Lattice closures of polyhedra

Given P⊂ℝ^n , a mixed-integer set P^I=P∩ (ℤ^t×ℝ^n-t ), and a k -tuple of n -dimensional integral vectors (π _1, … , 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 π _1^Tx,… ,π ^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≥ 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.