On disjunction convex hulls by lifting

TL;DR

This paper investigates the convex hull of disjunctions of polytopes using extended-variable formulations, revealing conditions under which full big-M lifting suffices and providing insights into the polyhedral structure and facet enumeration.

Contribution

It characterizes when full optimal big-M lifting describes the convex hull for disjunctions of polytopes and introduces conditions where MIR inequalities are more effective.

Findings

Full big-M lifting describes the convex hull for $d extless=2$.

For $d extgreater=3$, full big-M lifting does not always suffice.

All facets of the convex hull can be enumerated in polynomial time for fixed $d$.

Abstract

We study the natural extended-variable formulation for the disjunction of $n+1$ polytopes in $\mathbb{R}^d$. We demonstrate that the convex hull $D$ in the natural extended-variable space $\mathbb{R}^{d+n}$ is given by full optimal big-M lifting (i) when $d\leq 2$ (and that it is not generally true for $d\geq 3$), and also (ii) under some technical conditions, when the polytopes have a common facet-describing constraint matrix, for arbitrary $d\geq 1$ and $n\geq 1$. We give a broad family of examples with $d\geq 3$ and $n=1$, where the convex hull is not described after employing all full optimal big-M lifting inequalities, but it is described after one round of MIR inequalities. Additionally, we give some general results on the polyhedral structure of $D$, and we demonstrate that all facets of $D$ can be enumerated in polynomial time when $d$ is fixed.