Containment problems for polytopes and spectrahedra

TL;DR

This paper investigates the computational complexity of containment problems between polytopes and spectrahedra, providing complexity classifications and semidefinite conditions for certification, with exact characterizations in key cases.

Contribution

It extends complexity results to polytope/spectrahedron containment and develops semidefinite criteria that can certify containment, including exact conditions in important scenarios.

Findings

NP-hardness results for various containment problems

Semidefinite conditions can certify containment in many cases

Exact semidefinite characterizations for certain containment problems

Abstract

We study the computational question whether a given polytope or spectrahedron $S_A$ (as given by the positive semidefiniteness region of a linear matrix pencil $A(x)$) is contained in another one $S_B$. First we classify the computational complexity, extending results on the polytope/polytope-case by Gritzmann and Klee to the polytope/spectrahedron-case. For various restricted containment problems, NP-hardness is shown. We then study in detail semidefinite conditions to certify containment, building upon work by Ben-Tal, Nemirovski and Helton, Klep, McCullough. In particular, we discuss variations of a sufficient semidefinite condition to certify containment of a spectrahedron in a spectrahedron. It is shown that these sufficient conditions even provide exact semidefinite characterizations for containment in several important cases, including containment of a spectrahedron in a polyhedron. Moreover, in the case of bounded $S_A$ the criteria will always succeed in certifying containment of some scaled spectrahedron $\nu S_A$ in $S_B$.