I was wondering what the covering number of a positive semidefinite cone is. Consider the semidefinite optimization program \begin{align} \max\langle \mathbf{C}, \mathbf{X} \rangle~~\text{subject to}~~ \langle \mathbf{A}_i, \mathbf{X} \rangle \leq b_i, i= 1,\ldots, m, \mathbf{X} \succeq \mathbf{0}. \end{align} Is it possible to find finite number of elements $\{ \mathbf{V}_k \}_{k=1}^N$ in the SDP cone \begin{align} \left\{ {\bf {X} }\in \mathbb{R}^{n\times n}\colon \langle \mathbf{A}_i, \mathbf{X} \rangle \leq b_i, i= 1,\ldots, m, \mathbf{X} \succeq \mathbf{0} \right\} \end{align} such that $$\max_{1\leq k \leq N} \langle \mathbf{V}_k, \mathbf{C}\rangle $$ is a good approximation of the original SDP optimization program?

In linear programming, we know the answer is positive. We can just search over all the vertices of the constraint and get the exact objective value.