Bilinear Relaxation Technique: Theoretical Results and Solution …plaza.ufl.edu/artyom/Presentations/SupConf06.pdf · 2007. 12. 25. · CPLNF Problem Solution Algorithm FCNF Problem

CPLNF Problem Solution Algorithm FCNF Problem Discussion

Bilinear Relaxation Technique:Theoretical Results and Solution Algorithm

Artyom Nahapetyan

University of FloridaDepartment of Industrial and Systems Engineering

February 22, 2006


1 CPLNF ProblemMathematical FormulationBilinear Relaxation Technique

2 Solution AlgorithmDynamic Cost Updating ProcedureDynamic Slope Scaling ProcedureNumerical Experiments

3 FCNF ProblemMathematical Formulationε-ApproximationAdaptive Dynamic Cost Updating ProcedureNumerical Experiments

4 Discussion


Mathematical Formulation

Concave Piecewise Linear Network Flow Problem

Problem: CPLNF

Let G (N,A) represent a network where N and A are the sets ofnodes and arcs, respectively, and fa(xa) is the cost function of arca.

minx

∑a∈A

fa(xa)

s.t.Bx = b

xa ∈ [λ0a, λnaa ] ∀a ∈ A

B - node-arc incident matrix of the network Gfa(xa) - concave piecewise linear functions



Arc Cost Function

Function fa(xa)

fa(xa) =

c1a xa + s

1a xa ∈ [λ0a, λ1a)

c2a xa + s2a xa ∈ [λ1a, λ2a)

· · · · · ·cnaa xa + s

naa xa ∈ [λna−1a , λnaa ]

c1a > c2a > · · · > cnaa



Arc Cost Function

Function fa(xa)

fa(xa) =

c1a xa + 0 xa ∈ [0, λ1a)c2a xa + s

2a xa ∈ [λ1a, λ2a)

· · · · · ·cnaa xa + s

naa xa ∈ [λna−1a , λnaa ]

c1a > c2a > · · · > cnaa



Arc Cost Function

Function fa(xa)



Arc Cost Function

Function fa(xa)

fa(xa) =

c1a xa + 0(= f

1a (xa)) xa ∈ [0, λ1a)

c2a xa + s2a (= f

2a (xa)) xa ∈ [λ1a, λ2a)

· · · · · ·cnaa xa + s

naa (= f

naa (xa)) xa ∈ [λna−1a , λnaa ]



Arc Cost Function

Function fa(xa)

fa(xa) = mink∈Ka{f ka (xa)} = min

k∈Ka{cka xa + ska }

Ka = {1, 2, . . . , na}



Arc Cost Function

Function fa(xa)



Application

Discount Component in the Cost



Mixed Integer Formulation

Problem: CPLNF-IP

minx ,y

∑a∈A

∑k∈Ka

cka xka +

∑a∈A

∑k∈Ka

ska yka

Bx = b∑k∈Ka

xka = xa

∑k∈Ka

λk−1a yka ≤ xa ≤

∑k∈Ka

λkayka ,

∑k∈Ka

yka = 1

xka ≤ Myka , xka ≥ 0, yka ∈ {0, 1}


Bilinear Relaxation Technique

Relaxation

Problem:

CPLNF-R

minx ,y

∑a∈A

∑k∈Ka

cka xka +

∑a∈A

∑k∈Ka

ska yka

Bx = b∑k∈Ka

xka = xa

∑k∈Ka


∑k∈Ka

λkayka ,

∑k∈Ka

yka = 1

xka = xayka , x

ka ≥ 0, yka ≥ 0



Relaxation

Problem:

CPLNF-R

minx ,y

∑a∈A

∑k∈Ka

cka xka +

∑a∈A

∑k∈Ka

ska yka

Bx = b

∑k∈Ka


∑k∈Ka

λkayka ,

∑k∈Ka

yka = 1

xka = xayka , y

ka ≥ 0



Relaxation

Problem: CPLNF-R

minx ,y

∑a∈A

∑k∈Ka

cka yka

xa + ∑a∈A

∑k∈Ka

ska yka

Bx = b

∑k∈Ka


∑k∈Ka

λkayka ,

∑k∈Ka

yka = 1

yka ≥ 0



Theoretical Results

Lemma

Any feasible vector of the CPLNF-IP problem is feasible to theCPLNF-R

Lemma

Any local optimum of the CPLNF-R problem is either feasible tothe CPLNF-IP or leads to a feasible vector of CPLNF-IP with thesame objective function value.

Theorem

A global optimum of the CPLNF-R problem is a solution or leadsto a solution of the CPLNF-IP .



Economical Interpretation

Problem: CPLNF-R

minx ,y

∑a∈A

∑k∈Ka

[cka xa + s

ka

]yka =

∑a∈A

∑k∈Ka

f ka (xa)yka

Bx = b∑k∈Ka


∑k∈Ka

λkayka

∑k∈Ka

yka = 1

yka ≥ 0


Dynamic Cost Updating Procedure




4 Discussion



Two Problems

CPLNF-R

minx ,y

∑a∈A

∑k∈Ka

cka yka

xa + ∑a∈A

∑k∈Ka

ska yka

Bx = b∑k∈Ka


∑k∈Ka

λkayka

∑k∈Ka

yka = 1

yka ≥ 0



Two Problems

LP(y) (y is fixed)

minx

∑a∈A

∑k∈Ka

cka yka

xaBx = b

xa ∈ [0, λnaa ]



Two Problems

CPLNF-R

minx ,y

∑a∈A

∑k∈Ka

[cka xa + s

ka

]yka

Bx = b

∑k∈Ka


∑k∈Ka

λkayka

∑k∈Ka

yka = 1

yka ≥ 0



Two Problems

LP(x) (x is fixed)

miny

∑a∈A

∑k∈Ka

[cka xa + ska ]y

ka

∑k∈Ka


∑k∈Ka

λkayka

∑k∈Ka

yka = 1

yka ≥ 0

The solution of the problem is a binary vector

Can be solved using a search technique



Dynamic Cost Updating Procedure (DCUP)

DCUP: Iteratively Solves LP(x) and LP(y)

Step 1: Let y0 denote the initial vector of yk0a , where y10a = 1 and

yk0a = 0, ∀k ∈ Ka, k 6= 1. m← 1.

Step 2: Let xm = argmin{LP(ym−1)}, andym = argmin{LP(xm)}.

Step 3: If ym = ym−1 then stop. Otherwise, m← m + 1 and goto Step 2.



Theoretical Results

Theorem

Let (x∗, y∗) be the solution returned by DCUP. If y∗ is a uniquesolution of the LP(x∗) problem then (x∗, y∗) is a local minimum ofCPLNF-R.

Theorem

Given any initial binary vector y0, DCUP converges in a finitenumber of iterations.


Dynamic Slope Scaling Procedure

DSSP

Kim D., Pardalos P., “A Solution Approach to the FixedCharged Network Flow Problems Using a Dynamic SlopeScaling Procedure”, Operations Research Letters, 24, pp.195-203, 1999.

Kim D., Pardalos P., “Dynamic Slope Scaling and TrustInterval Techniques for Solving Concave Piecewise LinearNetwork Flow Problems”, Networks, 35(3), pp. 216-222,2000.



DSSP



An Alternative Formulation

minx

FT (x)x

s.t.Bx = b

xa ∈ [0, λnaa ]

where F (x) is the vector of functions

Fa(xa) =

{fa(xa)

xaxa > 0

M xa = 0=

c1a xa ∈ (0, λ1a]c2a +

s2axa

xa ∈ (λ1a, λ2a]· · · · · ·cnaa +

snaaxa

xa ∈ (λna−1a , λnaa ]M xa = 0

.



Theoretical Results

Theorem

The solution provided by the DSSP is the solution of the followingproblem

“find feasible x∗ such that FT (x∗)(x − x∗) ≥ 0, ∀xa ∈ [0, λnaa ],Bx = b”,



Theoretical Results

Theorem

The solution provided by the DSSP is the solution of the followingproblem

“find feasible x∗ such that FT (x∗)(x − x∗) ≥ 0, ∀xa ∈ [0, λnaa ],Bx = b”,

User Equilibrium problem: FT (xu)x ≥ FT (xu)xuSystem Optimum problem: FT (x)x ≥ FT (x s)x s


Numerical Experiments

Test Problems

30 problem sets

Network size (nodes-arcs-supply/demand nodes): 12-35-2,20-100-3, 40-300-4, 100-2000-20, and 200-5000-50.Demand: U[10,20], U[20,30], or U[30,40]Number of linear pieces: 5 or 10.

30 problems per problem set.

In total - 900 problems.



Results

DCUP better than DSSP - 50% of test problems .

DSSP better than DCUP - 20% of test problems .

30% of problems are the same.

Relative Error - 1-2%.

DCUP is 2-5 times faster than DSSP.






4 Discussion



Fixed Charge Network Flow Problem

Problem: FCNF

minx

f (x) =∑a∈A

fa(xa)

s.t.Bx = b

xa ∈ [0, λa] ∀a ∈ A

where

fa(xa) =

{caxa + sa xa ∈ (0, λa]0 xa = 0


ε-Approximation

ε-Approximation


ε-Approximation

ε-Approximation

minx

φε(x) =∑a∈A

φεaa (xa)

s.t.

Bx = b

xa ∈ [0, λa] ∀a ∈ A.

where ε denotes the vector of εa.


ε-Approximation

Theoretical Results

Definition

X = {x |Bx = b, xa ∈ [0, λa],∀a ∈ A},V (X ) set of vertexes of X ,

xε = argmin{φε(x) : x ∈ X} and x∗ = argmin{f (x) : x ∈ X}

Theorem

For all ε such that εa ∈ (0, λa], ∀a ∈ A, φε(xε) ≤ f (x∗).

Theorem

Let δ = min{xva |xv ∈ V (x), a ∈ A, xva > 0}. For all ε such that∀a ∈ A, εa ∈ (0, δ], φε(xε) = f (x∗).


Adaptive Dynamic Cost Updating Procedure

Adaptive Dynamic Cost Updating Procedure(ADCUP)



Test Problems

36 problem sets

Network size (nodes-arcs-supply/demand nodes): 20-100-3,40-300-4, 100-1000-10, and 150-3000-15.Variable cost: U[1,5], U[10,20], or U[30,40]Fixed cost: U[50,100], U[100,200], or U[200,400]

30 problems per problem set.

In total - 1080 problems.



Results: Small Networks

ADCUP better than DSSP - 57% of test problems .

DSSP better than ADCUP - 8% of test problems .


Relative error % (ADCUP,DSSP).

Fixed CostVariable cost U[50,100] U[100,200] U[200,400]

U[1,5] (2.5, 9.7) (3.2, 12.3) (3.9, 11.8)U[10,20] (0.2, 0.7) (0.5, 2.7) (1.3, 5.7)U[30,40] (0.1, 0.3) (0.2, 0.8) (0.7, 1.8)

CPU time - ADCUP is 2-3 times faster than DSSP.



Results: Large Networks

ADCUP better than DSSP - 94% of test problems .

DSSP better than ADCUP - 2% of test problems .


(DSSP − ADCUP)/ min{DSSP,ADCUP}.Fixed Cost

Variable cost U[50,100] U[100,200] U[200,400]U[1,5] 15.5 18.8 16.9

U[10,20] 1.2 3.4 7.1U[30,40] 0.4 0.9 2.3

CPU time - ADCUP is 5-20 times faster than DSSP.


Conclusion

The bilinear relaxation can be very useful for finding anapproximate solution in both problems.

DSSP provides an equilibrium type of solution.

DCUP guaranties convergence to a local minimum of therelaxation problem.

Applications:

Cutting plain algorithmBranch-and-Bound algorithm

CPLNF ProblemMathematical FormulationBilinear Relaxation Technique

Solution AlgorithmDynamic Cost Updating ProcedureDynamic Slope Scaling ProcedureNumerical Experiments

FCNF ProblemMathematical Formulation-ApproximationAdaptive Dynamic Cost Updating ProcedureNumerical Experiments

Discussion

Documents

Bilinear Relaxation Technique: Theoretical Results and Solution …plaza.ufl.edu/artyom/Presentations/SupConf06.pdf · 2007. 12. 25. · CPLNF Problem Solution Algorithm FCNF Problem