Attractive Mathematical
Representations Of Decision
Problems
Warren Adams
11/04/03
1
Research Interests
Design and implementation of solution
strategies for difficult (nonconvex)
decision problems.
Theoretical development.
Algorithmic design.
Computer implementation.
2
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.
3
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?
4
What Is An Attractive Formulation?
Since linear programming relaxations are
often used to approximate difficult
problems, formulations that have tight
continuous relaxations are desirable.
5
Fixed Charge Network Flow
(A classic example)
shipment cost
fixed cost
14
supply
12
1
1
demand
1
3
2
3
3
6
2
1
6
8
12
2
2
1
6
Standard Representation
minimize
subject to
14 x1  8 x 2  y1  2 y 2  y 3  6 y 4  2 y5  y 6
y1  y 2  y 3
 12 x1
y 4  y5  y 6
 y4
y1
3
 y5
y2
y3
0  x1  1
 12 x 2
3
 y6
6
0  x2  1
y1 , y 2 , y 3 , y 4 , y5 , y 6  0
x1 , x 2 binary
7
Standard Representation
Optimal relaxed value = 24.5.
shipment cost
fixed cost
14
supply
12
1
1
x1=1/4
3
demand
1
3
2
3
3
6
2
1
6
8
12
x2=3/4
2
2
1
3
6
8
Enhanced Representation
minimize
subject to
14 x1  8 x 2  y1  2 y 2  y 3  6 y 4  2 y5  y 6
y1  y 2  y 3
 12 x1
y 4  y5  y 6
 y4
y1
3
 y5
y2
3
 y6
y3
0  x1  1
 12 x 2
6
0  x2  1
y1  3x1
y 2  3x1
y 3  6 x1
y 4  3x2
y5  3x 2
y6  6 x2
y1 , y 2 , y 3 , y 4 , y5 , y 6  0
x1 , x 2 binary
9
Enhanced Representation
Optimal relaxed value =29.
shipment cost
fixed cost
14
supply
12
1
x1=1
1
3
2
3
demand
1
3
2
3
3
6
1
6
6
8
12
x2=0
2
2
1
10
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.
11
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.
12
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.
13
Reformulation-Linearization
Technique (RLT)
minimize ctx + dty
subject to Ax + By >= b
0=< x =<1
x binary
y >= 0
14
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.
15
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.
16
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.
17
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.
18
Level-0 Representation
x2
x2<=1 (1/2, 1)
(0, 1)
2x1+2x2<=3
(1, 1/2)
x1>=0
x1<=1
x1
(0, 0)
x2>=0
(1, 0)
XP0={(x1, x2): 2x1+2x2<=3, 0<=x1<=1, 0<=x2<=1}
19
Level-1 Representation
x2
(0, 1)
0.5x1+x2<=1
(2/3, 2/3)
x1>=0
x1+0.5x2<=1
x1
(0, 0)
x2>=0
(1, 0)
XP1={(x1, x2): x1+0.5x2<=1, 0.5x1+x2<=1, x1>=0, x2>=0}
20
Level-2 Representation
x2
(0, 1)
x1+x2<=1
x1>=0
x1
(0, 0)
x2>=0
(1, 0)
XP2={(x1, x2): x1+x2<=1, x1>=0, x2>=0}
21
Case Study: Quadratic 0-1
Knapsack Problem
minimize ctx + xtDx
subject to atx<=b
x binary
Capital budgeting problems.
Approximates related problems.
22
Computational Flavor
Problem Size
10
20
30
40
50
60
70
80
90
100
Classic Formulation
Nodes CPU Time
0
0
45
0
421
0
3,899
2
7,043
4
146,430
119
92,967
99
1,232,794
1,519
****
****
****
****
Level-1 Formulation
Nodes CPU Time
8
0
44
0
102
0
826
1
771
1
2,559
3
4,465
5
8,676
9
57,730
73
59,001
94
Averages of ten problems solved using CPLEX 8.0.
**** Average solution time exceeded the 35,000 CPU second limit.
23
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.
24
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.
25
Research Needs
• Wish to conduct collaborative,
interdisciplinary research that blends these
optimization tools with decision problems
arising in electric power systems.
• Eager for discussions!
26