A Strengthened SDP Relaxation for Quadratic Optimization Over the Stiefel Manifold

Abstract

We study semidefinite programming (SDP) relaxations for the NP-hard problem of globally optimizing a quadratic function over the Stiefel manifold. We introduce a strengthened relaxation based on two recent ideas in the literature: (i) a tailored SDP for objectives with a block-diagonal Hessian; and (ii) the use of the Kronecker matrix product to construct SDP relaxations. Using synthetic instances on five problem classes, we show that, in general, our relaxation significantly strengthens existing relaxations, although at the expense of longer solution times.