On finitely generated closures in the theory of cutting planes

TL;DR

This paper generalizes a result in cutting-plane theory, showing conditions under which the intersection of certain convex sets remains a rational polyhedron, with implications for mixed-integer optimization.

Contribution

It provides a short proof of a generalized theorem and characterizes when the max-facet-width is bounded within a class of lattice-free polyhedra.

Findings

The intersection of R_L(P) is a rational polyhedron under bounded max-facet-width.

A characterization of boundedness of max-facet-width is provided.

Results have applications in cutting-plane methods for mixed-integer optimization.

Abstract

Let $P$ be a rational polyhedron in $\mathbb{R}^d$ and let $\mathcal{L}$ be a class of $d$-dimensional maximal lattice-free rational polyhedra in $\mathbb{R}^d$. For $L \in \mathcal{L}$ by $R_L(P)$ 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 $L \in \mathcal{L}$ is bounded from above by a constant independent of $L$, then $\bigcap_{L\in \mathcal{L}} R_L(P)$ 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 $\mathcal{L}$. The presented results are motivated by applications in cutting-plane theory from mixed-integer optimization.