lower bounds for quadratic fractional functionsferrari.dmat.fct.unl.pt/cva2008/t_amaral.pdf ·...
TRANSCRIPT
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)
5/48CVA2008 – FCT UNL Caparica
• 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
6/48CVA2008 – FCT UNL Caparica
• Infeasible linear systems
Production planning Product specificationsProfits and costsMarketingLabour
Production problem
Infeasibility analysis
7/48CVA2008 – FCT UNL Caparica
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
9/48CVA2008 – FCT UNL Caparica
Find a solution of a close feasible system
• Infeasible linear systems
10/48CVA2008 – FCT UNL Caparica
Ax (≤&=) b(A+H)x (≤&=) b+p
Minimize Ψ(H,p)subject to
x ∈X
• Infeasible linear systems
11/48CVA2008 – FCT UNL Caparica
(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
14/48CVA2008 – FCT UNL Caparica
• 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
17/48CVA2008 – FCT UNL Caparica
• Infeasible linear systems
18/48CVA2008 – FCT UNL Caparica
• 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.
19/48CVA2008 – FCT UNL Caparica
• 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)
20/48CVA2008 – FCT UNL Caparica
• Infeasible linear systems
Quadratic/Quadratic
Sum of Quadratic/Quadratic
21/48CVA2008 – FCT UNL Caparica
Why preserve the structure of zeros?
• Infeasible linear systems
22/48CVA2008 – FCT UNL Caparica
• Infeasible linear systems
hij=0 if aij=0
23/48CVA2008 – FCT UNL Caparica
• Infeasible linear systems
24/48CVA2008 – FCT UNL Caparica
• Infeasible linear systems
KKT conditions
25/48CVA2008 – FCT UNL Caparica
• Infeasible linear systems
26/48CVA2008 – FCT UNL Caparica
• Infeasible linear systems
27/48CVA2008 – FCT UNL Caparica
• Infeasible linear systems
28/48CVA2008 – FCT UNL Caparica
• Infeasible linear systems
WHY GLOBAL OPTIMIZATION ?
29/48CVA2008 – FCT UNL Caparica
• Infeasible linear systems
SUM OF FRACTIONAL QUADRATIC FUNCTIONS with linear constraints
30/48CVA2008 – FCT UNL Caparica
Upper Bound
Lower Bounds
Optimal Value
31/48CVA2008 – FCT UNL Caparica
Branch & Bound
Node poblems
Branching Rule
Upper bound
Lower bound problem
Node picking strategy
• Global optimization method
32/48CVA2008 – FCT UNL Caparica
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
33/48CVA2008 – FCT UNL Caparica
Lower bound
• Global optimization method
34/48CVA2008 – FCT UNL Caparica
• Global optimization method
35/48CVA2008 – FCT UNL Caparica
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
36/48CVA2008 – FCT UNL Caparica
xs
y=xs2
lsk usk
y=δskxs+βs
k
y
xs*
• Global optimization method
37/48CVA2008 – FCT UNL Caparica
xj
y=xj2
ljk=ljk+1 xj* ujk=uj
k+2
y=δjkxj+βj
k
y
Branching Rule A and B
• Global optimization method
38/48CVA2008 – FCT UNL Caparica
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
39/48CVA2008 – FCT UNL Caparica
Nodes inspection
• Global optimization method
40/48CVA2008 – FCT UNL Caparica
Convergence
• Global optimization method
41/48CVA2008 – FCT UNL Caparica
Pentium IV (Intel), CPU 3 GHZ, 2GB RAM, LINUXGAMS - MINOS
Computational Experience
• Computational experience
42/48CVA2008 – FCT UNL Caparica
{ }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
49/48CVA2008 – FCT UNL Caparica
• 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