Lattice closures of polyhedra

Given P⊂ Rⁿ, a mixed-integer set P^I= P∩ (Z^t× R^{n - t}), and a k-tuple of n-dimensional integral vectors (π₁, … , π_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 π1Tx,…,πkTx 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≥ 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.