On finitely generated closures in the theory of cutting planes

Abstract

Let P be a rational polyhedron in Rd and let L be a class of d-dimensional maximal lattice-free rational polyhedra in Rd. For L∈L by RL(P) we denote the convex hull of points belonging to P but not to the interior of L. Andersen, Louveaux and Weismantel showed that if the so-called max-facet-width of all L∈L is bounded from above by a constant independent of L, then ⋂L∈LRL(P) is a rational polyhedron. We give a short proof of a generalization of this result. We also give a characterization for the boundedness of the max-facet-width on L. The presented results are motivated by applications in cutting-plane theory from mixed-integer optimization.