgorithmis basedonthesolution

proceduresof linearprogramming(pivoting,Bland’s rule,HungarianMethods,criss-crossmethod),however

thismethoddoesnotrequiretheenlargementof thebasictableauasFrank-Wolfe methoddoes.It canbecon-

sideredasafeasiblepointactive-setmethod.Thisalgorithmis astraightforwardgeneralizationof Klafszky’s

andTerlaky’s Hungarianmethod.It hasnearlythe samestructureasRitter’s algorithm(which is basedon

conjugatedirections),but it doesnot useconjugatedirections.


H. Theil and C. van De Panne. Quadraticprogrammingas an extensionof conventional

quadraticmaximization.ManagementScience, 7, 1–20,1960.

A. L. Tits andJ.L. Zhou.A simple,quadraticallyconvergentinterior-pointalgorithmfor linear-

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M. J.Todd.Linearandquadraticprogrammingin orientedmatroids.Journalof CombinatorialTheory, SeriesB, 39(2), 105–133,1985.

Abstract. Duality theoremsof linearandquadraticprogrammingareprovenconstructively in thecombina-

torial settingof orientedmatroids.Oneversionof thealgorithmfor linearprogramminghasthe interesting

featureof maintainingfeasibility. The developmentof the quadraticprogrammingduality result suggests

thestudyof propertiesof squarematricessuchassymmetryandpositive semidefinitenessin thecontext of

orientedmatroids.

M. J.ToddandY. Wang.A projectivealgorithmfor convex quadraticprogramming.Technical

report,Schoolof OperationsResearchand IndustrialEngineering,Cornell University,

Ithaca,New York, USA, 1991.

O. N. Tokareva. Theestimatesof therateof convergenceof algorithmsbasedon barrierfunc-

tions with the useof quadraticprogrammingproblems. Cybernetics, 17(5), 688–695,

1981.

J. A. Tomlin andM. A. Saunders.Stablereductionto KKT systemsin barriermethodsfor

linearandquadraticprogramming.ResearchReportRJ10039,IBM AlmadenResearch

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J. J. Torsti. Inversionof heatcapacityby quadraticprogrammingin thecasesof KCl andCu.AnnalesAcademiaeScientiarumFennicae, SeriesAVI (Physica), 372, 1–20,1971.

Abstract. By usingexperimentalspecificheatdataandby separatingthespecificheatinto two parts,lattice

specificheatandanharmoniccontributions,a methodhasbeenpresentedfor the calculationof the lattice

vibration spectrumand of the anharmoniccontributions to the specificheatwith their confidencelimits.

The calculatedfrequency spectrumof KCl was in qualitative agreementwith the frequency spectrumby

Copley et al. which correspondedto the inelasticneutronscatteringmeasurements.The highestpeakin

the spectrawasat 4 6 & 1012 s' 1 Likewise, the calculatedfrequency spectrumof Cu was found to be in

agreementwith Varshni’s andShukla’s theoreticalspectrum(abstr. A8094 of 1966)which is basedon the

axialsymmetricmodelfor latticedynamics.Thehighestpeakwasat6 3 & 1012 s' 1 If thevolumedependence

of the frequenciesis taken into account,the resultsfor both KCl andCu arein betteragreementwith the

experimentaldatathan when using the harmonicapproximation.The confidencelimits of the frequency

spectrawerecomparatively largeat higherfrequencies.By contrast,thelimits of coefficientsof anharmonic

contributionsto thespecificheatwerenarrow.

J.J.TorstiandA. M. Aurela. A fastquadraticprogrammingmethodfor solvingill-conditioned

systemsof equations.Journal of MathematicalAnalysisandApplications, 38(1), 193–

204,1972.

T. B. Trafalis and N. P. Couellan. Neural-network training via quadraticprogramming. In

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E. Triantaphyllou.A quadraticprogrammingapproachin estimatingsimilarity relations.IEEETransactionson FuzzySystems, 1(2), 138–145,1993.

Abstract. The problemof estimatinghow similar N objectsarewhenthey arecomparedwith eachother

is investigated,usingcomparative judgmentsof all possiblepairsof the N objectsas data.The pairwise

comparisonsfocuson thesimilarity relationsinsteadof therelative importanceof eachobject.A quadratic

programmingmodelis alsoproposed.It processesthesimilarity-basedpairwisecomparisonsanddetermines

the similarity relationsamongthe N objects.The modelhaslinear constraints;thereforeit canbe solved

easilyby transferringit into asystemof linearequations.

T. Tsuchiya.Globalconvergenceof theaffine scalingalgorithmfor primal degeneratestrictlyconvex quadraticprogrammingproblems. Annalsof Operations Research, 46–47(1–4), 509–539,1993.

Abstract. Dealswith global convergenceof the affine scalingalgorithmfor strictly convex QP problems

satisfyinga dual nondegeneracy condition.By meansof the local Karmarkarpotentialfunctionwhich was

successfullyappliedto demonstrateglobal convergenceof the affine scalingalgorithmfor LP, the author

shows globalconvergenceof thealgorithmwhenthe step-size1/8 is adoptedwithout requiringany primal

nondegeneracy condition.

A. W. Tucker. A least-distanceapproachto quadraticprogramming.In G. B. DantzigandA. F.

Veinott, eds, ‘Lectutesin Applied MathematicsII’, number1 in ‘Mathematicsof the

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Island,USA, 1968.

M. Tuma.LargeandSparseQuadratic Programming. PhDthesis,UIVT CSAV, Prague,Czeck

Republic,1989.

M. Tuma. A quadraticprogrammingalgorithmfor large andsparseproblems. Kybernetika,27(2), 155–167,1991.

Abstract. Describesa primal feasibleconvergent algorithmfor the quadraticprogrammingproblem.It is

especiallydesignedto copewith thelargeandsparseinstancesof thistypewith theirstructuralandnumerical

algorithmic specialties.It makes useof improvementsthat have not beenusedfor the generalquadratic

programmingor thathave notbeenusedsofar.

G. Uebe. Note on the paperof H. Amato andG. Mensch: rank restrictionon the quadratic

form in indefinitequadraticprogramming.Zeitschrift fur OperationsResearch, SerieA

(Theorie), 16(3), 113–114,1972.

H. Valiaho.New algorithmsfor quadraticprogramming.CommentationesPhysicoMathemat-icae, 36(9), 111–136,1970.

Abstract. A new classof algorithmsfor quadraticprogrammingis constructed,applyingthetheoryof step-

wiseregressionanalysisintroducedin Valiaho(1969),A SyntheticApproachto StepwiseRegressionAnal-

ysis.Thebasisof thealgorithmsis thefactthata feasiblevectoris thesolutionif andonly if theLagrangians

attachedto thebindingconstraintsarenegative. Thebasicalgorithmstartsfrom the freeminimum.At any

stage,aviolatedconstraintis turnedinto adesideratumandtheweightof thisdesideratumis increasedgrad-

ually from 0 to infinity , meanwhileremoving thosebindingconstraints,theLagrangiansattachedto which

becomezero.A modifiedalgorithmis alsoproposedleadingto thesolutionmorerapidly thanthebasical-

gorithm.As furtherdevelopments,interval andsimpleconstraintsareconsidered.A procedurefor changing

the quadraticprogrammingproblemis introduced,makingit possibleto copewith semidefinitecases.Fi-

nally anew algorithmfor linearprogrammingis constructedbasedon thefactthatlinearprogrammingis an

elementaryparticularcaseof quadraticprogramming.


H. Valiaho. Themethodof approximateconstraintsfor quadraticprogramming.Commenta-tionesPhysicoMathematicae, 41(3), 257–280,1971a.

Abstract. Thefundamentalsof themethodof approximateconstraintsarederivedwithout appealingto the

terminologyof regressionanalysis.Somepreviouserrorsarecorrected,andsomecontributionsmade:The

connectionbetweenpre- andpost-loopsis revealed;algorithmsarederived without usingdesiderata;the

effectof changingaconstrainton thesolutionis examined;Houthakker’s capacitymethodis carriedthrough

makinguseof theschemeof theMethodof ApproximateConstraints:a specialcaseof indefinitequadratic

programmingis solved.

H. Valiaho. Noteson somealgorithmsfor quadraticprogramming.CommentationesPhysicoMathematicae, 41(3), 247–255,1971b.

Abstract. A monotonicversionof thebasicalgorithmis introduced,usingnon-diagonalpivotal operations.

This approachmakes it possibleto derive the basicalgorithmwithout using the concepts’modified con-

straint’ or ’desideratum’.The efficienciesof somealgorithmsfor quadraticprogrammingarecomparedin

thelight of two numericalexamples.

H. Valiaho. On thedefinity of quadraticformssubjectto linearconstraints.Journal of Opti-

mizationTheoryandApplications, 38(1), 143–145,1982.

H. Valiaho.A unifiedapproachto one-parametricgeneralquadraticprogramming.Mathemat-ical Programming, 33(3), 318–338,1985.

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problemwhereall thecoefficientsarelinearor polynomialfunctionsof a scalarparameter. Thelocal mini-

mumvectorandthelocal minimumvaluearedeterminedexplicitly asrationalfunctionsof theparameter. A

numericalexampleis given.

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H. Valiaho.A new proof for thecriss-crossmethodfor quadraticprogramming.Optimization,25(4), 391–400,1992.

Abstract. In aworking paperChang(1979)hasintroduceda simplemethodfor thelinearcomplementarity

probleminvolving anonnegativedefinitematrix,basedonprincipalpivotingandtheleast-index ruleof Bland

(1977).Thismethodhastwo importantspecialcases,namelythecriss-crossmethodsfor linearprogramming

andfor convex quadraticprogramming,inventedlaterindependentlyby Terlaky andKlafszky. Theproof of

Changis very complicated.Terlaky andKlafszky provide muchsimplerproofsfor the criss-crossmethod.

We proposeanothersimpleproof for thecriss-crossmethodfor quadraticprogramming.It appearsfrom an

exampleby Roos(1990)thatthenumberof pivotsrequiredby thecriss-crossmethodmaygrow exponentially

with thenumberof theconstraintsof theproblem.We usetwo additionalexamples,dueessentiallyto Fathi

(1979)andMurty (1988),to analysetheworst-casebehaviour of themethod.

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nonconvex quadraticprogramming.Hestatesthatafterafinite numberof stepsthealgorithmdetectsthatthe

problemis infeasibleor hasanunboundedsolution,or findsatruelocalminimum.Wegiveacounterexample

whichshows thatcycling canoccurin Mraz’s algorithm.

C. vanDe Panne.A non-artificialSimplex methodfor quadraticprogramming.Report6229,

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of large-scalequadraticprogramming(QP)problems.A review of therecentliteratureonPE/MBFmethods

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large-scalebound-constrainedproblems.Theproposedalgorithmperformstwo typesof iterations:anouter

iterationin which theLagrangemultipliersof theboundsareadjusted,andaninneriterationfor thesolution

of anequalityconstrainedsubproblem.The inner iterationsolvesa modifiedproblem,containingPE/MBF

termsfor theboundsin theobjective andis subjectto equalityconstraintsonly. Theequalityconstraintsare

handleddirectly via theuseof additionalLagrangemultipliersduringtheinneriterationandthus,insteadof

anunconstrainedproblem,theinneriterationsolvesamodifiedequalityconstrainedproblem.Any inequality

constraints,other thanbounds,are formulatedasequalitiesvia the useof slackvariables.Computational

resultsshow thismethodto bepromising,andmotivatefurtherinvestigationfor thegeneralcaseof nonlinear

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previousexperiencein very large-scalebound-constrainedproblems.Theproposedalgorithmperformstwo

typesof iterations:anouteriterationin which theLagrangemultipliersof theboundsareadjusted,andan

inner iterationfor thesolutionof anequalityconstrainedsubproblem.Theinner iterationsolvesa modified

problem,containingpenalty/modifiedbarriertermsfor theboundsin theobjective, andis subjectto equality

constraintsonly. Theequalityconstraintsarehandleddirectly via theuseof additionalLagrangemultipliers

duringtheinneriterationandthus,insteadof anunconstrainedproblem,theinneriterationsolvesamodified

equalityconstrainedproblem.Any inequalityconstraints,otherthanbounds,areformulatedasequalitiesvia

the useof slackvariables.Computationalresultsshow this methodto be promising,andmotivate further

investigationfor thegeneralcaseof nonlinearprogrammingproblems.

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Abstract. We considerε-approximationschemesfor indefinitequadraticprogramming.We arguethatsuch

anapproximationcanbefoundin polynomialtime for fixedε andt, wheret denotesthenumberof negative

eigenvaluesof the quadraticterm.Our algorithmis polynomial in 1 ε for fixed t, andexponentialin t for

fixedε. Wenext look at thespecialcaseof knapsackproblems,showing thatamoreefficient (polynomialin

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formationto convert theprobleminto anoptimizationin thenon-negative orthant.Theoptimizationcanthen

besolvedby optimizationsin subspaces.Thealgorithmis shown to alwaysdecreasetheobjective function

andto converge in a finite numberof iterations.The algorithm is easilyprogrammedon a computer. An

example,previously usedby others,is solvedwith this techniquein a fewer numberof iterations.An illus-

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quadraticprogrammingschemearediscussed.

S. A. Vinogradov andD. F. Wilson. Phosphorescencelifetime analysiswith a quadraticpro-grammingalgorithmfor determiningquencherdistributions in heterogeneoussystems.BiophysicalJournal, 67(5), 2048–2059,1994.

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beendeveloped.This methodis basedon decompositionof thedatavectorto a linearly independentsetof

exponentialsandusesquadraticprogrammingprinciplesfor χ2 minimization.Solutionof theresultingalgo-

rithm requiresa finite numberof calculations(it is not iterative) andis computationallyfastandrobust.The

algorithmhasbeentestedonvarioussimulateddecaysandfor analysisof phosphorescencemeasurementsof

experimentalsystemswith discretedistributionsof lifetimes.Critical analysisof theeffectof signal-to-noise

on the resolvingcapabilityof the algorithmis presented.This techniqueis recommendedfor resolutionof

thedistributionsof quencherconcentrationin heterogeneoussamples,of whichoxygendistributionsin tissue

is animportantexample.Phosphorsof practicalimportancefor biologicaloxygenmeasurements:Pd-meso-

tetra (4-oarboxyphenyl) porphyrin (PdTCPP)and Pd-meso-porphyrin(PdMP) have beenusedto provide

experimentaltestof thealgorithm.


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nentspectrophotometryfor the casewhenan unknown additive is presentin the investigatedobjectalong

with componentshaving known spectra.The methodsof linear (LP) or quadratic(QP) programmingare

usuallyusedfor thedeterminationof thecontentin suchmixturesof componentswith known spectra.In the

presentwork acomparative numericalinvestigationof thepossibilitiesof themethodsof LP andQPfor the

solutionof theindicatedproblemis carriedout.

Y. H. Wan. On the averagespeedof Lemke’s algorithmfor quadraticprogramming.Mathe-maticalProgramming, 35(2), 236–246,1986.

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Abstract. A recurrentneuralnetwork for solvingquadraticprogrammingproblemswith equalityconstraints

is presented.The proposedrecurrentneuralnetwork is asymptoticallystableandableto generateoptimal

solutionsto quadraticprogramswith equalityconstraints.An opampbasedanaloguecircuit realisationof the

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grammingproblemswith uniquesolutions.Comparedwith theBouzerdoum-Pattisonnetwork (1993),there

is no needfor choosingtheself-feedbackor lateralconnectionmatricesin thepresentnetwork. Moreover,

thesizeof thedualnetwork is lessthanthatof theoriginalproblem.

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Abstract. This paperpresentsanew algorithmof theinterior pointquadraticprogrammingwhichcansolve

powersystemoptimizationproblemswith profoundlylesscomputationalefforts.Theproposedalgorithmhas

thefollowing two specialfeatures.First, it is basedonthepath-following interiorpointalgorithm,thesearch

directionof which is taken asNewton direction, thushaving quadraticconvergence.In the secondplace,

it solvesdirectly a symmetricindefinitesysteminsteadof reducingit to the usualsystemwith a positive-

definitematrix.Underthis artifice,thealgorithmavoidstheformationof ADAT andgeneratesfewer fill-ins

thanthecaseof factorizingthepositivedefinitesystemmatrixfor largescalesystems,realizingits impressive

speed-up.Sincethe formula of the interior point methodhave beendeducedmoregenerally, the proposed

algorithmcanstartfrom eithera feasible(interiorpoint)or aninfeasiblepoint(noninteriorpoint).Numerical

resultsontheIEEEtestsystemsandaJapanese344bussystem(2100variables,2088equalityconstraintsand

1400inequalityconstraints)haveverifiedthattheproposedalgorithmpossessesenoughrobustnessandneeds

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in this paper. The original problemis divided into many independentsubproblemat an initial point, anda

searchdirectionis obtainedby solvingeachof thesubproblem,aswell asanew iterative point is determined

suchthat the valueof objective function is decreasing.The convergenceof the algorithmis proved under

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satisfy, eachattributeand/ortherelative importanceof theattributesareonly roughlyspecified.In particular,

wewill work with verbalspecificationsof thesequantities,like low, medium,andhigh.Thesespecifications

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The characteristicsrequiredof a generalizedanalogare detailedand a suitableform of implementation

is described.Circuits implementingthe generallinearly constrainedquadraticprogramaredescribedand

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quadraticprogramming).Theequationssetuparesuitablefor computer-basedsolution.TheLP/PROTRAN

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theCholesky factorizationof thereducedHessianmatrix. It is shown how thesecanbeimplementedin such

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Abstract. Two classesof high-performanceneuralnetworks for solving linearandquadraticprogramming

problemsare given. We prove that the new systemconvergesglobally to the solutionsof the linear and

quadraticprogrammingproblems.In a neuralnetwork, network parametersareusuallynot specified.The

proposedmodelscanovercomenumericaldifficulty causedby neuralnetworkswith network parametersand

obtaindesiredapproximatesolutionsof thelinearandquadraticprogrammingproblems.

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theseproblems:it avoids theparameterturningproblem,it is capableof achieving theexactsolutions,and

it usesonly simplehardware in which no analogmultipliers for variablesare required.Furthermore,the

network solvesboththeprimalproblemsandtheirdualproblemssimultaneously.

Y. Xia and J. Wang. Primal neuralnetworks for solving convex quadraticprograms. In‘IJCNN’99. InternationalJointConferenceonNeuralNetworks.Proceedings,IEEE,Pis-cataway, NJ,USA’, Vol. 1, pp.582–587,1999.

Abstract. We proposetwo primal neuralnetworkswith globally exponentialstability for solvingquadratic

programmingproblems.Both the self-feedbackandlateralconnectionmatricesin the presentnetwork are

comparedwith the Bonzerdoum-Pattisonnetwork (1993).Moreover, the sizeof the proposednetworks is

sameasthatof theoriginal problem,smallerthanthatof primal-dualnetworks.

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simultaneously.

Z. Xia. The ABS classandquadraticprogrammingII: Generalcontraints. TechnicalReport91-5,Departmentof Mathematics,Statistics,ComputerScienceandApplications,Uni-versityof Bergamo,Italy, 1991a.

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lemswith equalityandinequalityconstraints.

Z. Xia. Quadraticprogrammingvia the ABS algorithmI: Equalitycontraints.TechnicalRe-port 91-1, Departmentof Mathematics,Statistics,ComputerScienceandApplications,Universityof Bergamo,Italy, 1991b.

Abstract. Somemethodsfor solving quadraticprogrammingproblemswith equalityconstraintsarerefor-

mulatedin termsof the ABS classof algorithms.This paperprovidesthe backgroundfor further studyin

applyingtheABS algorithmsto solve quadraticprogrammingandnonlinearlyconstrainedprogramming.

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Abstract. In this paper, a new algorithmfor thesolutionof nonconvex quadraticprogrammingproblemsis

proposed.Comparedwith themethodof Best(1984),themainimprovementof this methodis theintroduc-

tion of ”projectedcorrectdirection” in searchdirection,which maydropor addmany active constraintsat

eachiteration.Theconvergenceof themethodis provenunderthenonconvexity condition.Its specialcases

arealsodeveloped,andtheir theoreticalanalysisis given.Thenumericalexamplesshow its effectiveness.

W. L. Xu andA. G. Journel. Histogramandscattergramsmoothingusingconvex quadraticprogramming.MathematicalGeology, 27(1), 83–103,1995.


Abstract. An algorithm,basedonconvex quadraticprogramming,is proposedto smoothsamplehistograms.

Theresultingsmoothedhistogramremainscloseto theoriginal histogramandhonorstargetmeanandvari-

ance.Thealgorithmis extendedto smoothsamplescattergrams.Theresultingsmoothedscattergramremains

closeto theoriginal scattergramshapeandthe two previously smoothedmarginal distributionsandhonors

thetargetcorrelationcoefficient.

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Abstract. For original paper, seeM. Forti andA. Tesi, ibid., 42, pp. 354–66(1995).This letter makesthe

following comments:1) theassumptionof all neuronactivation functionsto vanishat the origin, which is

utilizedin theproofof theresultimplying theexistenceanduniquenessof thenetwork equilibriumpoint,can

beactuallyomitted;2) in theinfinite sectorcase,theresultof globalasymptoticstability (GAS)remainstrue

with respectto theclassof increasing(not necessarilystrictly) activations,asin thefinite sectorcase.Con-

sequently, a resultaboutabsolutestability (ABST) of neuralnetworks,which canrepresenta generalization

of theexisting relatedones,is alsoobtained.

H. Yabe,S. Takahashi,andN. Yamaki. Application of resolution-typequasi-Newton updateformula for serialquadraticprogramming.15th NumericalAnalysisSymposiumNihonUniv, Tokyo,Japan, pp.60–63,1986.

Abstract. Amongvariousnumericalsolutionsfor constrainedminimizationproblems,serialquadraticpro-

gramminghasbeenrecentlyattractinginterest.Thoughthis is aneffective method,it is necessaryto maintain

thepartialquadraticprogrammingproblemasa convex quadraticprogrammingproblemin a narrow sense,

in orderto secureglobal convergence.The authorsproposean algorithmsuchthat the partial problemal-

waysbecomesa convex quadraticprogrammingproblemin a narrow senseby a sortof relaxationof secant

conditions.In thealgorithmformingprocess,theresolution-typequasi-Newton updateformulais discussed.

Thisresolution-typeformulacanbeadaptedwell to theGoldfarb-Idnanimethodto solvethepartialquadratic

programmingproblem.

A. B. Yadykin. A parametricmethod for quadraticprogrammingproblemswith semi-definitequadraticforms. Zhurnal Vychislitel’noi Matematikii Matematicheskoi Fiziki,15(6), 1436–1446,1975.

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original problems.Theexistenceof a continuoussegmentally-lineartrajectorytowardstheminimumis es-

tablished.Theconstructionof this trajectoryis given.

A. B. Yadykin. Methodof factoringin solutionof problemsof parametricquadraticprogram-ming. AutomationandRemoteControl, 38(11),1687–1695,1977.

Abstract. A techniquefor transformingtheCholesky factors(LDLT expansion)of a positive semi-definite

matrixis considered.Thetechniqueisusedfor solvingaclassof parametricquadraticprogrammingproblems

of highdimensionality.

Y. Yajima andT. Fujie. A polyhedralapproachfor nonconvex quadraticprogrammingprob-lemswith box constraints.Journal of Global Optimization, 13(2), 151–170,1998. Seealso,ResearchReportson MathematicalandComputingSciences,SeriesB (OperationsResearch),number323,pages1–23,1996.

Abstract. We apply a linearizationtechniquefor nonconvex quadraticproblemswith box constraints.We

show thatcuttingplanealgorithmscanbedesignedto solve theequivalentproblemswhichminimizea linear

functionoveraconvex region.Weproposeseveralclassesof valid inequalitiesof theconvex regionwhichare

closelyrelatedto theBooleanquadricpolytope.Wealsodescribeheuristicproceduresfor generatingcutting

planes.Resultsof preliminary computationalexperimentsshow that our inequalitiesgeneratea polytope

which is a fairly tight approximationof theconvex region.


E. Yamakawa andM. Fukushima.A block-parallelconjugategradientmethodfor separablequadraticprogrammingproblems.Journalof theOperationsResearch Societyof Japan,39(3), 407–427,1996.

Abstract. For a large-scalequadraticprogrammingproblemwith separableobjective function,a variantof

the conjugategradientmethodcaneffectively be appliedto the dual problem.In this paper, we consider

a block-parallelmodificationof the conjugategradientmethod,which is suitablefor implementationon a

parallelcomputer. More precisely, the methodproceedsin a block Jacobimannerandexecutesthe conju-

gategradientiterationto solve quadraticprogrammingsubproblemsassociatedwith respective blocks.We

implementthemethodon a connectionmachinemodelCM-5 in thesingle-programmultiple-datamodelof

computation.We reportsomenumericalresults,which show that theproposedmethodis effective particu-

larly for problemswith someblockstructure.

E. Yamakawa, E. Maki, andM. Fukushima. An asynchronousblock-parallelalgorithm forseparablequadraticprogrammingproblems. Transactionsof the Institute of Systems,Control andInformationEngineers, 10(5), 248–258,1997.

Abstract. An asynchronousmodificationof theblock-parallelalgorithmis presentedfor quadraticprogram-

ming problemswith separableobjective functions.In particular, an implementationstrategy is proposedto

effectively decentralizethecommunicationoverheadassociatedwith a specificprocessorof a parallelcom-

puter. Throughextensive numericalexperimentson the ConnectionMachineCM-5, the efficiency of the

proposedmethodis examinedfor largescaleproblemswith someblockstructure.

C.W. Yang.An impactanalysisof aquadratic-programmingmodel—acaseof theAppalachian

coalmarketwith thepollutiontax. In ‘Useof Computersin theCoalIndustry’,Vol. 1983,

pp.665–669,1983.

E. K. YangandJ. W. Tolle. A classof methodsfor solving large,convex quadraticprograms

subjectto box constraints.MathematicalProgramming, 51(2), 223–228,1991.

Y. Ye.Furtherdevelopmentontheinterioralgorithmfor convex quadraticprogramming.Work-

ingpaper, Deparmentof EngineeringEconomicSystems,StanfordUniversity, California,

USA, 1987a.

Y. Ye. Interior pointalgorithmsfor linear, quadratic andlinearly constrainedconvex program-

ming. PhDthesis,EngineeringandEconomicSystemsDepartment,StanfordUniversity,

California94305,USA, 1987b.

Y. Ye. An extensionof Kamarkar’s algorithm and the trust region methodfor quadratic-programming.In N. Megiddo,ed.,‘Progressin MathematicalProgramming’,pp.49–63,SpringerVerlag,Heidelberg, Berlin, New York, 1989.

Abstract. An extensionof Karmakar(1984)’s algorithmandthetrust region methodis developedfor solv-

ing quadraticprogrammingproblems.This extensionis basedon theaffine scalingtechnique,followed by

optimizationover a trustellipsoidalregion. It createsa sequenceof interior feasiblepointsthatconverge to

theoptimalfeasiblesolution.Theinitial computationalresultsreportedheresuggestthepotentialusefulness

of this algorithmin practice.

Y. Ye. Interior-point algorithmsfor quadraticprogramming. In S. Kumar, ed., ‘RecentDe-

velopmentsin MathematicalProgramming’,GordonandBreachScientific Publishers,

Philadelphia,1991.

Y. Ye. On affine scalingalgorithmsfor nonconvex quadraticprogramming. MathematicalProgramming, 56(3), 285–300,1992.


Abstract. We investigatethe useof interior algorithms,especiallythe affine-scalingalgorithm, to solve

nonconvex—indefiniteor negative definite—quadraticprogramming(QP)problems.Althoughthenoncon-

vex QPwith a polytopeconstraintis a ”hard” problem,we show that theproblemwith anellipsoidalcon-

straint is ”easy”. When the ”hard” QP is solved by successively solving the ”easy” QP, the sequenceof

pointsmonotonicallyconverge to a feasiblepoint satisfyingboth the first andthe secondorderoptimality

conditions.

Y. Ye. Approximatingquadraticprogrammingwith boundconstraints.Workingpaper, Depart-mentof ManagementSciences,Universityof Iowa, IowaCity, USA, 1997a.

Abstract. We considerthe problemof approximatingthe global maximumof a quadraticprogramwith n

variablessubjectto boundconstraints.Basedontheresultsof GoemansandWilliamson(1996)andNesterov

(1997),weshow thata4 7 approximatesolutioncanbeobtainedin polynomialtime

Y. Ye. Indefinitequadraticprogramming.In ‘Interior-PointAlgorithm: TheoryandAnalysis’,

chapter9.4–9.5,pp.310–331.J.Wiley andSons,New York, USA, 1997b.

Y. Ye. On the complexity of approximatinga kkt point of quadraticprogramming.Workingpaper, Departmentof ManagementScience,Universityof Iowa, IowaCity, USA, 1997c.

Abstract. We presenta potentialreductionalgorithmto approximatea Karush-Kuhn-Tucker (KKT) point

of generalquadraticprogramming.We show that the algorithmis a fully polynomial-timeapproximation

scheme,andits running-timedependency on accuracy ε 0 1 is O ! 1 ε log 1 ε log log 1 ε !$ , com-

paredto the previously best-known resultO ! 1 ε 2 . Furthermore,the limit of theKKT point satisfiesthe

second-ordernecessaryoptimalityconditionof beinga local minimizer.

Y. Ye and E. Tse. An extensionof Karmarkar’s projective algorithm for convex quadratic

programming.MathematicalProgramming, 44(2), 157–179,1989.

T. L. Yen, C. K. Chi, andC. S. Wu. A quadraticprogrammingmethodfor interconnectioncrosstalkminimization.In ‘ISCAS’99. Proceedingsof the1999IEEEInternationalSym-posiumon CircuitsandSystemsVLSI, IEEE,Piscataway’, Vol. 6, pp.270–273,1999.

Abstract. Adjacentwires in a VLSI chip becomecloserasthefabricationtechnologyrapidly evolves.Ac-

cordingly it becomesimportantto considercrosstalkcausedby thecouplingcapacitancebetweenadjacent

wires in thelayoutdesignfor thefastandsafeVLSI circuits.Thecrosstalkis a functionof couplinglength

anddistance.Thecouplinglengthcanbereducedby segmentrearrangementtechnique.This paperpresents

a crosstalkminimizationtechniqueby adjustingthespacebetweenadjacentwires.An exampleis shown in

thispaperto demonstratetheeffectivenessof thetechnique.

C.M. Ying andB. Joseph.Performanceandstabilityanalysisof LP-MPCandQP-MPCcascadecontrolsystems.AICHEJournal, 45(7), 1521–1534,1999.

Abstract. Model predictive control (MPC) is usedextensively in industryto optimally controlconstrained,

multivariableprocesses.Fornonsquaresystems(with moreinputsthanoutputs),extradegreesof freedomcan

beusedto dynamicallydrivetheprocessto its economicoptimumoperatingconditions.Thisis accomplished

by cascadinga local linearprogramming(LP) or quadraticprogramming(QP)controllerusingsteady-state

models.Sucha cascadecontrolscheme(LP-MPCor QP-MPC)continuouslycomputesandupdatestheset

pointsusedby thelower-level MPCalgorithm.While thismethodologyhasbeenin useby industryfor many

years,its propertieshavenot beenaddressedin theliterature.Thepropertiesof suchcascadedMPC systems

areanalyzedfrom the point of view of implementationstrategies, stability properties,andeconomicand

dynamicperformance.Sometheoreticalresultsonstabilityarederivedalongwith acasestudyinvolving the

Shellcontrolproblem.


C. M. Ying, S. Voorakaranam,and B. Joseph. Analysis and performanceof the LP-MPCandQP-MPCcascadecontrol system. In ‘Proceedingsof the 1998AmericanControlConference.AmericanAutom. Control Council, Evanston,IL, USA’, Vol. 2, pp. 806–810,1998.

Abstract. The objective of this paperis to addresssomeof the theoreticalandpracticalissuesrelatedto

the implementationof two stagemodel predictive control (MPC) schemessuchas linear program-MPC

andquadraticprogram-MPC.SinglestageQDMC hasthe drawbacksof steadystateoffsets, inconsistent

set-pointsanddoesnot take into accounteconomicobjectives.Theadditionof anotherstageon the top of

QDMC is usedto addresstheseproblems.Resultsof theapplicationof two stageMPCto theShellChallenge

problemarepresented.A comparisonis madewith thesinglestageconventionalQDMCandsomeinteresting

observationsarereported.Stability anddynamicperformanceissuesarealsoconsideredin this paper.

D. B. Yudin. Stochasticquadraticprogramming.Engineering-Cybernetics, 3, 1–7,1969.

Abstract. Two modelsof controlunderconditionsof incompleteinformationarepresented.Economicmeth-

odsfor thesolutionof thecorrespondingstochasticquadraticprogrammingproblemsaregiven.

S. Zahl. A deformationmethodfor quadraticprogramming.Journal of the RoyalStatistical

Society, 26B, 141–160,1964.

S. Zahl. Supplementto ’a deformationmethodfor quadraticprogramming’. Journal of the

RoyalSatisticalSociety, 26B, 166–168,1965.

A. Zelner. Linear regressionwith inequalityconstraintson thecoefficients: anapplicationof

quadraticprogrammingandlineardecisionrules.Report6109,EconomicInstituteof the

NetherlandsSchoolof Economics,Rotterdam,Netherlands,1961.

H. Zhang,W. X. Zhong, and Y. Gu. A combinedparametricquadraticprogramminganditerationmethodfor 3-D elastic-plasticfrictional contactproblemanalysis. ComputerMethodsin AppliedMechanicsandEngineering, 155(3–4),307–324,1998.

Abstract. The paperis concernedwith the formulation and numericalrealizationof elastic-plasticfric-

tionalcontactproblems.Comparedwith two-dimensionalelastic-plasticcontactproblems,three-dimensional

elastic-plasticcontactproblemswith friction aremoredifficult to dealwith, becauseof the unknown slip

directionof the tangentialforce andexhaustingcomputingtime when the doublenonlinearproblemsare

considered.In order to avoid thesedifficulties, a combinedPQP(parametricquadraticprogramming)and

iterationmethodis constructed.The stiffnessmatrix of a 3D contactelementis introducedby meansof a

penaltyfunctionexpressionof thecontactpressureandfrictional force.Thecontactconditionandtheflow

rule areexpressedby thesameform asin a non-associatedplasticflow problem,andthepenaltyfactorscan

beeliminatedby usinga definitenumericaltechnique.Theiterationalgorithm,whosefunctionsareto give

a precisediscretizationof spaceCoulomb’s friction law, is usedalongwith thePQPmethodandalleviates

thedifficulty of unknown slip directionandcutsdown computingcosts.Also, theadditionalcomplementary

conditionsarediscussed,so that thenon-physicalray solutionscausedby theoriginal mathematicalmodel

canbeeliminated.Numericalexamplesaregivento demonstratethevalidity of thepresentmethod.

X. S. ZhangandH. C. Zhu. A neuralnetwork modelfor quadraticprogrammingwith simpleupperand lower boundsand its applicationto linear programming. In D. Z. Du andX. S. Zhang,eds,‘AlgorithmsandComputation.5th InternationalSymposium,ISAAC’94 Proceedings.Springer-Verlag,Berlin, Germany’, pp.119–127,1994.

Abstract. We put forwarda neuralnetwork modelfor quadraticprogrammingproblemswith simpleupper

and lower boundsandanalyzethe propertiesof solutionsobtainedby the model. It is shown that linear

programmingproblemscanbetransferredinto suchquadraticprogrammingproblemsandcanbesolvedby

themodel.


W. X. ZhongandS. M. Sun. A finite elementmethodfor elasto-plasticstructuresandcontactproblemsby parametricquadraticprogramming. International Journal for NumericalMethodsin Engineering, 26(12),2723–2738,1988.

Abstract. Thestiffnessmatrix of acontactelementis introducedby meansof apenaltyfunctionexpression

of thecontactpressureandfrictional force.Thecontactconditionandtheflow ruleareexpressedby thesame

form asin anon-associatedplasticflow problem.A unifiedPQP(parametricquadraticprogramming)model

relatedto contactproblemsaswell asto elasto-plasticstructuresis constructed.A seriesof PQPformulae

for contactproblemsandelastic-plasticstructuresis derived in the text, andsomenumericalexamplesare

illustratedaswell.

W. X. ZhongandS. M. Sun. A parametricquadraticprogrammingapproachto elasticcon-tact problemswith friction. Computers and Structures, 32(1), 37–43,1989. Seealso,ComputerModelling in OceanEngineering.1:651-658,1988.

Abstract. A numericalprocedureisdevelopedfor thesolutionof contactproblemswith friction. Thestiffness

matrix of the contactelementis introducedby meansof the penaltyfunction expressionsof the contact

pressureandfrictional force.Furthermore,the parametricquadraticprogramming(PQP)methodbasedon

thefinite elementmethodfor frictional contactproblemsis obtained,in analogywith that for elasto-plastic

problems.Applying somespecialtechniques,the penaltyparameterscanbe eliminated.The accuracy and

efficiency of themethodhasbeenverifiedin severalapplications.

W. X. ZhongandR. L. Zhang.Quadraticprogrammingwith parametricvectorin plasticityand

geomechanics.In ‘NUMETA 87: Transient/DynamicAnalysisandConstitutive Laws

for EngineeringMaterials’,Vol. 2, pp.595–602,1987.

W. X. ZhongandR. L. Zhang. Parametricvariationalprinciplesandtheir quadraticprogram-ming solutionsin plasticity. ComputersandStructures, 30(4), 887–896,1988.

Abstract. Two parametricvariationalprinciples,theparametricminimumpotentialenergy principleandthe

parametricminimumcomplementaryenergy principle,arepresented.Theseprinciplescanbeusedto solve

incrementalproblemsin plasticityandgeomechanicsin a directway andproblemswherethematerialsare

inconsistentwith the Drucker postulateof stability, suchas in nonassociatedflow or softeningproblems.

Examplesof applicationsincludeproblemssolvedby bothanalyticalandparametricquadraticprogramming

approaches.

C. Y. Zhu. On the primal-dualsteepestdescentalgorithmfor extendedlinear-quadraticpro-gramming.SIAMJournalon Optimization, 5(1), 114–128,1995.

Abstract. The aim of this paperis two-fold. First, new variantsareproposedfor the primal-dualsteepest

descentalgorithmasduein the family of primal-dualprojectedgradientalgorithmsdevelopedby Zhu and

Rockafellar[SIAM J.Optim.,3 (1993),pp.751–783]for large-scaleextendedlinear-quadraticprogramming.

Thevariantsincludea secondupdateschemefor theiterates,wheretheprimal-dualfeedbackis arrangedin

a new pattern,aswell asalternatives for the ”perfect line search”in the original versionof the reference.

Second,new linear convergenceresultsare proved for all thesevariantsof the algorithm, including the

original versionasa specialcase,without theadditionalassumptionsusedby Zhu andRockafellar. For the

variantswith thesecondupdatescheme,a muchsharperestimationfor the rateof convergenceis obtained

dueto thenew primal-dualfeedbackpattern.

K. ZimmermannandW. Achtziger. On timeoptimalpiecewiselinearschedulesfor LSGP-and

LGPS-partitioningsof arrayprocessorsviaquadraticprogramming.Technicalreport200,

Instituteof AppliedMathematics,Univeristyof Erlangen-Nuremberg, Germany, 1996a.

K. ZimmermannandW. Achtziger. Synthesizingregulararraysfrom singleaffine recurrences

via quadraticandbranchingparametriclinearprogramming.In ‘Parcella’96, Akademie

Verlag,Leipzig’, pp.224–231,1996b.


K. ZimmermannandW. Achtziger. On time optimal implementationof uniform recurrencesonto arrayprocessorsvia quadraticprogramming. Journal of VLSI SignalProcessingSystemsfor Signal,Image, andVideoTechnology, 19(1), 19–38,1998.

Abstract. Many importantalgorithmscanbe describedby n-dimensionaluniform recurrences.The com-

putationsarethenindexedby integral vectorsof lengthn andthedatadependenciesbetweencomputations

canbedescribedby thedifferencevectorof thecorrespondingindexeswhichareindependentof theindexes.

Thispaperaddressesthefollowing optimizationproblem.Givenann-dimensionaluniformrecurrencewhose

computationindexesaremappedby a linear function onto the processorsof anarrayprocessorembedded

in k-space(1 k n). Find anoptimal linear functionfor thecomputationindexes.We studya continuous

approximationof this problemby passingfrom linear to quasi-lineartiming functions.Theresultantprob-

lem formulationis thena quadraticprogrammingproblemwhich canbesolvedby standardalgorithmsfor

quadraticor generalnonlinearoptimizationproblems.We demonstratetheeffectivenessof our approachby

severalnontrivial testexamples.

G. Zioutas,L. Camarinopoulos,and E. Bora Senta. Robust autoregressive estimatesusingquadraticprogramming.EuropeanJournal of Operational Research, 101(3), 486–498,1997.

Abstract. Therobustestimationof autoregressive parametersis formulatedin termsof aquadraticprogram-

ming problem.This article’s main contribution is to presentan estimatorthat down weightsboth typesof

outliers in time seriesand improves the forecastingresults.New robust estimatesareyieldedby combin-

ing optimally two weight functionssuitablefor innovation andadditive outliers in time series.The tech-

niquewhich is developedhereis basedon an approachof mathematicalprogrammingapplicationto * p-

approximation.The behavior of the estimatorsare illustratednumerically, underthe additive outlier gen-

eratingmodel.Monte Carlo resultsshow that the proposedestimatorscomparedfavorably with respectto

M-estimatorsandboundedinfluenceestimators.Basedon theseresultswe concludethat onecanimprove

therobustpropertiesof AR(p) estimatorsusingquadraticprogramming.

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