Vertices of Spectrahedra arising from the Elliptope, the Theta Body, and Their Relatives

Abstract

Abstract. Utilizing dual descriptions of the normal cone of convex optimization problems in conic form, we characterize the vertices of semidefinite representations arising from Lovász theta body, generalizations of the elliptope and related convex sets. Our results generalize vertex characterizations due to Laurent and Poljak from the 1990's. Our approach also leads us to nice characterizations of strict complementarity and to connections with some of the related literature.

A Lower Bound on the Positive Semidefinite Rank of Convex Bodies

“…It is known (see, e.g., [15, Theorem 2.2.5]) that any convex set has at most a countable number of vertices. Vertices of spectrahedra arising from combinatorial optimization problems have been studied in [10,4]. The next theorem gives an upper bound on the number of vertices of any spectrahedral shadow.…”

Combinatorial and geometric dualities in graph homomorphism optimization problems

“…Our next proposition combines Proposition 5.16 with a well-known linear function which maps the boolean quadric polytope to the cut polytope CUT ±1 V . Following [29], define…”