ESI 6448 Discrete Optimization Theory

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ESI 6448Discrete Optimization Theory

• Lecture 21

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Finiteness of fractional dual algorithm

• Make the following choices in the fractional dual
  algorithm for IP (a) Choose the source row to
  be the first row with a noninteger ai0.
  (b) Use the lexicographic version of the dual
  simplex algorithm.Then, assuming that
  the original problem has an upper bound on the
  cost z of a feasible solution, the algorithm
  terminates with an integer solution in a finite
  number of steps or finds that there is no
  feasible integer solution to the original problem.

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Disjunctive inequalities

-x13x2 ? 7
(5, 4)
(10, 4)
P 2
(4, 2)
P 1
(0, 1)
(5, 0)
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Disjunctive inequalities

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Disjunctive procedure

• Disjunctive procedure (D-inequalities)

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D-inequalities

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Basic Mixed Integer Inequalities

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Mixed integer rounding inequalities

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MIR procedure

• MIR procedure (MIR inequalities)

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Gomory Mixed Integer Cut

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Example

(20/7, 3)
(2, 1)
(2, 1/2)
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Strong valid inequalities

• Gomory FCPA
• terminates in a finite of steps
• practical?
• Find strong valid inequalities
• more effective
• leads to a stronger formulation

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Dominance

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Example

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Polyhedra

• We assume P ? Rn contains n linearly independent
  directions, i.e. n1 affinely independent points.
• Such a P is full-dimensional.
• If P is full-dimensional, it has a unique minimal
  descriptionP x ? Rn aix ? bi for i 1, ,
  mwhere each inequality is unique to within a
  positive multiple.

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Faces

• x1, , xk ? Rn are affinely independent
  iffx2x1, ,xkx1 are linearly independent
  iff(x1, 1), , (xk, 1) ? Rn1 are linearly
  independent
• The dimension of P, dim(P), max of affienly
  independent points in P 1
• P ? Rn is full-dimensional iff dim(P) n
• F defines a face of P if F x ? P ?x ?0
  for some valid inequality ?x ? ?0 of P
• F is a facet of P if F is a face and dim(F)
  dim(P) 1
• If F x ? P ?x ?0 is a face of P, the
  valid inequality ?x ? ?0 is said to represent or
  define the face.

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Facets

• If P is full-dimensional, a valid inequality ?x ?
  ?0 is necessary in the description of P iff it
  defines a facet of P.
• P ? R2 described byx1 ? 2x1 x2 ?
  4x1 2x2 ? 10x1 2x2 ? 6x1 x2 ? 2x1
  ? 0 x2 ? 0

x1 2x2 ? 10
x1 2x2 ? 6
x1 ? 0
(2, 2)
(0, 2)
x1 x2 ? 4
x1 ? 2
x1 x2 ? 2
(2, 0)
x2 ? 0
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Today

• More inequalities
• D-inequalities
• MIR inequalities
• Strong valid inequalities
• Dominance
• Polyhedra, faces, facets