Lattice closures of polyhedra

Given P ⊂ R, a mixed integer set P I = P ∩ (Z × 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 P I obtained by taking the convex hull of points x in P for which π 1 x, . . . , π T k x 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, Kannan, and Schrijver (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.