Hidden Convexity, Optimization, and Algorithms on Rotation Matrices

This paper studies hidden convexity properties associated with constrained optimization problems over the set of rotation matrices SO ( n ) . Such problems are nonconvex because of the constraint X ∈ SO ( n ) . Nonetheless, we show that certain linear images of SO ( n ) are convex, opening up the possibility for convex optimization algorithms with provable guarantees for these problems. Our main technical contributions show that any two-dimensional image of SO ( n ) is convex and that the projection of SO ( n ) onto its strict upper triangular entries is convex. These results allow us to construct exact convex reformulations for constrained optimization problems over SO ( n ) with a single constraint or with constraints defined by low-rank matrices. Both of these results are maximal in a formal sense. Funding: A. Ramachandran was supported by the H2020 program of the European Research Council [Grant 805241-QIP]. A. L. Wang was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant OCENW.GROOT.2019.015 (OPTIMAL)]. K. Shu was supported by the Georgia Institute of Technology (ACO-ARC fellowship).