Chapter 6: Semidefinite Representability

Abstract

It is natural to ask which convex optimization problems can be formulated as semidefinite programs. If such a formulation exists, how can we find it? The answer to these questions is equivalent to finding an exact representation of a convex set as a spectrahedron or projected spectrahedron. Whenever this can be done, we say that the convex set has a semidefinite representation or it is semidefinite representable. IntroductionWe begin by examining the question of when a convex set S is a spectrahedron. Since a spectrahedron is defined by a linear matrix inequality, the points on the boundary of S must satisfy a polynomial equation given by the determinant of its linear pencil. Therefore, only convex sets whose boundaries have a polynomial description can be spectrahedra. In fact, being a spectrahedron is even more restrictive and we will examine some of these restrictions in this chapter. In particular, we will present a complete characterization of two-dimensional spectrahedra due to Helton and Vinnikov. In higher dimensions a full characterization of which convex sets are spectrahedra is unknown.The class of projected spectrahedra is considered next. We will provide some natural necessary conditions for a set to be a projected spectrahedron. Deriving sufficient conditions brings us to explicit construction methods for semidefinite representations. A general technique for constructing such representations and approximations of a convex set S given by polynomial equations and inequalities is to use moments. The basic idea is that we introduce an independent variable for † Jiawang Nie was supported by NSF grants DMS-0757212 and DMS-0844775.

Spectrahedral Shadows and Completely Positive Maps on Real Closed Fields

“…Remark 3.16. A closed convex cone K ⊆ R n is a spectrahedral shadow if and only if its dual cone is a spectrahedral shadow (Exercise 6.23 in [Nie13]). The dual cone of P + (S) is the closed conical hull of the image of R n under the monomial map given by the monomials from S.…”