• 论文 •

### 核范数和谱范数下广义Sylvester方程最小二乘问题的有效算法

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. 华南师范大学数学科学学院, 广州 510631
• 收稿日期:2015-12-22 出版日期:2017-05-15 发布日期:2017-07-18
• 基金资助:

国家自然科学基金资助项目（11561015，11671158），广西自然科学基金资助项目（2016GXNSFAA380074，2016GXNSFFA380009）.

Li Jiaofen, Song Dandan, Li Tao, Li Wen. AN EFFICIENT METHOD FOR SOLVING GENERALIZED SYLVERSTER EQUATION MINIMIZATION PROBLEM UNDER THE NUCLEAR AND SPECTRAL NORM[J]. Mathematica Numerica Sinica, 2017, 39(2): 129-150.

### AN EFFICIENT METHOD FOR SOLVING GENERALIZED SYLVERSTER EQUATION MINIMIZATION PROBLEM UNDER THE NUCLEAR AND SPECTRAL NORM

Li Jiaofen1, Song Dandan1, Li Tao1, Li Wen2

1. 1. School of Mathematics and Computational Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin 541004, China;
2. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
• Received:2015-12-22 Online:2017-05-15 Published:2017-07-18

minXS||∑i=1NAiXBi-C||q

In this paper,we are concerned with the following generalized Sylvester equation least squares problem of the form
minXS||∑i=1NAiXBi-C||q,
,where||.||stands for the Schatten q-norm,which defined as||M||qq=∑i=1nσiq(M) and σi(M)(i=1,...,n) be the singular values of M∈Rn×n,S be the closed convex set.Some special types of this problem can be applied in image processing and control theory.An inexact version of alternating direction method (ADM) with truly implementable inexactness criteria is proposed for solving this problem under the nuclear norm and spectrum norm,namely q=1,+∞,combining with the Singular Value Threshold algorithm,Moreau-Yosida regularization algorithm,Spectral Projection algorithm and LSQR algorithm to deal with the generated subproblems.Numerical experiments are performed to illustrate the feasibility and efficiency of the proposed algorithm with randomly generated data.

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