1

Attractive Mathematical Representations Of Decision

Problems

Warren Adams11/04/03

2

Research Interests

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

Theoretical development.Algorithmic design.Computer implementation.

3

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.

4

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?

5

What Is An Attractive Formulation?

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

6

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

7

Standard Representation

binary , 0 , , , ,,

1 0 1 0 6 3 3

12 12 subject to

262814 minimize

21

654321

21

63

52

41

2654

1321

65432121

xxyyyy yy

xxyy

yyyy

xyyyxyyy

yyyyyyxx

8

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

9

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

xxyyyy yy

xyxyxyxyxyxy

xxyy

yyyy

xyyyxyyy

yyyyyyxx

10

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

11

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.

12

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.

13

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.

14

Reformulation-Linearization Technique (RLT)

minimize ctx + dtysubject to Ax + By >= b

0=< x =<1 x binary y >= 0

15

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.

16

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.

17

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.

18

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.

19

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}

20

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}

21

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}

22

Case Study: Quadratic 0-1 Knapsack Problem

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

Capital budgeting problems.Approximates related problems.

23

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.

24

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.

25

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.

26

Research Needs

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

• Eager for discussions!