1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03
Jan 03, 2016
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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)
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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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yy
xyyy
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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
yy
yy
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!