Hidden Convexity, Optimization, and Algorithms on Rotation Matrices

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-07-01 DOI:10.1287/moor.2023.0114

Akshay Ramachandran, Kevin Shu, Alex L. Wang

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引用次数: 0

Abstract

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).

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旋转矩阵的隐凸性、优化和算法

本文研究与旋转矩阵集合上的约束优化问题相关的隐凸性质[公式：见正文]。由于[公式：见正文]的约束，这类问题是非凸的。然而，我们证明了[公式：见正文]的某些线性图像是凸的，从而为这些问题提供了可证明的凸优化算法的可能性。我们的主要技术贡献表明，[公式：见正文]的任何二维图像都是凸的，而且[公式：见正文]对其严格上三角项的投影也是凸的。这些结果使我们能够为[公式：见正文]上的约束优化问题构建精确的凸重构，这些问题具有单一约束或由低阶矩阵定义的约束。这两个结果在形式意义上都是最大的：A. Ramachandran 得到了欧洲研究理事会 H2020 计划 [805241-QIP] 的资助。A. L. Wang 得到了 Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant OCENW.GROOT.2019.015 (OPTIMAL)] 的资助。K. Shu得到了佐治亚理工学院（ACO-ARC奖学金）的资助。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.