A new semidefinite programming relaxation scheme for a class of quadratic matrix problems

We consider a special class of quadratic matrix optimization problems which often arise in applications. By exploiting the special structure of these problems, we derive a new semidefinite relaxation which, under mild assumptions, is proven to be tight for a larger number of constraints than could be achieved via a direct approach. We show the potential usefulness of these results when applied to robust least-squares and sphere-packing problems.