Currently, I am studying Lovasz's semidefinite program (SDP) to calculate $\alpha(G)$ for perfect graphs. And I encountered a paper$^\color{magenta}{\star}$ by Silva & Tunçel. A vertex of a convex set is an extreme point of that set whose normal cone is full-dimensional. I am very curious why it's in general important to study vertices of the spectrahedron instead of extreme points. What information can we get from the vertices?
I know in linear programming, vertices coincide with extreme points, and we can always find an optimal point that is an extreme point. But for SDP, do we still have guarantees like this? Can we understand the shape or get characterizations of all optimal solutions to the SDP?
$\color{magenta}{\star}$ Marcel K. de Carli Silva, Levent Tunçel, Vertices of spectrahedra arising from the elliptope, the theta body, and their relatives, SIAM Journal on Optimization, 2015. [arXiv]
