Consider the following optimization problem: $\max \sum_{j=1}^k \lambda_j(X)$ subject to $v_j^TXv_j \leq c_j$, $X \geq 0$. $\lambda_j(X)$ is the $j$th largest eigenvalue of the positive semidefinite matrix $X$ of dimension $n\times n$, $v_j$ are some given vectors and $c_j$ some given positive numbers. $k$ is between $1 \leq k < n$. $T$ denotes transpose. All variables are real-valued. Are there any theoretical results about the optimal matrix $X$ for this problem? Is it true that the optimal solution has to be either at the vertices of our polyhedron (defined by the linear inequalities) or at one of the (unique) intersections between some of the hyperplanes and the positive semidefinite cone? We know that the sum of the largest eigenvalues is a convex function on the elements of $X$, so this is about maximizing a convex function over the convex set that is defined by intersecting the positive semidefinite cone with some hyperplanes. Grateful for any hints or references.