Lattice closures of polyhedra

Abstract

Given \(P\subset {\mathbb {R}}^n\), a mixed-integer set \(P^I=P\cap ({\mathbb {Z}}^{t}\times {\mathbb {R}}^{n-t}\)), and a k-tuple of n-dimensional integral vectors \((\pi _1, \ldots , \pi _k)\) where the last \(n-t\) entries of each vector is zero, we consider the relaxation of \(P^I\) obtained by taking the convex hull of points x in P for which \( \pi _1^Tx,\ldots ,\pi ^T_kx\) are integral. We then define the k-dimensional lattice closure of \(P^I\) to be the intersection of all such relaxations obtained from k-tuples of n-dimensional vectors. When P is a rational polyhedron, we show that given any collection of such k-tuples, there is a finite subcollection that gives the same closure; more generally, we show that any k-tuple is dominated by another k-tuple coming from the finite subcollection. The k-dimensional lattice closure contains the convex hull of \(P^I\) and is equal to the split closure when \(k=1\). Therefore, a result of Cook et al. (Math Program 47:155–174, 1990) implies that when P is a rational polyhedron, the k-dimensional lattice closure is a polyhedron for \(k=1\) and our finiteness result extends this to all \(k\ge 2\). We also construct a polyhedral mixed-integer set with n integer variables and one continuous variable such that for any \(k < n\), finitely many iterations of the k-dimensional lattice closure do not give the convex hull of the set. Our result implies that t-branch split cuts cannot give the convex hull of the set, nor can valid inequalities from unbounded, full-dimensional, convex lattice-free sets.