Sum of Squares Certificates for Containment of $\mathcal{H}$-polytopes in $\mathcal{V}$-polytopes

TL;DR

This paper investigates sum of squares certificates for polytope containment problems, showing finite convergence of a semidefinite hierarchy under certain conditions, thus providing a computational approach to a classical co-NP-complete problem.

Contribution

It introduces a semidefinite hierarchy based on sum of squares certificates for deciding polytope containment, with proven finite convergence under mild conditions.

Findings

Hierarchy converges finitely under explicit conditions

First hierarchy step suffices for large containment cases

Provides a computational method for a co-NP-complete problem

Abstract

Given an $\mathcal{H}$-polytope $P$ and a $\mathcal{V}$-polytope $Q$, the decision problem whether $P$ is contained in $Q$ is co-NP-complete. This hardness remains if $P$ is restricted to be a standard cube and $Q$ is restricted to be the affine image of a cross polytope. While this hardness classification by Freund and Orlin dates back to 1985, for general dimension there seems to be only limited progress on that problem so far. Based on a formulation of the problem in terms of a bilinear feasibility problem, we study sum of squares certificates to decide the containment problem. These certificates can be computed by a semidefinite hierarchy. As a main result, we show that under mild and explicitly known preconditions the semidefinite hierarchy converges in finitely many steps. In particular, if $P$ is contained in a large $\mathcal{V}$-polytope $Q$ (in a well-defined sense), then containment is certified by the first step of the hierarchy.