ESI 6448 Discrete Optimization Theory

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ESI 6448Discrete Optimization Theory

• Lecture 24

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Last class

• Strong valid inequalities
• Convex hull proofs

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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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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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Convex hull proofs

• Show that P x ? Rn Ax ? b describes conv(X)
  for X ? Zn
• A has a special structure such as TU.
• There is no fractional extreme point of P.
• Any LP relaxation over P has optimal integral
  solution.
• There are primal-dual feasible solutions w/ same
  value.
• Every facet-defining inequality for conv(X) is
  identical to one of the inequalities defining P.
• Optimal solutions of any optimization problem are
  on faces of P.
• Ax ? b forms a TDI system.
• Projection from an extended formulation

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Example

• X (x, y) ? Rm ? B ?mi1 xi ? my,
  xi ? 1 for i 1, , m
• P (x, y) ? Rm ? R xi ? y, for i 1, , m,
  y ? 1
• Show that P describes conv(X).

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0-1 knapsack inequalities

• Assume are positive and b gt 0.
• N 1, , n
• A set C ? N is a cover if ?j?C aj gt b.C is
  minimal if C \ j is not a cover for any j ? C.
• C is a cover iff xC ? X, where xiC 1 if i ? C,
  0 o.w.
• If C ? N is a cover for X, the cover inequality
  is valid for X.

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Example

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Extended cover inequalities

• Strengthen the basic cover inequalities
• If C is a cover for X, the extended cover
  inequality is valid for X,
  whereE(C) C ? j aj ? ai for all i ? C

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Example

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Strengthening cover inequalities

• Strongest, nonredundant inequalities
• Minimal cover inequality is facet-defining.
• Lifting minimal cover inequality gives a
  facet-defining inequality.
• Lifting
• Find best possible values for ?j for j ? N \ C
  s.t. is
  valid for X.

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Example

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Procedure to lift cover inequalities

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Example

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Strength of cover inequalities

• For a minimal cover C for X,

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Separation for cover inequalities

• Separation problem For a given nonintegral
  point x, is there a cover inequality that cuts
  off x.
• If ? ? 1, x satisfies all the cover
  inequalitiesIf ? lt 1 with optimal solution zR,
  the cover inequality ?j?R xj ? R 1 cuts
  off x by an amount 1 ?.

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Example

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Mixed 0-1 inequalities

• X can be viewed as the feasible region of a
  simple fixed charge flow network
• A set C C1 ? C2 w/ C1 ? N1, C2 ? N2 is a
  generalized cover for X if ?j?C1 aj ?j?C2 aj
  b ? with ? gt 0.? is called the (cover-)excess.

b
0 ? xj ? ajyj j ? N1
0 ? xj ? ajyj j ? N2
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Flow cover inequalities

• Example

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Today

• 0-1 knapsack inequalities
• cover inequalities
• lifting
• facet-defining inequality
• separation
• Mixed 0-1 inequalities
• flow cover inequalities