Hidden Convexity, Optimization, and Algorithms on Rotation Matrices
研究旋转矩阵约束优化问题中的隐藏凸性，发现某些线性映射是凸的，从而可用凸优化算法求解，对机器人、计算机视觉等领域有用。
This paper studies hidden convexity properties associated with constrained optimization problems over the set of rotation matrices [Formula: see text]. Such problems are nonconvex because of the constraint [Formula: see text]. Nonetheless, we show that certain linear images of [Formula: see text] 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 [Formula: see text] is convex and that the projection of [Formula: see text] onto its strict upper triangular entries is convex. These results allow us to construct exact convex reformulations for constrained optimization problems over [Formula: see text] 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).