• 论文 •

广义特征值极小扰动问题的一类黎曼共轭梯度法

1. 1. 桂林电子科技大学数学与计算科学学院, 桂林 541004;
2. 云南大学数学与统计学院, 昆明 650500;
3. 桂林电子科技大学国际学院, 桂林 541004;
4. 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, 广西应用数学中心 (桂林电子科技大学), 广西自动检测技术与仪器重点实验室, 桂林 541004
• 收稿日期:2021-03-25 出版日期:2022-11-14 发布日期:2022-11-08
• 通讯作者: 周学林,Email:zhouxuelin0309@163.com.
• 基金资助:
国家自然科学基金资助项目（12261026，11961012，12201149），广西自然科学基金资助项目（2016GXNSFAA380074，2017GXNSFBA198082），广西科技基地和人才专项（2021AC06001），2022年桂林电子科技大学校级研究生创新项目（2022YCXS142），广西自动检测技术与仪器重点实验室基金（YQ21103，YQ22106）资助.

Kong Lingchang, Wei Keyang, Zhou Xuelin, Li Jiaofen. A RIEMANNIAN CONJUGATE GRADIENT APPROACH FOR SOLVING THE GENERALIZED EIGENVALUE PROBLEM WITH MINIMAL PERTURBATION[J]. Mathematica Numerica Sinica, 2022, 44(4): 508-533.

A RIEMANNIAN CONJUGATE GRADIENT APPROACH FOR SOLVING THE GENERALIZED EIGENVALUE PROBLEM WITH MINIMAL PERTURBATION

Kong Lingchang1, Wei Keyang1, Zhou Xuelin2,3, Li Jiaofen4

1. 1. School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China;
2. School of Mathematics and Statistics, Yunan University, Kunming 650000, China;
3. College of International Exchange, Guilin University of Electronic Technology, Guilin 541004, China;
4. School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Center for Applied Mathematics of Guangxi (GUET), Guangxi Key Laboratory of Automatic Detecting Technology and Instruments, Guilin University of Electronic Technology, Guilin, 541004, China
• Received:2021-03-25 Online:2022-11-14 Published:2022-11-08

This paper presents an efficient approach for solving a kind of complex product manifold constrained matrix least squares problem, which derived from the $l$ parameterized generalized eigenvalue problem for nonsquare matrix pencils with minimal perturbation. Different from the existing work, the paper directly focuses on the complex problem model, combining the geometric properties of the considered complex product manifold and basing on the modified Fletcher-Reeves nonlinear conjugate gradient method on Euclidean space, a class of Riemannian nonlinear conjugate gradient algorithm is designed for solving the underlying problem, and the global convergence analysis is given. Numerical experiments and numerical comparisons are given to show that the proposed algorithm converges faster than the existing algorithm with parameter $l=1$, and can get the same accuracy as the existing algorithm with parameter $l=n$. Detailed comparisons with some latest methods, including some other gradient-based methods, some Riemannian second-order algorithms, and two non-manifold optimization algorithms are also provided to show the merits of the proposed approach.

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