A Strengthened SDP Relaxation for Quadratic Optimization Over the Stiefel Manifold

TL;DR

This paper proposes a new, strengthened SDP relaxation method for quadratic optimization over the Stiefel manifold, improving solution quality at the cost of increased computational time.

Contribution

It introduces a novel SDP relaxation combining block-diagonal Hessian tailoring and Kronecker products, enhancing existing relaxations for this NP-hard problem.

Findings

Significantly improves relaxation strength on synthetic instances

Outperforms previous relaxations in solution quality

Increases computational time due to added complexity

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; (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.