Journal of the Operations Research Society of China ›› 2020, Vol. 8 ›› Issue (2): 199-248.doi: 10.1007/s40305-020-00295-9

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  • 收稿日期:2019-06-12 修回日期:2019-12-10 出版日期:2020-06-30 发布日期:2020-07-07
  • 通讯作者: Zai-Wen Wen, Jiang Hu, Xin Liu, Ya-Xiang Yuan E-mail:wenzw@pku.edu.cn;jianghu@pku.edu.cn;liuxin@lsec.cc.ac.cn;yyx@lsec.cc.ac.cn

A Brief Introduction to Manifold Optimization

Jiang Hu1, Xin Liu2,3, Zai-Wen Wen1, Ya-Xiang Yuan2   

  1. 1 Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China;
    2 State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
    3 University of Chinese Academy of Sciences, Beijing 100190, China
  • Received:2019-06-12 Revised:2019-12-10 Online:2020-06-30 Published:2020-07-07
  • Contact: Zai-Wen Wen, Jiang Hu, Xin Liu, Ya-Xiang Yuan E-mail:wenzw@pku.edu.cn;jianghu@pku.edu.cn;liuxin@lsec.cc.ac.cn;yyx@lsec.cc.ac.cn
  • Supported by:
    Xin Liu's research was supported in part by the National Natural Science Foundation of China (No. 11971466), Key Research Program of Frontier Sciences, Chinese Academy of Sciences (No. ZDBS-LY-7022), the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences and the Youth Innovation Promotion Association, CAS.
    ai-Wen Wen's research was supported in part by the the National Natural Science Foundation of China (Nos. 11421101 and 11831002), and the Beijing Academy of Artificial Intelligence.
    Ya-Xiang Yuan's research was supported in part by the National Natural Science Foundation of China (Nos. 11331012 and 11461161005).

Abstract: Manifold optimization is ubiquitous in computational and applied mathematics, statistics,engineering,machinelearning,physics,chemistry,etc.Oneofthemainchallenges usually is the non-convexity of the manifold constraints. By utilizing the geometry of manifold, a large class of constrained optimization problems can be viewed as unconstrained optimization problems on manifold. From this perspective, intrinsic structures, optimality conditions and numerical algorithms for manifold optimization are investigated. Some recent progress on the theoretical results of manifold optimization is also presented.

Key words: Convergence, First-order-type algorithms, Manifold optimization, Retraction, Second-order-type algorithms

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