  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;
(ii) and the use of the Kronecker matrix product to construct SDP relaxations.
Using synthetic instances on four problem classes, we show that, in general,
our relaxation significantly strengthens existing relaxations, although at the
expense of longer solution times.
