Attractive Mathematical Representations Of Decision Problems

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Attractive Mathematical Representations Of
Decision Problems

• Warren Adams
• 11/04/03

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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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Standard Representation
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Standard Representation

• Optimal relaxed value 24.5.

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Enhanced Representation
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Enhanced Representation

• Optimal relaxed value 29.

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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 gt b
• 0lt x lt1
• x binary
• y gt 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
  iltj, 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 gt b
• x binary
• y gt 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 n1
  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
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Level-1 Representation
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Level-2 Representation
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Case Study Quadratic 0-1 Knapsack Problem

• minimize ctx xtDx
• subject to atxltb
• x binary
• Capital budgeting problems.
• Approximates related problems.

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Computational Flavor

• Problem 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!