On the Polyhedrality of the Chvatal-Gomory Closure

TL;DR

This paper characterizes when the Chvatal-Gomory closure of certain convex sets is finitely generated and polyhedral, unifying previous results and extending understanding of the closure's structure.

Contribution

It provides an equivalent condition for the finite generation of the CG closure and characterizes its polyhedrality for a broad class of convex sets.

Findings

CG closure is polyhedral iff the recession cone is rational polyhedral.

Unifies existing results for rational polyhedra and compact convex sets.

Generalizes the conditions for polyhedrality of CG closures.

Abstract

In this paper, we provide an equivalent condition for the Chvatal-Gomory (CG) closure of a closed convex set to be finitely-generated. Using this result, we are able to prove that, for any closed convex set that can be written as the Minkowski sum of a compact convex set and a closed convex cone, its CG closure is a rational polyhedron if and only if its recession cone is a rational polyhedral cone. As a consequence, this generalizes and unifies all the currently known results, for the case of rational polyhedron and compact convex set.