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    On convergence in mixed integer programming

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    Let P⊆Rm+n{P \subseteq {\mathbb R}^{m+n}} be a rational polyhedron, and let P I be the convex hull of P∩(Zm×Rn){P \cap ({\mathbb Z}^m \times {\mathbb R}^n)} . We define the integral lattice-free closure of P as the set obtained from P by adding all inequalities obtained from disjunctions associated with integral lattice-free polyhedra in Rm{{\mathbb R}^m} . We show that the integral lattice-free closure of P is again a polyhedron, and that repeatedly taking the integral lattice-free closure of P gives P I after a finite number of iterations. Such results can be seen as a mixed integer analogue of theorems by Chvátal and Schrijver for the pure integer case. One ingredient of our proof is an extension of a result by Owen and Mehrotra. In fact, we prove that for each rational polyhedron P, the split closures of P yield in the limit the set P
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