Exposed faces of semidefinitely representable sets

TL;DR

This paper investigates the geometric structure of semidefinite representable sets, showing that the moment matrix method can only produce representations if all faces of the set are exposed, thus providing a necessary condition.

Contribution

It establishes a necessary condition for semidefinite representability using Lasserre's moment matrix method, linking face exposure to the method's applicability.

Findings

All faces of spectrahedra are exposed.

Lasserre's method requires all faces to be exposed.

Provides a necessary condition complementing existing sufficient conditions.

Abstract

A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite. Motivated by the fact that diagonal LMIs define polyhedra, the solution set of an LMI is called a spectrahedron. Linear images of spectrahedra are called semidefinite representable sets. Part of the interest in spectrahedra and semidefinite representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming, like one can do on polyhedra by linear programming. It is known that every face of a spectrahedron is exposed. This is also true in the general context of rigidly convex sets. We study the same question for semidefinite representable sets. Lasserre proposed a moment matrix method to construct semidefinite representations for certain sets. Our main result is that this method can only work if all faces of the considered set are exposed. This necessary condition complements sufficient conditions recently proved by Lasserre, Helton and Nie.