Lattice closures of polyhedra

Given P⊂Rn, a mixed-integer set PI=P∩(Zt×Rn−t), and a k-tuple of n-dimensional integral vectors (π1,…,πk) where the last n−t entries of each vector is zero, we consider the relaxation of PI obtained by taking the convex hull of points x in P for which πT1x,…,πTkx are integral. We then define the k-dimensional lattice closure of PI 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 PI 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.