lower bounds for quadratic fractional functionsferrari.dmat.fct.unl.pt/cva2008/t_amaral.pdf ·...

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

3/48CVA2008 – FCT UNL Caparica

• 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

4/48CVA2008 – FCT UNL Caparica

• Fractioal Quadratic Problem

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)

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• Fractioal Quadratic Problem

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

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• Infeasible linear systems

Production planning Product specificationsProfits and costsMarketingLabour

Production problem

Infeasibility analysis

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mmibxamibxa

ii

ii

,,1,,,1,

0

0

L

L

+===≤

update of old models

integration of partial models

unrealistic definitions.

• Infeasible linear systems

8/48CVA2008 – FCT UNL Caparica

Remove constraints

mmibxamibxa

ii

ii

,,1,,,1,

0

0

L

L

+===≤

• Infeasible linear systems

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Find a solution of a close feasible system

• Infeasible linear systems

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Ax (≤&=) b(A+H)x (≤&=) b+p

Minimize Ψ(H,p)subject to

x ∈X

• Infeasible linear systems

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(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

• Infeasible linear systems

12/48CVA2008 – FCT UNL Caparica

(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)

(∞)

• Infeasible linear systems

13/48CVA2008 – FCT UNL Caparica

(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

• Infeasible linear systems

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• Infeasible linear systems

15/48CVA2008 – FCT UNL Caparica

(A+H)x (≤&=) b+p

Minimize Ψ(H,p)

(H,p)=W Ψ(W)= ΣΣ(wij)2

i,j(F)

• Infeasible linear systems

16/48CVA2008 – FCT UNL Caparica

- 0.1365 - 0.1613 0.0522

- 0.0714 - 0.0844 0.0273

- 0.1065 - 0.1259 0.0407

(H,p)=

• Infeasible linear systems

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• Infeasible linear systems

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• Infeasible linear systems

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.

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• Fractioal Quadratic Problem

Linear/Linear

Quadratic/Linear

Quadratic/Quadratic

Sum of ratios

Aplications and caracterization

Charnes-Cooper (1962)

Cambini (2002)

Tuy, Konno (2004)

Yamamoto, Konno (2007)

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• Infeasible linear systems

Quadratic/Quadratic

Sum of Quadratic/Quadratic

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Why preserve the structure of zeros?

• Infeasible linear systems

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• Infeasible linear systems

hij=0 if aij=0

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• Infeasible linear systems

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• Infeasible linear systems

KKT conditions

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• Infeasible linear systems

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• Infeasible linear systems

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• Infeasible linear systems

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• Infeasible linear systems

WHY GLOBAL OPTIMIZATION ?

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• Infeasible linear systems

SUM OF FRACTIONAL QUADRATIC FUNCTIONS with linear constraints

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Upper Bound

Lower Bounds

Optimal Value

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Branch & Bound

Node poblems

Branching Rule

Upper bound

Lower bound problem

Node picking strategy

• Global optimization method

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k

k k

Node problem

• Global optimization method

l2≤ x2≤u2k k

ln≤ xn≤unk k

ls≤ xs≤usk k

l1≤ x1≤u1k k

ls≤ xs≤usK+1 K+1

ls≤ xs≤usK+2 K+2

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Lower bound

• Global optimization method

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• Global optimization method

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l2≤ x2≤u2k k

ln≤ xn≤unk k

ls≤ xs≤usk k

l1≤ x1≤u1k k

ls≤ xs≤usK+1 K+1

ls≤ xs≤usK+2 K+2

Branching

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xs

y=xs2

lsk usk

y=δskxs+βs

k

y

xs*

• Global optimization method

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xj

y=xj2

ljk=ljk+1 xj* ujk=uj

k+2

y=δjkxj+βj

k

y

Branching Rule A and B

• Global optimization method

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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

• Global optimization method

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Nodes inspection

• Global optimization method

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Convergence

• Global optimization method

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

Computational Experience

• Computational experience

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{ }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

43/48CVA2008 – FCT UNL Caparica

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

• Computational experience

44/48CVA2008 – FCT UNL Caparica

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

• Computational experience

45/48CVA2008 – FCT UNL Caparica

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

• Computational experience

46/48CVA2008 – FCT UNL Caparica

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

• Computational experience

47/48CVA2008 – FCT UNL Caparica

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

• Computational experience

48/48CVA2008 – FCT UNL Caparica

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

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• 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