ESI 6448 Discrete Optimization Theory

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ESI 6448Discrete Optimization Theory

• Lecture 20

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Last class

• Fractional dual algorithm for IP
• Using Gomory cuts
• Finite of steps
• Lexicography
• More inequalities
• Disjunctive inequalities

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Fractional dual algorithm for IP

• Initialization.
• Solve LP and get optimal sol x
• Iteration. (reoptimization)?
• Choose a source row i (ai0 lt 0).
• Add the Gomory cut using the row i and a
  corresponding basic variable s.
• Apply the dual simplex algorithm.
• If dual is unbounded, stop w/ empty sol.
• Let x be the new optimal sol from the
  reoptimization.
• If x is integral, stop w/ optimal integer sol
  x.

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Example

x2
3
2
1
x1
3
2
1
0
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Finiteness of fractional dual algorithm

• To prove the finiteness,
• no cycle should be generated
• anticycling rules for simplex
• Lexicographical pivoting rule
• lexicography ordering
• Blands rule
• subscript ordering

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Lexicography

• lex-min choice will be unique

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Anticycling

• Lexicographic pivoting rule
• Entering column any j s.t. cj is negative
• Exiting column for each i s.t. ui gt 0, divide
  i-th row by ui and choose the lex-min row l.
• Blands rule (smallest subscript pivoting rule)?
• Entering column smallest j s.t. cj is negative.
• Exiting column out of all ties, choose the
  smallest l.
• first row is selected.

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Lexicography for primal simplex

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Blands rule for primal simplex

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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 constraints

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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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Today

• Finiteness of fractional dual algorithm for IP
• Finite of steps using lexicography
• More inequalities
• D-inequalities
• MIR inequalities