Consider this to be our SDP problem:
Minimize $\langle C, X \rangle$ such that
- $\langle A_i, X \rangle \ge b_i$ for all $i \in [m]$ and
- $X \succcurlyeq 0$.
For SDPs, is there a relationship between the extreme points of the feasible region and the optimal solutions? In particular, does at least one optimal solution lie at an extreme point? (I know this is true for LPs, but does it extend to SDPs?)
For reference, I am asking because I am looking at two proofs (1 and 2) of a theorem by Barvinok and Pataki, and they formulate their statements of the theorem in different ways. In particular, the difference makes me suspect there is some relationship between extreme points and optima.
Edit: By extreme point I mean vertex.