Part II General Integer Programming II.1 The Theory of Valid Inequalities 1

 Let S = {x  Z + n : Ax  b} P = {x  R + n : Ax  b} S = P  Z n  Have max{cx: x  S} = max{cx: x  conv(S)}. How can we construct inequalities describing conv(S)? Use integrality and valid inequalities for P to construct valid inequalities for S.  Def: Valid inequalities  x   0 and  x   0 are said to be equivalent if ( ,  0 ) = ( ,  0 ) for some > 0.  x   0 dominates or is stronger than  x   0 if they are not equivalent and there exists  > 0 such that    and  0   0. A maximal valid inequality is one that is not dominated by any other inequality.  A maximal inequality for S defines a nonempty face of conv(S), but not conversely. Integer Programming

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Integer Rounding Integer Programming

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Chvatal-Gomory (C-G) Rounding Method Integer Programming

Optimizing over the First Chvátal closure Integer Programming

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 If we find good but not necessarily optimal solutions to the MIP, we find very effective valid inequalities. Also heuristic methods to find good feasible solutions to the MIP are helpful.  MIP model may not be intended as computational tools to solve real problems. But we can examine the strength of rank-1 C-G inequalities to describe the convex hull of S for various problems.  For some structured problems, e.g. knapsack problem, the separation problem for the first Chvatal closure may have some structure which enables us to handle the problem more effectively. Integer Programming

Modular Arithmetic Integer Programming

Disjunctive Constraints Integer Programming

Integer Programming

Integer Programming

Boolean Implications Integer Programming

Geometric or Combinatorial Implication Integer Programming