1 attractive mathematical representations of decision problems warren adams 11/04/03
DESCRIPTION
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.TRANSCRIPT
![Page 1: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/1.jpg)
1
Attractive Mathematical Representations Of Decision
Problems
Warren Adams11/04/03
![Page 2: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/2.jpg)
2
Research Interests
Design and implementation of solution strategies for difficult (nonconvex) decision problems.
Theoretical development.Algorithmic design.Computer implementation.
![Page 3: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/3.jpg)
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.
![Page 4: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/4.jpg)
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?
![Page 5: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/5.jpg)
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.
![Page 6: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/6.jpg)
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
![Page 7: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/7.jpg)
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
![Page 8: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/8.jpg)
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
![Page 9: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/9.jpg)
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
![Page 10: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/10.jpg)
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
![Page 11: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/11.jpg)
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.
![Page 12: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/12.jpg)
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.
![Page 13: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/13.jpg)
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.
![Page 14: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/14.jpg)
14
Reformulation-Linearization Technique (RLT)
minimize ctx + dtysubject to Ax + By >= b
0=< x =<1 x binary y >= 0
![Page 15: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/15.jpg)
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.
![Page 16: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/16.jpg)
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.
![Page 17: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/17.jpg)
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.
![Page 18: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/18.jpg)
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.
![Page 19: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/19.jpg)
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}
![Page 20: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/20.jpg)
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}
![Page 21: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/21.jpg)
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}
![Page 22: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/22.jpg)
22
Case Study: Quadratic 0-1 Knapsack Problem
minimize ctx + xtDx subject to atx<=b x binary
Capital budgeting problems.Approximates related problems.
![Page 23: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/23.jpg)
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.
![Page 24: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/24.jpg)
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.
![Page 25: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/25.jpg)
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.
![Page 26: 1 Attractive Mathematical Representations Of Decision Problems Warren Adams 11/04/03](https://reader035.vdocument.in/reader035/viewer/2022070605/5a4d1acf7f8b9ab059970c25/html5/thumbnails/26.jpg)
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!