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 $\text{SO}(n)$. Such problems are nonconvex because of the constraint $X\in \text{SO}(n)$. Nonetheless, we show that certain linear images of $\text{SO}(n)$ 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 $\text{SO}(n)$ is convex and that the projection of $\text{SO}(n)$ onto its strict upper triangular entries is convex. These results allow us to construct exact convex reformulations for constrained optimization problems over $\text{SO}(n)$ 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).