Abstract
We consider mixed-integer linear programs where free integer variables are expressed in terms of nonnegative continuous variables. When this model only has two integer variables, Dey and Louveaux characterized the intersection cuts that have infinite split rank. We show that, for any number of integer variables, the split rank of an intersection cut generated from a rational lattice-free polytope L is finite if and only if the integer points on the boundary of L satisfy a certain “2-hyperplane property.” The Dey–Louveaux characterization is a consequence of this more general result.
Highlights
IntroductionWe consider mixed integer linear programs with equality constraints expressing m ≥ 1 free integer variables in terms of k ≥ 1 nonnegative continuous variables
In this paper, we consider mixed integer linear programs with equality constraints expressing m ≥ 1 free integer variables in terms of k ≥ 1 nonnegative continuous variables.k x = f + rjsj j=1 (1) x ∈ Zm s ∈ Rk+.The convex hull R of the solutions to (1) is a corner polyhedron [11, 12]
There is a facet-defining inequality for R that cannot be deduced from a finite recursive application of the split closure operation. This inequality is an intersection cut generated from a maximal lattice-free triangle L with integer vertices and rays going into its corners
Summary
We consider mixed integer linear programs with equality constraints expressing m ≥ 1 free integer variables in terms of k ≥ 1 nonnegative continuous variables. There is a facet-defining inequality for R that cannot be deduced from a finite recursive application of the split closure operation This inequality is an intersection cut generated from a maximal lattice-free triangle L with integer vertices and rays going into its corners. Given a polytope L ⊂ Rm containing f in its interior, let LB be the convex hull of the boundary points for the rays r1, . The L-cut has finite split rank if and only if LB has the 2-hyperplane property This corollary is a direct generalization of the characterization of Dey and Louveaux for m = 2, as triangles of Type 1 do not have the 2-hyperplane property whereas all other lattice-free polytopes in the plane do, as one can check using Lovasz’ [13] characterization of maximal lattice-free convex sets in the plane.
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