On the polyhedrality of cross and quadrilateral closures

Split cuts form a well-known class of valid inequalities for mixed-integer programming problems. Cook et al. (Math Program 47:155–174, 1990) showed that the split closure of a rational polyhedron P is again a polyhedron. In this paper, we extend this result from a single rational polyhedron to the union of a finite number of rational polyhedra. We then use this result to prove that cross cuts yield closures that are rational polyhedra. Cross cuts are a generalization of split cuts introduced by Dash et al. (Math Program 135:221–254, 2012). Finally, we show that the quadrilateral closure of the two-row continuous group relaxation is a polyhedron, answering an open question in Basu et al. (Math Program 126:281–314, 2011).