Lattice closures of polyhedra

Given $$P\subset {\mathbb {R}}^n$$ P ⊂ R n , a mixed-integer set $$P^I=P\cap ({\mathbb {Z}}^{t}\times {\mathbb {R}}^{n-t}$$ P I = P ∩ ( Z t × R n - t ), and a k -tuple of n -dimensional integral vectors $$(\pi _1, \ldots , \pi _k)$$ ( π 1 , … , π k ) where the last $$n-t$$ n - t entries of each vector is zero, we consider the relaxation of $$P^I$$ P I obtained by taking the convex hull of points x in P for which $$ \pi _1^Tx,\ldots ,\pi ^T_kx$$ π 1 T x , … , π k T x are integral. We then define the k -dimensional lattice closure of $$P^I$$ 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 $$P^I$$ P I and is equal to the split closure when $$k=1$$ 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 $$k=1$$ k = 1 and our finiteness result extends this to all $$k\ge 2$$ k ≥ 2 . We also construct a polyhedral mixed-integer set with n integer variables and one continuous variable such that for any $$k < n$$ 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.