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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2024, Vol. 41 ›› Issue (3): 507-524.doi: 10.3969/j.issn.1005-3085.2024.03.010

• • 上一篇    下一篇

多元统计分析中一类矩阵迹函数极小化问题的分裂迭代法

段  强1,  周学林1,2,  李姣芬1,3   

  1. 1. 桂林电子科技大学数学与计算科学学院，广西高校数据分析与计算重点实验室，桂林 541004
    2. 云南大学数学与统计学院，昆明 650000
    3. 广西应用数学中心(桂林电子科技大学)，桂林 541004
  • 收稿日期:2021-06-19 接受日期:2021-08-13 出版日期:2024-06-15 发布日期:2024-08-15
  • 通讯作者: 周学林 E-mail: zhouxuelin0309@163.com
  • 基金资助:
    国家自然科学基金(12261026;12361079;11961012;12201149)；广西自然科学基金(2023GXNSFAA026067)；桂林电子科技大学研究生创新教育计划(2022YXW01; 2022YCXS142)；广西自动检测技术与仪器重点实验室基金(YQ23104; YQ22106).

Splitting Iterative Methods for Minimizing a Class of Matrix Trace Function in Multivariate Statistical Analysis

DUAN Qiang1,  ZHOU Xuelin1,2,  LI Jiaofen1,3   

  1. 1. Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004
    2. School of Mathematics and Statistics, Yunan University, Kunming 650000
    3. Center for Applied Mathematics of Guangxi, Guilin University of Electronic Technology, Guilin 541004
  • Received:2021-06-19 Accepted:2021-08-13 Online:2024-06-15 Published:2024-08-15
  • Contact: X. Zhou. E-mail address: zhouxuelin0309@163.com
  • Supported by:
    The National Natural Science Foundation of China (12261026; 12361079; 11961012; 12201149); the Natural Science Foundation of Guangxi (2023GXNSFAA026067); the Innovation Project of Guilin University of Electronic Technology Graduate Education (2022YXW01; 2022YCXS142); the Guangxi Key Laboratory of Automatic Detecting Technology and Instruments (YQ23104; YQ22106).

摘要： 研究了来源于多元统计分析中的一类含列正交约束的矩阵迹函数极小化模型，该模型的特殊形式广泛应用于多维标度分析中DEDICOM模型和正交INDSCAL模型最小二乘拟合等问题中。结合变量分裂构造了几类经典的基于分裂的不可行迭代算法求解该约束迹函数极小化模型，并给出算法外层迭代框架和内层子问题的具体求解方案。数值实验验证了算法的有效性。

关键词: 正交分裂, 矩阵迹函数, 正交约束, 增广拉格朗日方法

Abstract: In this paper, we considered a class of matrix trace function minimization problem under orthogonal constraints which arise in multivariate statistical analysis. Serval special forms of the considered problem model are widely used in the least square fitting of DEDICOM model and orthogonal INDSCAL model in multidimensional scaling analysis. Combining with orthogonal splitting techniques, several classical unfeasible iterative algorithms for solving manifold optimization problems are constructed to solve the underlying problem, and the iterative framework of these algorithms and the specific solution scheme of the generated subproblems are given. Some numerical tests are given to show the efficiency of the proposed methods.

Key words: orthogonal splitting, matrix trace function, orthogonal constraint, augmented Lagrangian method

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