Attractive Mathematical Representations Of Decision Problems

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Attractive Mathematical Representations Of Decision

Problems

Warren Adams11/04/03

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Research Interests

Design and implementation of solution strategies for difficult (nonconvex) decision problems.

Theoretical development.

Algorithmic design.

Computer implementation.

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Significance & Impact

This talk summarizes a new, powerful procedure for constructing attractive formulations of optimization problems. The formulations generalize dozens of published papers. Striking computational successes have been realized on various problem types.

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Formulation Can Matter!

• Although more than one mathematical representation can accurately depict the same physical scenario, the choice of formulation can critically affect the success of solution strategies.

• What is an attractive formulation?

• How to obtain an attractive formulation?

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What Is An Attractive Formulation?

Since linear programming relaxations are often used to approximate difficult problems, formulations that have tight continuous relaxations are desirable.

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Fixed Charge Network Flow(A classic example)

1

shipment cost

2

6

1

2

1

fixed cost

12

1

2

3

2

1supply

12

demand

3

3

6

14

8

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Standard Representation

binary ,

0 , , , ,,

1 0 1 0

6

3

3

12

12 subject to

262814 minimize

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654321

21

63

52

41

2654

1321

65432121

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yyyy yy

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xyyy

xyyy

yyyyyyxx

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Standard Representation

Optimal relaxed value = 24.5.

x1=1/4

3

3

1 6

1

shipment cost

2

6

1

2

fixed cost

12

1

2

3

2

1supply

12

demand

3

3

6

14

8

x2=3/4

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Enhanced Representation

binary ,

0 , , , ,,

6 3 3

6 3 3

1 0 1 0

6

3

3

12

12 subject to

262814 minimize

21

654321

262524

131211

21

63

52

41

2654

1321

65432121

xx

yyyy yy

xyxyxy

xyxyxy

xx

yy

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xyyy

xyyy

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Enhanced Representation

Optimal relaxed value =29.

x1=1

3

3

1

6

1

shipment cost

2

6

1

2

fixed cost

12

1

2

3

2

1supply

12

demand

3

3

6

14

8

x2=0

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In General, How To Obtain Attractive Formulations?

Attractive formulations for special problem classes can be found in the literature, but no general (encompassing) schemes exist.

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A New Perspective

• Historic reasoning. Convert to linear form, making any needed substitutions and/or transformations. Avoid nonlinearities.

• Newer reasoning. Construct nonlinearities. Then convert to linear form, using the nonlinearities to yield superior representations.

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A Method For Obtaining Attractive Formulations

• Reformulate the problem by incorporating additional variables and nonlinear restrictions that are redundant in the original program, but not in the relaxed version.

• Linearize the resulting program to obtain the problem in a different variable space.

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Reformulation-Linearization Technique (RLT)

minimize ctx + dty

subject to Ax + By >= b

0=< x =<1

x binary

y >= 0

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RLT: A General Approach To Attractive Formulations (Level-1)

• Reformulation. Multiply each constraint by product factors consisting of every 0-1 variable xi and its complement 1- xi. Apply the binary identity xi xi = xi for each i.

• Linearization. Substitute, for each (i,j) with i<j, a continuous variable wij for every occurrence of xixj or xjxi, and, for each (j,k), a continuous variable vjk for every occurrence of xjyk.

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Linearized Problem (Level-1)

minimize ctx + dty subject to Ax + By + Dw +Ev >= b x binary y >= 0

The linearized problem is equivalent to the original program in that for any feasible solution to one problem, there is a feasible solution to the other problem with the same objective value.

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Relaxation Strength?

The weakest level-1 representations tend to dominate alternate formulations available in the literature, even for select problems having highly-specialized structure!

As a result, we have been able to solve larger problems than previously possible.

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A Hierarchy Of Relaxations

By changing the product factors, an n+1 hierarchy of relaxations emerges, with each level at least as tight as the previous level, and with an explicit algebraic characterization of the convex hull available at the highest level.

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Level-0 Representation

x1>=0

x2>=0

x2<=1

x1<=1

2x1+2x2<=3

(0, 0) (1, 0)

(0, 1)

(1/2, 1)

(1, 1/2)

x2

x1

XP0={(x1, x2): 2x1+2x2<=3, 0<=x1<=1, 0<=x2<=1}

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Level-1 Representation

0.5x1+x2<=1

(2/3, 2/3)

x1+0.5x2<=1x1>=0

x2>=0(0, 0) (1, 0)

(0, 1)

x2

x1

XP1={(x1, x2): x1+0.5x2<=1, 0.5x1+x2<=1, x1>=0, x2>=0}

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Level-2 Representation

x1+x2<=1

x1>=0

x2>=0(0, 0) (1, 0)

(0, 1)

x2

x1

XP2={(x1, x2): x1+x2<=1, x1>=0, x2>=0}

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Case Study: Quadratic 0-1 Knapsack Problem

minimize ctx + xtDx subject to atx<=b x binary

Capital budgeting problems.Approximates related problems.

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Computational FlavorProblem Size Classic Formulation Level-1 Formulation

Nodes CPU Time Nodes CPU Time 10 0 0 8 0 20 45 0 44 0 30 421 0 102 0 40 3,899 2 826 1 50 7,043 4 771 1 60 146,430 119 2,559 3 70 92,967 99 4,465 5 80 1,232,794 1,519 8,676 9 90 **** **** 57,730 73 100 **** **** 59,001 94

Averages of ten problems solved using CPLEX 8.0.**** Average solution time exceeded the 35,000 CPU second limit.

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Computational Successes

• Electric Distribution System Design.• Reliable Water Distribution Networks.• Engineering and Chemical Process

Design Problems.• Time-Dynamic Power Distribution. • Water Resources Management.• Quadratic Assignment Problem.• Capital Budgeting Problems.

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Ongoing Research

• Discrete variable problems. Generalizing the product factors to Lagrange interpolating polynomials.

• Balancing problem size and relaxation strength.

• Generating new families of inequalities.

• Applying functional product factors.

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Research Needs

• Wish to conduct collaborative, interdisciplinary research that blends these optimization tools with decision problems arising in electric power systems.

• Eager for discussions!