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.