ESI 6448 Discrete Optimization Theory

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ESI 6448Discrete Optimization Theory

• Lecture 22

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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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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.
• x1, , xk ? Rn are affinely independent
  iffx2x1, ,xkx1 are linearly independent
  iff(x1, 1), , (xk, 1) ? Rn1 are linearly
  independent

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Faces

• If x ? Rn Ax b ? ?, the maximum number of
  affinely independent solutions to Ax b is n 1
  rank(A).
• 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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Strong inequalities

• How to prove the strength of valid inequalities
• show that
• an inequality defines a facet of convex hull, or
• a set of inequalities describes the convex hull
• Assume conv(P) is full-dimensional
• If P is full-dimensional, a valid inequality ?x ?
  ?0 is necessary in the description of P iffthere
  are n affinely independent points of P satisfying
  ?x ?0 .

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Property of facets

• Let (A, b) be the equality set of P ? Rn and
  let F x ? P ?x ?0 be a proper face of P,
  theni) F is a facet of P iffii) If ?x ?0, ? x
  ? F then (?, ?0) (?? uA, ??0ub) for
  some ? ? R and u ? RM.
• Let P ? Rn be full-dimensional and let F x ? P
  ?x ?0 be a face of P and let X x ? F ?x
  ?0, where X ? n, theni) F is a facet of P
  iffii) If ?x ?0, ? x ? X then (?, ?0) ?(?,
  ?0) for some ? ? R.

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Facet proofs

• Given P ? Zn and a valid inequality ?x ? ?0 for
  P, we can show that (?, ?0) defines a facet of
  conv(P) by
• a) finding n points x1, , xn ? P satisfying ?x
  ?0 and then proving that xis are affinely
  independent, or
• b) (indirect way)i) Select t ? n points x1, ,
  xt ? P satisfying ?x ?0. Suppose all xis
  lie on a generic hyperplane ?x ?0.ii) Solve
  for k 1,,t.iii) If the only
  solution is (?, ?0) ?(?, ?0) for ? ? 0, then
  (?, ?0) is facet-defining.

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Example

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Today

• Strong valid inequalities
• Dominance
• Polyhedra, faces, facets
• Facet convex hull proofs