矩阵不等式约束下矩阵方程最小二乘问题的增广Lagrangian方法

数学物理学报 ›› 2017, Vol. 37 ›› Issue (3): 562-576.

矩阵不等式约束下矩阵方程最小二乘问题的增广Lagrangian方法

李姣芬¹, 宋丹丹¹, 周学林², 邢雨蒙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广西桂林 541004;
2. 桂林电子科技大学教务处广西桂林 541004

收稿日期:2016-09-12 修回日期:2017-01-17 出版日期:2017-06-26 发布日期:2017-06-26
通讯作者: 李姣芬,E-mail:lijiaofen603@guet.edu.cn E-mail:lijiaofen603@guet.edu.cn
基金资助:
国家自然科学基金（11301107，11261014，11561015）和广西自然科学基金（2016GXNSFAA380074，2016GXNSFFA380009）

Augmented Lagrangian Method for Matrix Equation Least-squares Problem Under a Matrix Inequality Constraint

Li Jiaofen¹, Song Dandan¹, Zhou Xuelin², Xing Yumeng¹

1. School of Mathematics and Computational Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guangxi Guilin 541004;
2 Academic Affairs Office, Guilin University of Electronic Technology, Guangxi Guilin 541004

Received:2016-09-12 Revised:2017-01-17 Online:2017-06-26 Published:2017-06-26
Supported by:
Supported by the NSFC (11301107, 11261014, 11561015) and the Natural Science Foundation of Guangxi Province (2016GXNSFAA380074, 2016GXNSFFA380009)

摘要/Abstract

摘要：

称X∈R^m×n为实（R，S）对称矩阵，若满足X=RXS，其中R∈R^m×m和S∈R^n×n为非平凡实对合矩阵，即R=R^-1≠±I_m，S=S^-1≠±I_n.该文将优化理论中求凸集上光滑函数最小值的增广Lagrangian方法应用于求解矩阵不等式约束下实（R，S）对称矩阵最小二乘问题，即给定正整数m，n，p，t，q和矩阵A_i∈R^m×m，B_i∈R^n×n（i=1，2，…，q），C∈R^m×n，E∈R^p×m，F∈R^n×t和D∈R^p×t，求实（R，S）对称矩阵X∈R^m×n且在满足相容矩阵不等式EXF ≥ D约束下极小化||A_iXB_i-C||，其中EXF ≥ D表示矩阵EXF-D非负，||·||为Frobenius范数.该文给出求解问题的矩阵形式增广Lagrangian方法的迭代格式，并用数值算例验证该方法是可行且高效的.

关键词: 矩阵不等式, 最小二乘问题, 实(R,S)对称矩阵, 增广Lagrangian方法

Abstract:

We say that a matrix X∈R^m×n is real (R, S) symmetric matrix if X=RXS, where R∈R^m×m and S∈R^n×n are real nontrivial involutions; thus R=R^-1≠±I_m, S=S^-1≠±I_n. In this paper we apply the augmented Lagrangian method, for minimizing general smooth functions on convex sets in optimization theory, to solve the (R, S) symmetric matrix least squares problem under a linear inequality constraint. That is, given positive integers m, n, p, t, q, matrices A_i∈R^m×m, B_i∈R^n×n (i=1,2,…,q), C∈R^m×n, E∈R^p×m, F∈R^n×t and D∈R^p×t, find a (R, S) symmetric matrix X∈R^m×n that minimize||A_iXB_i-C||under matrix inequality constraint EXF ≥ D, where EXF-D means that matrix EXF -D nonnegative. We present matrix-form iterative format basing on the augmented Lagrangian method to solve the proposed problem and give some numerical examples to show that the iterative method is feasible and effective.

Key words: Matrix inequality, Least-squares problem, (R,S) Symmetric matrix, Augmented Lagrangian method

中图分类号:

O241.2

李姣芬, 宋丹丹, 周学林, 邢雨蒙. 矩阵不等式约束下矩阵方程最小二乘问题的增广Lagrangian方法[J]. 数学物理学报, 2017, 37(3): 562-576.

Li Jiaofen, Song Dandan, Zhou Xuelin, Xing Yumeng. Augmented Lagrangian Method for Matrix Equation Least-squares Problem Under a Matrix Inequality Constraint[J]. Acta mathematica scientia,Series A, 2017, 37(3): 562-576.

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