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Index

Page numbers in italics referto principal references.

absolute ly continuous , 214absorbing set , 244abstract linear program, 110,

111Accessibility lemma, 7, 83act ive

const raint , 30set , 29, 98, 126, 160, 176

adjoint, 3, 12ffprocess , 115-122

affinecombination, 7fun ction, 3, 203

conjugate , 79hull , 7minorant , see minorant ,

affineset , 7

normals to , 19almost homogeneous, 81amenable, ix, 228-232analyt ic centre, 62approximate

crit ical point, see crit icalpoint, approximate

selection, 191-193approximately convex, 224arit hme t ic-geome t ric mean, 5, 12Asplund, 221att ainment, 4

and lower semi continuity,248

289

at tainment (cont .)du al,90in best approximat ion, see

distance fun ct ionattainment

in Fenchel problems, 52in LP and SDP, 109-113primal ,90quadratic program, 175, 205

Bairecategory theorem , 199, 244space, 239

ball , 1Banach

cont raction principle,179- 187

in metric space, 250space, 239-250

barreled, 239, 244base, see cone, base forBasic separation theorem , see

separ ation, BasicBau schke, Hein z, viiiBayes condit ional probability

rule, 86best approximation, 19BFGS update, 21, 62biconjugate, see Fenchel

biconjugateof process , 119

bipolarcone, see cone, bipolarset, 67, 69, 118

Birkhoff's theorem , 10, 12, 74

290

Bish op -Phelps theorem, 249, 250Bol tzm ann-Shan non entropy,

55, 245Bolzano

- Poinoare-Mirandatheorem, 206

- Weierst rass theorem, 3, 4,248

Borel measurable, 214bornology, 240Borsuk, 182

- Ulam theorem , 187-189Borwein- Preiss var iation al

principle, 250Bose-Einst ein entropy, 55Bouli gand, 140boundary, 2

properties, 142, 242bounded , 3ff

converge nce theorem, 215convex fun ctions, 242level set, see level set ,

boundedprocess, 117-122range, 210- 211set of multipliers, 162subd ifferent ial map, 242

Boyd , ixBregm an distance, 39broad crit ical cone, 174-177Brouwer , 182

fixed point theorem, 79,179-192, 206, 209-211

in Banach space, 250Browder-Kirk theorem , 183Bronsted-Rockafellar theorem ,

250

calculus of vari ati ons, viiiCanad ian , ixCaratheo dory's theorem, 25, 75,

225Cartesian product , Iff

tange nt cone to , 141

Index

Cauchy- Schwarz ineq uality, 1,10, 12, 63

and steepest desce nt , 31Cellina approximate selection

t heorem , 191-193,250cent ral path, 112

for LP, 112for SDP, 113

chain rul e, 52, 151, 166, 217,228, 234

change of vari able theorem,180-183

Ch ebyshev set , ix , 218-227Chi, Lily, see HedgehogClarke, ix

directional de rivat ive,124-144

normal cone , 139, 218calc ulus , 158versus lim iting, 170

subdifferential, 124-152 , 218and gene ric

di fferenti ability, 197and prox imal normals,

223as a cusco , 190, 194intrinsic, 133-135, 145,

149, 213-216of composit ion, 151of sum , 125

subgradient, 124unique, 132-134

tangent cone, 137-141, 170and met ric regu lar ity, 167and transvers ality,

158-159closed

fun cti on , see lowersemicontinuous vers usclosed fun ction

graph t heo rem, 118, 249images , 194level set , 76

Index

closed (cont.)multifunction, see

multifunction, closedrange, 241set, 2ffsubdifferential, see

subdifferential, closedmultifunction

subspace, 241closure

of a function, 78of a set , 2

codimensioncountable, 244finite, 246

coercive, 20multifunction, 202, 207

cofinite, 83Combettes, Patrick, viiicompact, 3ff

and Ekeland principle, 157convex hull of, 26convex sets, 67countably, 248general definition, 186, 191images, 194in infinite dimensions, 239level sets, see level set ,

compactoperator, 249polyhedron, 98range of multifunction, 194unit ball, 241weakly, see weakly compact

compactly epi-Lipschitz , 248complement, orthogonal, 3complementarity

problem, 202-208semidefinite, see

semidefinitecomplementarity

strict , 234-237complementary slackness, 29, 43,

45, 113

291

complementary slackness (cont.)in cone programming, 112

complemented subspace, 239complete, 182, 239, 244

and Ekeland's principle, 250composing

convex functions , 6USC multifunctions, 194 ,

200concave

conjugate, 62function , 33

condition number , 122cone, 1, 9ff

and processes , 114base for , 60

compact, 81, 121, 141bipolar, 53, 54, 58, 67, 76,

110, 115, 139contingent, see contingent

conecritical, see critical conedual,27finitely generated, see

finitely generated conegenerating, 120infinite-dimensional, 249lattice, 9nonnegative, 245normal , see normal conenormality of, 120open mapping theorem, 85partial smoothness of, 234pointed, 54, 60, 72, 86 , 99 ,

120, 141polar , vii, 53, 54, 67, 69, 160

of sum and intersection,58

polyhedral , see polyhedralcone

program, 110-113, 115, 116pseudotangent , 142recession, 5, 6, 61, 83 , 143self-dual , see self-dual cone

292

cone (cont.)semidefinit e, see

semidefinite conesums, 58, 245support function of, 55tangent, see t angent conevariational inequality over ,

202conical absorption, 71conjugate, see Fenchel conjugateconnected, 180, 183constraint

active, 30equality, 29, 153, 160error, 168function , 29inequality, 15-32, 160

and partial smoothness,234

convex, 43in infinite dimensions, 248

linear, 16, 19, 21, 52, 53, 62,109, 162

in infinite dimensions, 247qualification, 30, 200, 234

equivalence of Slater andMangasarian et al, 45

in cone programming, 116infinite-dimensional, 250linear independence, 30,

160, 162, 176Mangasarian- Fromovitz,

30-32, 12~ 160-165Mangasarian-Fromowitz,

232Slater, 44-47, 90, 91, 109,

110, 162, 168contingent

cone , 138-144, 153-162,228to feasible region, 160,

232necessary condition, 139,

157, 160, 161sufficiency, 143

Index

continuity, 3ffabsolute, 214and bounded level sets, 78and maximality, 198and USC, 193generic, 199in infinite dimensions ,

239-250of convex fun ctions, 52,

65-69, 76, 241failure, 84univariate, 83

of extensions, 196of linear functionals , 241of multifunctions, 114of partial derivatives, 132of projection, 20, 201of selections, 191-199

continuously differentiable, seedifferentiability,continuous

contour, 82contraction, 179

Banach space, 250non-uniform, 183

control theory, vii , 112convergent subsequence, 3ffconvex

analysis, vii-ixinfinite-dimensional, 68,

249monotonicity via, 209-211polyhedral, 97

approximately, 224calculus, 52, 53, 56, 139

failure , 57sum rule, 52

combination, 2, 5ffconst raint, 43fffunction, 4, 33, 44ff

bounded, 242characterizations, 37composition, 6

Index

convex fun ction (cont .)condit ions for minimizer ,

16cont inuity of, see

continuity of convexfunctions

cr it ical points of, 16, 33difference of, 57 , 108, 219differentiabili ty of, 36directional derivat ive, 16examples, 39exte nded-valued, 33 , 43,

46ffHessian characteri zation ,

see Hessian, andconvexity

of matrices , 40on Banach space, 249recogni zin g, 37 , 38regu larity, 131, 138sy mmetric , 27

growth cond it ions , 7hull, 2, 5ff

and exposed points, 249and extreme p oints, 68and .Gordan 's theo rem, 23of lim iting subdifferenti al,

145, 149im age, 190-199log- , 41midpoint , 80multifunction, 114 , 115order- , see order-convexprocess, see processprogram, 43-48, 51ff

duality, 88Schur-, 25, 27, 38 , 108, 135set, 2ffspectra l fun ction, see

spect ral fun cti onstrictly, see strictl y convex,

essentia llysubd ifferentia l, 131ff

and limiting, 145

293

convex ified represent ative,209-210

convexity, see convexand continuity, see

continuity of convexfun cti ons

and differenti ability, 15and monotonicity, 129in linear spaces, 249in optimization, 33ffof Chebys hev sets, ix ,

220 -227core, 34

in infin ite dimensions, 244versus interior, 37, 67

cost funct ion , 204countable

basis, 244codime nsion, 244

countably compact, 248cover , 191, 248crit ical cone

broad, 174-177narrow, 172-177

crit ica l point , 16approximate, 17st rong , 234- 238unique, 19

curvature, 172cusco, 190- 211 ,219-223

DAD problems, 42, 62Davis ' theor em , 105, 106, 108DC function , see convex

fun ction, difference ofDebrunner-Flor exte ns ion

theorem , 209-211dens e

hyp erplane, 244range, 246, 248subspace, 245

der ivative, see differe nt iabilitydirectional , see direct ion al

de rivative

294

derivative (cont .)Frechet , see Frechet

derivativeGateaux, see

differentiability,Gat eaux

generalized, 123Had amard, see Had amard

derivativestrict , see strict derivativeweak Hadamard, see weak

Hadamard derivativedet erminant , 9, 163, 180, 183

order preservation, 107Deville-Godefroy-Zizler vari­

at ional principle, 250differentiability

and pseudoconvexity, 143bornological , 240cont inuous, 132-134, 157,

159, 164approximat ion by, 180

Frechet , see Frechetderivative

Gateaux, 15, 28, 61,130-136, 139, 240-245

generic, 197, 199of convex funct ions , 36, 82of dist ance funct ion , 57,

218,222of Lipschitz functions, ix,

133of spectral funct ions , 105of the conjugate , 225strict , see st rict derivativetwice, 172-176, 231, 233

differential inclusion , 249dimension, 68

infinite, see infinitedimensions

Dinicalc ulus, failure, 128derivative, 127, 214

Index

Dini (cont.)directional derivative, 145,

147and contingent cone , 137cont inuity, 128, 146Lipschitz case , 123, 129,

131subdifferential , 124, 125,

129, 131, 145of distan ce function, 169,

218surjective , 128

subgradient, 124, 146, 148exists densely, 135, 145,

150Dirac, see Fermi-Diracdirecti on al der ivative, 15, 61

and subgradients, 35, 123and tangent cone, 137Clarke, see Clarke

directional derivativeDini , see Dini

subdifferenti alMichel-Penot, see

Michel-Penotdi rectional derivative

of convex fun ction, 34- 42of max-functions, 28, 33, 38sublinear, 34, 123, 124, 137

disjoint op erator ranges, 248distance

Bregm an, 39from feasibility, 168function, 57, 133, 137-144,

218-226attainme nt, 241, 243, 248differ ent iability, 241di rectional derivative , 144regularity, 138, 246subdiffere nt ials, 169to level set, 171

to inconsistency, 119, 122divergence bounds, 63

Ind ex

domainof convex function , 33 , 44,

65of mu lt ifunction, 114of subdifferential, 35

not convex , 40po lyhedral, 97

doubly stochastic , 10, 12, 75pattern , 42

Dowker , 196dua l

attainment , see attainment,dual

cone, 27func t ion, 88linear progr am, 109, 202problem, 88

examples, 91solut ion, 82, 89, 90space, 239value , 52, 88-96

in LP and SD P , 109-113du ality, vii, 2, 76ff

cone program , see coneprogram

duali ty-based algorit hms , viiFenche l, see Fen chel duali tygap, 88-96

Duffin's, 46, 92in LP and SDP, 110-113,

203geometric progr amming, 103in convex programming, 88infinite-dimensional , 91, 249Lagr angian, see Lagrangian

dualityLP, vii, 25, 109-113, 202nonconvex, 93norm, 117process, 114-122quadratic progr amming, 205SDP, vii, 109-113st rict-smoot h , 78, 82

295

du ali ty (cont.)weak

cone program, 109, 110Fenchel, 52-53, 101Lagrangian , 88 , 91

Duffin 's duality gap , see du ali ty,gap , Duffin 's

efficient , 204eigenvalues , 9

derivati ves of, 135functions of, 104isotonicity of, 136lar gest , 162of operators, 249op timizati on of, 106subdifferenti als of, 135sums of, 108

eigenvector , 19, 163Einstein , see Bose- EinsteinEke land variat ional pri nciple, 17,

153-157, 179, 224, 225in metric space, 250

engineering , ixentropy

Boltzmann- Shann on , 55Bose- Einstein , 55Fermi- Dirac, 55maximum, 41, 56, 62

and DAD problems, 42and expected sur prise, 87

epi-Lipschitz-like, 248epigraph, 43ff

as multifunction graph , 193closed , 76, 81normal cone to , 47polyhedral , 97regula rity, 246suppo rt functi on of, 55

equilibrium , 193equivalent no rm , see norm ,

equivalentessent ially smooth, 37, 74, 80

conjugate, 78, 82

296

essentially smooth (cont.)log barriers, 51minimizers, 40spectral functions , 106

essent ially strictly convex, seestrictly convex,essentially

Euclidean space, 1-9, 239subspace of, 24

exact penalization, seepenalization, exact

existence (of optimal solution,minimizer) , 4, 79, 90ff

expected surprise, 86exposed point, 73

strongly, 249extended-valued, 145

convex functions , see convexfunction ,extended-valued

extensioncontinuous, 196maximal monotone, 210

extreme point, 67existence of, 73of polyhedron, 98set not closed, 73versus exposed point , 73

Fan- Kakutani fixed point

theorem, 190-201, 203inequality, 10-14, 104, 105minimax inequality, 205theorem, 10, 13

Farkas lemma, 23-25, 109, 160and first order conditions,

23and linear programming,

109feasible

in order complementarity,205

region, 29, 160

Index

feasible region (cont .)partly smooth, 234

solution, 29, 43, 110Fenchel, ix, 54

- Young inequality, 51,52,71, 105

-Young inequality, 225biconjugate, 49, 55, 76-85,

99, 105, 106, 126and duality, 89and smoothness, 225, 226

conjugate, 23, 49-63and Chebyshev sets, 221and duality, 88and eigenvalues, 104and subgradients, 51examples, 50of affine function, 79of composit ion, 93of exponent ial, 49, 56, 62,

63of indicator function , 55of quadratics, 55of value function, 89self-, 55strict-smooth duality, 78,

82transformations, 51

duality, 52-63, 73, 77, 81,102

and complementarity, 204and LP, 110-113and minimax, 201and relative interior, 74and second order

conditions, 174and strict separation, 70generalized, 102in infinite dimensions,

239,249linear constraints, 53, 62,

71, 100polyhedral, 100, 101symmetric, 62

Index

Fenchel d ua lity (cont.)versus Lagr angian, 93

problem , 52Fermi- Dirac entropy, 55F illmore-W illia ms theorem , 108finite cod imension, 246finite dimensions, 239- 250fini tely generated

cone, 25, 26, 97-99function, 97-101set, 97-101

first order cond it ion (s)and max-functions , 28- 32and the Farkas lemma, 23for optimality, 16Fri t z John, see Fritz John

condit ionsin infinite dimensions, 248Karush-Kuhn-Tucker , see

Karush - Kuhn- Tuckertheorem

linear constraints , 16, 19,21,42

necessary, 15, 16, 29, 139,160,174, 175

sufficient , 16Fish er informati on , 79, 84, 87F itz patrick fun ction, 209fixed point , 179-211

in infin ite dimensions , 250methods, 203property, 184theorem

of Brouwer , see Brouwerfixed point theor em

of Kakutani-Fan, seeKakutani-Fan fixedpoint t heorem

Fourier identification, 246Frechet derivati ve, 132-134, 153,

213-216and continge nt necessary

condition , 139, 157and inversion, 184-185

297

Frechet derivative (cont.)and mult ip liers , 163an d subderivatives , 152in constraint qualification ,

160in infini te d imensions,

240- 250Fri t z J ohn conditions , 29- 31,

130, 165and Gordan 's t heorem, 30nonsmooth, 127second or der , 175

Fubini 's theo rem, 214- 216fun ctional analys is, 239, 248fundam ental theorem of

ca lculus , 214, 216fur thest point , 73, 221, 226-227fuzzy sum rule, 146, 148, 150

G8, 191, 197gamma function , 41Gateaux

derivati ve, see de rivative ,Gateaux

different iable, seedifferentiability,Gateaux

gauge function , 66 , 71, 184genera lized

deriva ti ve, 123Hessian, 217J acobi an, 216-217

generated cuscos, 196gen erating cone , 120generic, 190

cont inuity , 199differenti abili ty, 197, 199single-value d , 199

geometric progr amming, 101,102

global minimizer, see minimizer ,global

Godefroy, see Deville-Godefroy­Zizler

298

Gordan's theorem, 23-27and Fritz John conditions,

30graph, 114, 190

minimal , 197normal cone to , 150of subdifferential , 145

Graves , 157Grossberg, see Krein-GrossbergGrothendieck space, 243growth condition, 4, 20

cofinite, 83convex, 7multifunction, 201, 208

Guignardnormal cone calculus, 158optimality conditions, 164,

177

Haberer , Guillaume, viiiHadamard, 182

derivative, 240-250inequality, 48, 163

Hahn- Banach extension, 55, 58

geometric version, 248- Kat et ov- Dowker sandwich

theorem, 196Hairy ball theorem, 186-187halfspace

closed, 3, 25ffin infinite dimensions, 246op en, 23, 25support function of, 55

Halmos, viiiHardy et al. inequality, 10-12Hedgehog theorem, 187hemicontinuous, 198Hessian, 17, 172-176

and convexity, 37, 38, 40generalized, 217

higher order optimalityconditions, 175

Hilbert space, 221, 239

Index

Hilbert space (cont.)and nearest points , 249

Hiriart.-Urruty, vii, 25Holder's inequality, 31, 41, 71homeomorphism, 182, 184homogenized

linear system, 109process , 120

hypermaximal, 198, 207, 208hyperplane, 2, 25ff

dense, 244separating, see separation

by hyperplanessupporting, 67, 122,

240-249

identity matrix, 9improper polyhedral function ,

101incomplete, 239inconsistent, 29, 111

distance to, 122indicator function , 33, 67, 137

limiting subdifferential of,146

subdifferential of, 37inequality constraint, see

const raint , inequalityinfimal convolution, 57, 137, 157infimum, 3ffinfinite-dimensional, viii , 79,

157 , 239-250interior, 2ff

relative, see relative interiortangent characterization,

170versus core, see core versus

interiorinterior point methods, vii-ix,

54, 79, 91, 162inverse

boundedness, 120function theorem, 159, 184,

235

Index

inverse (cant .)image, 3, 100Jacobian , 180multifunction , 114- 122

inversion, 221, 226Ioffe, 149isomet ric, 85, 86isotone, 6, 205

cont ingent cone , 141eigenvalues, 136tangent cone, 143

J acobian, generalized , 216-217James t heore m, 243, 249Jordan 's theorem, 163Josephson-Nissenzweig

sequence, 243theorem , 243

Kakut ani- Fan fixed point t heorem,

190- 201, 203minimax t heorem , 96, 206

Karu sh-Kuhn-Thckertheorem, 30-32, 130, 160

convex case, 43- 45, 131infini te-dimension al , 250nonsmooth, 127

vector, 47, 93Katetov, 196Kirchhoff 's law, 20Kirk , see Browder-KirkKlee cavern, 221Knaster- Kur atowski-Mazurkie­

wicz principle, 185 , 206Konig, 11Kr ein

- Grossberg t heore m, 120- Rut man theorem, 54, 158-Rutman t heo rem , 230

Kruger, 149Kuhn, see Karush-Kuhn-Tucker

Lagrange mult iplier , 17, 29-32,161

299

Lagrange multiplier (ca nt .)and second order

condit ions , 172-176and subgradients, 43bounded set, 162convex case, 43-47in infinit e d imensions, 249nonexistence, 46, 163, 185

Lagrangian , 29, 172- 176convex, 43, 88du ality, 88-96, 103

infinite-di mension al , 249linear progr amming, 109

necessary condit ions, seenecessary condit ions,Lagrange

sufficient conditions , 43- 48,107

Lambert W-function, 58lat tice

cone, 9ordering, 11, 203

Ledyaev, ixLegendre, 54Lemarechal, Claude, vii , viiilevel set , 3, 13

bounded , 4, 7, 69, 78, 83closed , 76compact, 20, 41, 51, 95, 154

of Lagrangian , 90, 91distan ce to , 168norm al con e to , 47, 171

Ley, Olivier , viiilimit (of seq uence of points) , 2limiting

mean value t heorem , 151norm al cone, see normal

cone, limi tingsubdifferent ial, 145-152

and regul arity, 166-171of compos it ion, 150of distance fun ction , 170sum rule, see nonsmooth

calculus

300

line segme nt, 2, 142lin eality space, 34linear

constraint , see constraint ,linear

fun ctionalcont inuous , 241discontinuous, 248

independence qualifi cation ,see constraintqualification, linearindependence

inequality constraints, 62map,3ff

as process, 120object ive , 109operator, 249progr amming (LP), vii, 54,

91abs t ract, 1l0, Ill , 202and Fenchel du ali ty,

1l0-1l3and processes, ll4and variational

inequaliti es, 203duali ty, see du ali ty, LPpenalized , 91, ll3, 162primal problem , 109

space, 249span , 2subs pace, 2

linearization, 153Lipschitz, ix , 65,66,68, 123-152,

155, 183, 213-217bomological derivatives, 240eigenvalues, 108, 135extension, 157gene ric differentiability, 197non- , 127perturbation, 250

Liustemik, 157t heorem, 156, 158, 160

via inverse funct ions, 159local minimizer , 15-19, 29ff

Index

local minimizer (cont .)strict , 174

localiza tion , 120locally bounded , 65, 66, 68, 78,

190, 194, 196-198locally Lipschitz, see LipschitzLoewn er orde ring , 9log, 5, 13, 49, 55, 62, 92, 104log barrier , 49log det , 13, 15, 20, 21, 32, 37, 40,

46, 48, 49, 55, 69, 92,104-106

log-convex , 41logarithmic homogeneity, 79, 82lower norm, ll7lower semi continuous, 37, 76-81 ,

101and attainment, 248and USC , 193approximate minimizers,

153ca lculus , 146-148gener ic cont inuity, 199in infinite dimensions , 239mul tifuncti on , 114sandwich theorem, 196value fun cti on , 89, 90, 95versus closed fun cti on , 76

LP, see linear programmingLSC (multi function) , ll4-119,

190, 192-196, 204Lucet , Yves , viii

Mangasarian- Fromovitzcons traint qualification,see const raintqu alifica tion ,Mangasarian­Fromovitz

manifold , 233-238mathem ati cal economics , viii ,

ll9, 193matrix , see also eigenvalues

analysis, 104

Index

matrix (cont.)completion , 21, 40optimization, 109

Max formula, 36- 42, 53, 61 , 116,123

and Lagrangian necessarycondit ions , 44

nonsmooth, 124, 125, 139relativizing, 42, 74

max-function(s)and first order condit ions,

28-32directional derivative of, 28subdifferential of, 47 , 59,

125Clarke, 129, 151limiti ng, 151, 171

maximal monotonicity, 190-211maximizer , vii , 3ffmaximum ent ropy, see entropy,

maximumMa zur kiewicz, see

Kn as ter-Kuratowski­Mazurkiewicz

mean value t heorem, 128, 136,217

infinite-d imensional, 250limi ting , 151

measure t heory, 213-217metric regularity, vii, 153-159,

183 , 184, 229and second order

conditions, 172- 173and sub differentials,

166-171in Banach space, 250in infinite dimensions , 239weak, 154-158

metric space , 250Michael select ion theorem ,

193-196, 204infinite-dimension al, 250

301

Michel-Penotdirectional derivative,

124-144subdifferential, 124-135subgrad ient , 124

unique, 130, 132midpoint convex , 80minimal

cusco, 219 -223graph, 197solut ion in orde r

compleme ntarity, 205minimax

convex-con cave, 95Fan 's inequality, 205Kakutani's theorem, 96, 206von Neumann 's theorem,

see von Neumannminimax t heorem

minimizer , vii , 3ffand differen ti ability, 15and exact penalization, 137app roximate , 153existence , see existenceglobal, 4, 16, 33fflocal, 15-19, 29ffnonexistence, 17of essentially smooth

fun ctions, 40strict , 174subdifferential zeroes, 35 ,

123minimum volume ellipsoid, 32,

40, 48Minkowski , 5, 101

theorem, 68, 73, 98 , 182converse, 73in infinit e dimens ions, 249

minorant , 76affine, 76, 79, 84, 100closed , 78

Miranda, see Bolz ano-Poincare­Miranda

monotonically re lated, 209-210

302

rnonotonicityand convexity, 129maximal. 190-211multifunction , 190-211of complementarity

problems, 203, 204of gradients, 40via convex analysis , 209-211

Mordukhovich , 149, 169Moreau, 54

- Rockafellar theorem ,78-83, 249

Motzkin-Bunt theorem, 221multicriteria optimization, 140multifunction, vii, 114-122 ,

190-211closed , 80

and maximal monotone,198

versus USC, 193set-valued map , 114subdifferential, 35

mult iplier , see Lagrangemultipli er

mult iva luedcomplementarity pro blem ,

202variational inequali ty, 200

narrow crit ical cone, 172-177Nash equilibrium, 204, 206near est point , 19, 24, 57, 182,

188and prox-regularity, 228and proximal normals ,

218-223and subdifferent ials, 169and variat iona l inequali ties,

200in epigraph, 135in infinit e dimensions , 239,

248in polyhedron, 62proj ection , 20, 211

Index

nearest point (cont.)selection, 193, 199un iqu e, see also Chebyshev

set , 228necessary condit ion(s) , 125, 138

and subdifferenti als , 123and sufficient, 175and variati on al inequalities,

200contingent , see cont ingent

necessary conditionfirst or der, see first order

condit ion (s), necessaryfor optimality, 16Fritz John, see Fritz John

condit ionsGui gnard , 164, 177high er order , 175Karush -Kuhn-Tucker , see

Karush- Kuhn- Tuckert heore m

Lagrange, 44-46, 49, 89,130, 131

nonsmooth, 126, 130, 139,145, 149, 151

limi ting and Clarke, 170second orde r, 172stronge r, 127, 145

neighbourhood , 2Nemirovski , ixNesterov , ixNewton-type methods, 172Nikodym, see Radon-NikodymNissenzweig, see

J osephson-Nissenzweignoncompact variat ional

inequality , 202nondifferenti able, 18, 33ffnonempty images , 114, 118nonexpansive, 180 , 182, 220

in Ban ach space, 250nonlinear

equation, 179progr am, 160, 177

Index

nonnegative cone , 245nonsmooth

analysis , viiffand metric regul arity, 157infinite-dimensional , 149Lipsch itz , 137

calculus, 125, 128, 139, 155and regul arity, 133equality in , 131failure , 145, 149fuzzy, 146infinite-dimensional , 250limiting, 145, 148-151 ,

167, 169mixed , 134nor me d fun ction, 166sum rule, 125

max formulae, see Maxformula , nonsmooth

necessary conditions, seenecessary condition(s) ,nonsmooth

optimization, seeoptimization,nonsmooth

regularity, see regularnorm, 1

-attaining, 240, 243 , 245,249

-preserving, 10, 12equivalent , 66, 69 , 192lower , 117of linear map, 117of process, 117-122smooth, 188strictly convex, 249subgradients of, 38top ology, 241-243upper , 117

normalproximal , ix, 218-223

normal cone, 15, 16, 18and polarity, 53and relative int erior, 74

303

normal cone (con t.)and subgradients, 37, 56and tange nt cone, 54Clarke, see Clarke normal

coneexamp les, 18limiting, 146, 166-171

and subdifferential , 150proximal , 218to epigraph, 47to graphs, 150to intersecti on , 56, 86to level sets, 47

normal mapping, 200, 203normal problem, 91normal vector, 15, 218normed space, 2:{9, 244null space, 3, 116, 117

objective fun ct ion , 29, 30fflinear , 109

one-sided approx imat ion , 35open, 2

functions and regularity,169, 183

mapping theorem, 71, 82 ,101, 110, 120

for cones, 85for processes, 118in Banach space, 249in infinite dimensions, 239

multifunction, 114-121operator

linear , 249op timal

cont rol, viiisolution , 4ffvalue, 51, 88-96, 100, 174

function , 43in LP and SDP, 109-113,

202primal , 52

optimality conditi ons, vii, 15-22and the Farkas lemma , 24

304

optimality conditions (cont.)and variational inequ alit ies ,

200first order, see first order

condition(s) foroptimality

higher order, 175in Fenchel problems, 56, 82necessary, see necessary

condit ion(s)nonsmooth, 123second order, see second

order conditi onssufficient, see sufficient

condit ion (s)optimizat ion, vii, 3ff

and calculus, 16and convexity, 33and nonlinear equations,

179computational, vii , 162, 172duality in, 76, 88infin ite-dimensional , viii, 79,

157linear , 109matrix, 109mult icriteria, 55nonsmooth, 28, 33, 123-152

infini te-dimensional , 250one-sided approximati on , 35problem , 4, 29ffsubgradient s in , 35, 123vector, 72, 140, 141

order-convex , 59-62, 72, 80, 108-reversing, 49-sublinear , 59- 62, 108, 121-theoreti c fixed point

results, 179complementarity, 203- 205epigraph, 121infimum, 60interval, 120preservation, 11, 72

Index

order preservat ion (cont.)of determinant , 107

statist ic, 129regular ity, 135subdiffer ential , 152

subgradients, 55, 60-62ordered spectral decomposit ion ,

10ordering , 9

lat ti ce, 11orthogonal

complement, 3invariance, 107matrix, 10, 182proj ection , 25similarity t ransformation,

107to subspace, 24

orthonormal basi s, 163

p-norm , 31, 71par acompact , 250Pareto minimization, 72, 204

proper , 141parti tion of uni ty, 191-195, 207partly smooth , ix , 233-238pen alizati on , 91, 113, 162

exact, 137-140, 155, 158,167, 229

qu ad rati c, 164Penot , see Miche l-Penotpermut ati on

matrix, 10, 27, 75, 108perturbati on , 43, 51ffPhelps, see Bishop-Phelpspiecewise linear , 184Poincar e, see Bolzano- Po incare-

Mirandapoin ted , see cone, pointedpointwise maximum, 79polar

calc ulus, 70, 117concave, 85cone, see con e, polar

Index

polar (cont.)set , 67, 69~70

polyhedralalgebr~ 100-101, 116calculus , 101complementarity problem ,

205cone , 98, 102, 110 , 113, 161Fenche l duality, 100fun cti on, 97-102mult ifunction, 114problem, 109, 110process, 116qu asi- , 175set, see pol yhedronvariat ional inequali ty, 203

polyh edron , 3, 9, 11, 58, 97- 102compact, 98in vector optimization, 141infinite-dimensional , 246nearest poin t in , 62partial smoothness of, 237po lyhedral set , 97tangent cone to, 101

polynomialnearest , 21

polytope, 55 , 97-99in infinite dimensions, 246

positive (semi)de finite, 9ffpos it ively hom ogeneous , 33P reiss, see Borwein-Proisspr imal

linear progr am, 109problem , 88recovering soluti ons, 82semidefi nite progr am, 111value, see op timal value

process, 114-122, 249product , see Cartesian produ ctprojecti on, see also nearest poin t

continuity, 223onto subspace, 24orthogonal, 25re laxed, 182

305

properfunct ion , 33, 44, 76, 97, 116P areto minimization , 141point , 142

prox-regular, 228-238proximal nor mal, ix , 218-223pseudoconvex

fun ction, 143set, 142, 143

P shenichn ii- Rockafellarcondit ions, 58

qu adrat icapproximation, 172-175conjugate of, 55path, 173penalizati on , 164program, 91, 175, 205

quasi relative interi or , 244, 248qu asiconcave, 205quasipolyh ed ral , 175quotient space, 247

Rademacher's theorem , ix, 133,197, 213-216, 218

Radon- Nikodyrn prop erty, 249Radst rom cancellation, 5range

closed , 241dense, see dense range

range of mult ifun ction , 114, 191,194, 201

bounded , 210-211rank-one, 122ray, 242, 247Rayleigh quotient , 19real function , 123recession

cone, see cone, recessionfunction, 83

reflexive Banach space, 239- 249regular , 130-136, 138

and generic differe nt iability,197

306

regul ar (cont .)and Rademacher 's theorem,

216regul arity

condit ion, 30, 44, 65, 100,160

epigraphical, 246metric, see metric regul ari typrox- , see prox-regulartange nt ial, see t an genti al

regularityrelative interior , 5-8, 173, 185

and cone calculus, 159and cone programming, 113and Fenchel duality, 74, 102and Max formula, 42calc ulus, 74in infinite dim ensions , 242,

248qu asi , 244, 248

relaxed projection , 182representa tive, 209-210reso lvent , 208retraction, 180, 183reversing, 206Riesz lemma, 188Robinson , 119, 157Rockafellar, vii- ix, 54, 58, 78,

119, 250Ru tman, see Krein-Rutman

saddlepoint, 95, 96, 201Sandwich theorem , 58, 210

Hahn-Katetov-Dowker, 196scalarization, 72, 140, 142Schur

-convexity, see convex,Schur-

space, 243Schwarz , see Cauchy- Schwa rzSDP, see semidefinite progr amsecond order condit ions , 17,

172-177, 237selection , 190-1 99

Index

self map, 179-1 88, 207in Banach space, 250

self-conjugacy, 55self-dual cone , 18, 53, 54, 85,

105, 111selfadjoint , 249semidefini te

complementarity , 108, 208cone, 9, 18, 53, 54, 104, 106,

109,233matrix, 9progr am (SDP) , vii, 54, 92,

109-113, 162central path, 113

Sendov, Hristo, viiisensitivity analysis, 233separable, 62 , 92

and sem icontinuity, 247Banach space, 241-245

separation, 2, 5, 25ffand bipolars , 54, 67and Gordan 's theorem, 23and Hahn-Banach , 248and scalar ization, 142Basic t heorem, 2, 17, 77by hyperplanes, 3in infinit e dimensions, 241nonconvex , 142st r ict, 70strong, 6

set-valued map, seemulti fun ction

Shannon, seeBoltzmann-Shannon

signal reconstruction, 79simplex, 66 , 79simultaneous orde red sp ectral

decomposition , 10, 105single-valued, 190, 197

gene ric, and maximalmon otonicity, 199

sing ular value, 13largest , 163

skew symmetric, 197

Index

Slater condition, see constraintqualification, Slater

smooth Banach space, 241solution

feasible, see feasible solutionoptimal, 4ff

solvability of variationalinequalities, 201-208

spectralconjugacy, 104, 106, 107decomposition, 10, 19differentiability, 105function , 104-108, 133

convex, 105, 106subgradients, 105, 107theory, 249

sphere, 180, 186-189inversion in , 226

square-root iteration , 11st able, 91

Clarke tangent cone, 138steepest descent

and Cauchy-Schwarz, 31Stella's variational principle, 250St ern, ixStiemke's theorem, 26Stone-Weierstrass theorem,

180-183strict

compleme nt ar ity, 234-237derivative, 132-134, 149,

150, 155-167generic, 197

local minimizer , 174separation, 70

strict-smooth duality, 78, 82strictly convex, 4, 38-41

and Hessian, 38conjugate, 78, 82essentially, 35, 40, 84log barriers, 51norm, 249power fun ction, 21spectral functions , 106

307

st rictly convex (cont.)unique minimizer, 19

strictly differentiable, see strictderivative

strongcrit ical point , 234-238

subadditive, 34subcover, 191subdifferential , see also

subgradient(s)and essential smoothness , 74bounded multifun ction , 242calculus, 123Clarke, see Clarke

subdifferentialclosed multifunction, 80,

134, 145, 149, 156, 167compact ness of, 66convex, see convex

subdifferentialDini, see Dini

subdifferentialdomain of, see domain of

subdifferentialin infinite dimensions, 250inverse of, 80limiting, see limiting

subdifferentialmaximality, 240Michel-Perrot , see

Michel-Penotsubdifferential

monotonicity, 190, 197, 198nonconvex, 123nonempty, 36 , 240of distance functions ,

219-223of eigenvalues, 135of polyhedral fun ction, 102on real line, 149smaller , 145support function of, 55versus derivative, 123

subgradient (s) , vii , 35

308

subgradient(s) (cont .)and conjugation, 51and Lagrange mul tipliers, 43and lower semi continuity, 81and normal cone, 37, 56at op timality, 35Clarke, see Clarke

subgradientcons t ru ct ion of, 35Dini, see Dini subgradi entexistence of, 36, 43, 53, 100,

116in infinite dimensions, 239Michel-Penot, see

Michel-Penotsubgradient

of convex functions, 33-42of distance functions,

219-223of max-functions, see

max-function,subdifferenti al of

of maximum eigenvalue , 38of norm, 38of polyhedral function , 98of spectral functions, see

spect ral subgradi entsorder , see order subgradientunique, 36, 242, 245

sublinear, 33, 35, 58, 66, 69, 85,107, 108, 137

and support functions, 77directional derivative, see

directional derivative,sublinear

everywhere-finite, 77order- , 59-62recession functions, 83

subspace, 2closed , 241compleme nted , 239countable-codimensiona l,

244dense, 245

Index

subspace (cont .)finite-codimensional, 246projecti on onto, 24sums of, see sum of

subspacessufficient condit ion(s)

and pseudoconvexity, 143first order , see first order

condit ion(s) , sufficientLagrangian , see Lagrangian

sufficient condit ionsnonsmooth, 149partly smooth, 234second order, 174

sumdirect , 3of cones, see cone sumsof sets , 1of subspaces, 245, 247rule

convex, see convexca lculus

nonsmooth, seenonsmooth calculus

sun, 220-224support fun ction (s), 55, 80, 82

and sublinear fun ctions, 77directional derivative,

124-128of subdifferentials, 125

support poin t , 240-245suppor t ing

functional , 240-245hyperplane, see hyperplane,

supportingsupremum, 3

norm, 243surjective

and growt h, 198, 207and maximal monotone,

198, 208Jacobian, 155, 156, 159,

166, 173, 176, 184linear map , 71, 101, 110

Index

surjective (cont.)process, 114-122

surprise, expected, 86symmetric

convex function, 27function, 104-108matrices, 9-14set , 108

tangency properties, 241tangent cone , 137-144

and directional derivatives,137

as conical approximat ion,137

calculus, 73, 86, 159, 228Clarke, see Clarke tangent

conecoincidence of Clarke and

contingent, 138convex, 54, 74, 138ideal , 143intrinsic descriptions, 138,

140to graphs, 141, 150to polyhedron, 101

t angent space, 157tangent vector field , 186tangential regularity, 138, 156,

158, 229, 233, 246Theobald's condition, 13theorems of the alternative,

23-27, 97Todd, Mike, viiitrace, 9transversality, 158, 164, 228-232,

234trust region, 93Tucker, see

Karush-Kuhn- TUckertwice differentiable, see

differentiability, twic e

Ulam, 182

309

uniformbounded ness theorem, 249convergence, 180, 195multipliers, 176

uniquefixed point, 179, 183minimizer, 19nearest point, 248subgradient, see

subgradient, uniqueupper norm, 117upper semicontinuity (of

multifunctions) , 117Urysohn lemma, 196USC (multifunction), 190-207

value function, 43-48, 52, 88-91,116, 119

polyhedral, 100Vandenberghe, ixvariational

inequality, 200-208principle, 17

in infinite dimensions,239, 250

of Ekeland, see Ekelandvariational principle

vector field, 186-187vector optimization, see

op timization, vectorVille's theorem, 26viscosity subderivative, 149, 151von Neumann, 11

lemma, 13minimax theorem, 79, 81,

201, 204

Wang, Xianfu, viiiweak

-star topology, 241-243duality, see duality, weakHadamard derivative,

240-241

310

weak (cont.)metric regul arity, see metric

regularity, weakminimum, 72topology, 241-243

weakly compac t , 243, 249and nearest points, 249

Weierstrass , see alsoBolzano-Weierstrass ,Stone-Weierstrass

proposition, 4, 17ffWets, vii-ixWeyl, 101Williams, see Fillmore-WilliamsWolenski , ix

Young, see Fenchel-Young

Zizler , see Devill e-Godefroy­Zizler

Zorn 's lemma, 190, 210

Index