Spectrahedral Containment and Operator Systems with Finite-Dimensional Realization

TL;DR

This paper investigates the theoretical foundations of a method for spectrahedral containment problems, revealing conditions for tightness and providing error bounds, with implications for optimization and convex geometry.

Contribution

It characterizes when the spectrahedral containment strengthening is tight, specifically for polyhedral cones, and extends existing error bounds in the field.

Findings

Strengthening is tight iff the polyhedral cone is a simplex.

Provides new error bounds for non-simplex cones.

Offers insights into finite-dimensional realizations of operator systems.

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 Nemirovksi. Their method provides a strengthening of the containment problem, that 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.