On convergence in mixed integer programming

Abstract

Let $${P \subseteq {\mathbb R}^{m+n}}$$ be a rational polyhedron, and let P I be the convex hull of $${P \cap ({\mathbb Z}^m \times {\mathbb R}^n)}$$ . 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 $${{\mathbb R}^m}$$ . 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 Chvatal 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 I .