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

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, focused on the dimension of the normal cone, also leads us to nice characterizations of strict complementarity and to connections with some of the related literature.