Download - Lower bounds for quadratic fractional functionsferrari.dmat.fct.unl.pt/cva2008/t_amaral.pdf · Lower bounds for quadratic fractional functions FCT Universidade Nova de Lisboa CMA

$: Lower bounds for quadratic fractional functionsferrari.dmat.fct.unl.pt/cva2008/t_amaral.pdf · Lower bounds for quadratic fractional functions FCT Universidade Nova de Lisboa CMA$
1

Paula Amaral

Lower bounds for quadraticfractional functions

FCT Universidade Nova de LisboaCMA (Centro de Matemática e Aplicações)

2/48CVA2008 – FCT UNL Caparica

• Fractional problems• Infeasible linear system • Global optimization method• Computational experience• Conclusions

OUTLINE


• Fractioal Quadratic Problem

Fractional programming problems

)()( min

Xx xgxf

∈

Stancu-Minasian (1999) – A fifth bibliography of fractional programmingSchaible (1981) – Fractional Programming: applications and algorithms

Engineering, business, finance, economics

performance / costincome / investementcost / time



Fractional programming problems

Fractional programming : a tool for the assessment of sustainabilityLara P. Stancu-Minasian I. (1999)

Maximizing predictability in the stock and bond marketsLo A., Mackinlay C. (1997)

Finantial planning with fractional goalsGoedhart M., Spronk J. (1995)

Discrete Fractional Programming techniques for location modelsBarros A. I. (1998)



Linear/Linear

Quadratic/Linear

Quadratic/Quadratic

Sum of ratios

Charnes-Cooper (1962)

Cambini (2002)

Tuy, Konno (2004)

Yamamoto, Konno (2007)

)()( min

Xx xgxf

∈

linear , quadratic , concave-convex, polynomial fractional programs


• Infeasible linear systems

Production planning Product specificationsProfits and costsMarketingLabour

Production problem

Infeasibility analysis


mmibxamibxa

ii

ii

,,1,,,1,

0

0

L

L

+===≤

update of old models

integration of partial models

unrealistic definitions.



Remove constraints

mmibxamibxa

ii

ii

,,1,,,1,

0

0

L

L

+===≤



Find a solution of a close feasible system



Ax (≤&=) b(A+H)x (≤&=) b+p

Minimize Ψ(H,p)subject to

x ∈X



(A+H)x (≤&=) b+p

Minimize Ψ(H,p)subject to

x ∈X

Ψ(H,p)=0 iff (H,p)=0

Ψ(H,p) ≥0



(A+H)x (≤&=) b+px ∈ X

Minimize Ψ(H,p)

(H,p)=W

Linear Programming ProblemVATOLIN-2000

Ψ(W)= Max |wij|i,j

Ψ(W)= ΣΣ|wij|i,j

Ψ(W)= Max Σ|wij|i j

(l∞)

(l1)

(∞)



(A+H)x (≤&=) b+p

Minimize Ψ(H,p)

(H,p)=W

Ψ(W)= Max |wij|i,j

Ψ(W)= ΣΣ|wij|i,j

Ψ(W)= Max Σ|wij|i j

(l∞)

(l1)

(∞)

wij=±α for i=1,…,m, j=1,..n+1

wij=±αi j=k0 j≠k



(A+H)x (≤&=) b+p

Minimize Ψ(H,p)

(H,p)=W Ψ(W)= ΣΣ(wij)2

i,j(F)



- 0.1365 - 0.1613 0.0522

- 0.0714 - 0.0844 0.0273

- 0.1065 - 0.1259 0.0407

(H,p)=




Total Least SquaresTotal Least Squaressubset constraints

Sabine van Huffel, The total least squares problem: computational aspects and analysis, Frontiers in Applied Mathematics, 9, SIAM, 1991.

Amaral P, Barahona P. Connections between the total least squares and the correction of an infeasible system of linear inequalities. Linear Algebra and Applications 2005; 395: 191-210.Amaral P, Barahona P., A framework for optimal correction of inconsistent linear constraints.Constraints 2005; 10: 67-86.

Amaral P, Júdice J, Sherali H D. A reformulation--linearization--convexification algorithm foroptimal correction of an inconsistent system of linear constraints.Computers and Operations Research 2008; 35: 1494-1509.



Linear/Linear

Quadratic/Linear

Quadratic/Quadratic

Sum of ratios

Aplications and caracterization

Charnes-Cooper (1962)

Cambini (2002)

Tuy, Konno (2004)

Yamamoto, Konno (2007)



Quadratic/Quadratic

Sum of Quadratic/Quadratic


Why preserve the structure of zeros?




hij=0 if aij=0



KKT conditions



WHY GLOBAL OPTIMIZATION ?



SUM OF FRACTIONAL QUADRATIC FUNCTIONS with linear constraints


Upper Bound

Lower Bounds

Optimal Value


Branch & Bound

Node poblems

Branching Rule

Upper bound

Lower bound problem

Node picking strategy

• Global optimization method


k

k k

Node problem


l2≤ x2≤u2k k

ln≤ xn≤unk k

ls≤ xs≤usk k

l1≤ x1≤u1k k

ls≤ xs≤usK+1 K+1



Lower bound



l2≤ x2≤u2k k

ln≤ xn≤unk k

ls≤ xs≤usk k

l1≤ x1≤u1k k



Branching


xs

y=xs2

lsk usk

y=δskxs+βs

k

y

xs*



xj

y=xj2

ljk=ljk+1 xj* ujk=uj

k+2

y=δjkxj+βj

k

y

Branching Rule A and B



xs

y=xj2

ljk=ljk+1

y=δjk+2xj+βj

k+2

ujk=uj

k+2xs*

=ujk+1

=ljk+2

y=δjkxj+βj

k

y=δjk+1xj+βj

k+1

y

Branching Rule A and B



Nodes inspection



Convergence



Pentium IV (Intel), CPU 3 GHZ, 2GB RAM, LINUXGAMS - MINOS

Computational Experience

• Computational experience


{ }610

,1−<

−UBMaxLBUB

Nodes generated in the tree 1000

Larger tolerances 10-ρ with 1 ≤ ρ ≤ 5

li=0 ui=li+t with t=1, 5, 50, 500

tolerances 10-ρ with 1 ≤ ρ ≤ 6


Tolerances

0

1

2

3

4

5

6

7

Galene

t Ite

st2

Itest6

Bgp

rtr Fore

st Wood

infe

Prob4

Prob5

Prob6

Prob7

Prob8

Prob9

Prob10

Prob

11

Prob12

Prob

13

Prob14

Prob

15

Prob16

Prob

17

Prob18

Prob

19

Prob20

Problems

10-ρ B&B_AB&B_B



CPU time

0.01

0.1

1

10

100

Galene

t Ite

st2

Itest6

Prob

4 Prob

5 Prob

6 Prob

7 Prob

8 Prob

9 Prob

10

Prob11

Prob

12

Prob13

Prob

14

Prob15

Prob

16

Prob17

Prob

18

Prob19

Prob

20

Problems

log(

cpu)

B&B_AB&B_B



Number of Iterations

1

10

100

1000

10000

100000

Galene

t Ite

st2

Itest6

Prob

4 Prob

5 Prob

6 Prob

7 Prob

8 Prob

9 Prob

10

Prob11

Prob

12

Prob13

Prob

14

Prob15

Prob

16

Prob17

Prob

18

Prob19

Prob

20

Problems

log(

iter)

B&B_AB&B_B



Iterations

1

10

100

1000

10000

1 2 3 4 5 6

10−ρ

log(

num

ber o

f ite

ratio

ns)

Itest2,t=1Itest2,t=5Itestt2,t=50Itest2,t=500



CPU, ITEST2

00.20.40.60.8

11.21.4

1 2 3 4 5 6

10−ρ

seco

nds t=1

t=5t=50t=500



Summary, conclusions and future work

• Fractional programming formulation of a zero preserving correction of a general inconsistent system of linear constraints.

• A Branch-and-Bound algorithm.• Linearization of the functions in each denominator of the

objective function.• Two different branching rule strategies.• The lower and upper bound for variables has a

significant impact.• The root node upperbounding procedure offers a strong

heuristic.• SDP relaxations

• Conclusions


• Improving lower bounds (SDP)

Luís M. FernandesInstituto Politécnico de Tomar andInstituto Telecomunicações, Coimbra, Portugal

Joaquim JúdiceDepartamento de Matemática, Universidade de Coimbra andInstituto Telecomunicações, Coimbra, Portugal

Hanif D. SheraliGrado Department of Industrial & Systems Engineering, Virginia Polytechnic Institute & State University, USA

Collaboration

Download - Lower bounds for quadratic fractional functionsferrari.dmat.fct.unl.pt/cva2008/t_amaral.pdf · Lower bounds for quadratic fractional functions FCT Universidade Nova de Lisboa CMA

Top Related