More on Disjunctive Convex Hulls of Generalized Cross Polytopes

Abstract

We continue the study of the convex hull $D\subset\mathbb{R}^{d+n}$ of a disjunction of $n+1$ polytopes in $\mathbb{R}^d$, managed by $n$ binary indicator variables. Previously, $D$ had been characterized for arbitrary polytopes when $d\in\{1,2\}$, for polytopes sharing a common constraint matrix (under a technical condition, including hyper-rectangles), and for two generalized cross polytopes ($n=1$). Here we give a complete description of $D$ for arbitrary $n$ and $d$ when all $n+1$ polytopes are generalized cross polytopes. Every vertical facet is described by a normalized support inequality, and such an inequality is facet describing if and only if its equality graph, a bipartite graph on the polytopes and the coordinates, is connected; all vertical facets are generated explicitly from feasible spanning trees. We characterize when such a facet is one of the previously known pairwise inequalities, and we show that pairwise inequalities suffice when $d\leq3$, but not in general. We establish a bijection between the vertical facets and the maximal cells of a regular subdivision of $\Delta_n\times\Delta_{d-1}$\,, which yields the sharp bound $n+1+2^d\binom{n+d-1}{n}$ on the number of facets of $D$, attained for generic data. We also give a polynomial-time separation algorithm that returns a violated facet, which is maximally violated among the normalized support inequalities. Computational experiments show that the facets that are not pairwise inequalities close a large part of the gap left by the pairwise inequalities.