Spectrahedral containment and operator systems with finite-dimensional realization

Abstract

Containment problems for polytopes and spectrahedra appear in various applications, such as linear and semidefinite programming, combinatorics, convexity, and stability analysis of differential equations. This paper explores the theoretical background of a method proposed by Ben-Tal and Nemirovski [SIAM J. Optim., 12 (2002), pp. 811–833]. Their method provides a strengthening of the containment problem, which is algorithmically well tractable. To analyze this method, we study abstract operator systems and investigate when they have a finite-dimensional concrete realization. Our results give some profound insight into their approach. They imply that when testing the inclusion of a fixed polyhedral cone in an arbitrary spectrahedron, the strengthening is tight if and only if the polyhedral cone is a simplex. This is true independent of the representation of the polytope. We also deduce error bounds in the other cases, simplifying and extending recent results by various authors.