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arXiv · 2304.08596

Hidden convexity, optimization, and algorithms on rotation matrices

Akshay Ramachandran·Kevin Shu·Alex L. Wang
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Abstract

This paper studies hidden convexity properties associated with constrained optimization problems over the set of rotation matrices $\text{SO}(n)$. Such problems are nonconvex due to 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 optimal in a formal sense.

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Akshay Ramachandran, Kevin Shu, Alex L. Wang. 2023-04-17. Hidden convexity, optimization, and algorithms on rotation matrices. https://arxiv.org/abs/2304.08596

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