English

运筹学学报

• 运筹学 • 上一篇    下一篇

正交约束优化问题的一阶算法

高斌1,2,*  刘歆1,2  袁亚湘2   

  1. 1. 中国科学院大学,  北京  100190   2. 中国科学院数学与系统科学研究院, 科学与工程计算国家重点实验室, 北京  100190
     
  • 收稿日期:2017-09-02 出版日期:2017-12-15 发布日期:2017-12-15
  • 通讯作者: 高斌 gaobin@lsec.cc.ac.cn
  • 基金资助:

    国家自然科学基金项目(Nos. 11471325, 91530204, 11622112, 11688101, 11331012, 11461161005), 中国科学院前沿科学重点研究计划 (No. QYZDJ-SSW-SYS010), 国家数学与交叉科学中心, 中国科学院科学与工程计算国家重点实验室

First-order algorithms for optimization problems with orthogonality constraints

GAO Bin1,2,*   LIU Xin1,2  YUAN Yaxiang2   

  1. 1. University of Chinese Academy of Sciences,  Beijing 100190, China 2. State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
  • Received:2017-09-02 Online:2017-12-15 Published:2017-12-15

PDF

840

摘要：

带有正交约束的矩阵优化问题在材料计算、统计及数据分析等领域中有着广泛的应用. 由于正交约束的可行域是~Stiefel~流形, 一直以来流形上的优化方法是求解这一问题的主要方法. 近年来, 随着实际应用问题所要求的变量规模的扩大, 传统的流形优化方法在计算上的劣势显现出来, 而一些迭代简单、收敛快的新算法逐渐被提出. 通过收缩方法、非收缩可行方法、不可行方法三个类别分别来介绍求解带有正交约束的矩阵优化问题的最新算法. 通过分析这些方法的主要特性, 以及应用问题的要求, 对这类问题算法设计的研究进行了展望.

关键词: 正交约束, Stiefel 流形, 收缩方法

Abstract:

Optimization problems with orthogonality constraints have a wide range of applications in the field of materials science, statistics and data science. Many optimization algorithms on manifold can be applied to this type of problems, since the feasible region of orthogonal constraint is known as Stiefel manifold.
In recent years, with the expansion of variable scale required by practical application, the limitations of existing methods on manifold are reflected in practice. On the other hand, some efficient approaches based on new concepts are proposed recently. In this paper, we briefly introduce the main classes of methods for optimization problems with orthogonality constraints including  retraction based method, non-retraction based method and infeasible method respectively. We also discuss the main characteristics of these approaches, the scenarios in which these approaches are suitable and the possible directions for further development.
 

Key words: orthogonality constraints, Stiefel manifold, retraction based method