Exposed Faces of Semidefinitely Representable Sets

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 semidefinitely representable sets. Part of the interest in spectrahedra and semidefinitely representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming, such as 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 semidefinitely representable sets. Lasserre proposed a moment matrix method to construct semidefinite representations for certain sets. Our main result is that this method can work only if all faces of the considered set are exposed. This necessary condition complements sufficient conditions recently proved by Lasserre, Helton, and Nie.