A Semidefinite Hierarchy for Containment of Spectrahedra

TL;DR

This paper introduces a hierarchy of semidefinite conditions to certify whether one spectrahedron is contained within another, improving upon previous criteria by leveraging moment relaxations and polynomial optimization techniques.

Contribution

It develops a novel hierarchical approach for spectrahedron containment certification that is stronger than existing single-criterion methods and extends to positivity of matrix space maps.

Findings

Hierarchy is strictly stronger than previous criteria.

Exactness results from solitary criteria carry over to the hierarchy.

Applicable to positivity and complete positivity of matrix maps.

Abstract

A spectrahedron is the positivity region of a linear matrix pencil and thus the feasible set of a semidefinite program. We propose and study a hierarchy of sufficient semidefinite conditions to certify the containment of a spectrahedron in another one. This approach comes from applying a moment relaxation to a suitable polynomial optimization formulation. The hierarchical criterion is stronger than a solitary semidefinite criterion discussed earlier by Helton, Klep, and McCullough as well as by the authors. Moreover, several exactness results for the solitary criterion can be brought forward to the hierarchical approach. The hierarchy also applies to the (equivalent) question of checking whether a map between matrix (sub-)spaces is positive. In this context, the solitary criterion checks whether the map is completely positive, and thus our results provide a hierarchy between positivity and complete positivity.