计算数学
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计算数学  2018, Vol. 40 Issue (4): 387-401    DOI:
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核范数和谱范数下广义Sylvester方程最小二乘问题的一类改进算法
蔡文银, 徐玲玲
南京师范大学数学科学学院, 南京 210023
AN IMPROVED ALGORITHM FOR LEAST SQUARES PROBLEM OF GENERALIZED SYLVESTER EQUATION UNDER NUCLEAR NORM AND SPECTRAL NORM
Cai Wenyin, Xu Lingling
Nanjing Normal University, School of Mathematical Sciences, Nanjing 210023, China
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摘要 在文献[10]中,作者从数值角度讨论核范数和谱范数下的广义Sylvester方程约束最小二乘问题
minXS||Σi=1NAiXBi-C||
的算法,其中S为闭凸集合.采用的数值算法是非精确交替方向法,并结合阈值算法、Moreau-Yosida正则化算法、谱投影算法、LSQR,SPG等算法求解相应子问题.本文在文献[10]的基础上,通过引入新变量,应用交替方向法简化子问题的求解,其中每个子问题都可以精确求解,更重要的是每个变量都具有显式的表达式.在理论方面我们证明了算法的收敛性,数值试验表明改进后的算法不管是在时间上还是在迭代步上,运行的结果得到很大的改善.
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关键词广义Sylvester方程   谱范数   核范数   交替方向法   矩阵的向量化     
Abstract: In the paper[10], the authors discussed the numerical method solving generalized Sylvester equation least square problems with the nuclear norm and spectral norm:
minXS||Σi=1NAiXBi-C||
,where XS is a closed convex set. They used inexact alternating direction method in combination with threshold algorithm, Moreau - Yosida regularization algorithm, spectrum projection algorithm,LSQR algorithm and SPG algorithm. Based on[10], we introduce a new variable and use the alternating direction method to simplify the algorithm. Each subproblem can be solved exactly. More important, each variable has its own explicit solution expression. We prove the convergence of the proposed algorithm. The numerical tests show that the improved algorithm can be improved greatly in both time and iteration.
Key wordsgeneralized Sylvester equation   spectral norm   nuclear norm   alternating direction method   matrix vector quantization   
收稿日期: 2018-03-07;
基金资助:

国家自然科学基金(No:11401314).

通讯作者: 徐玲玲,Email:xulingling@njnu.edu.cn     E-mail: xulingling@njnu.edu.cn
引用本文:   
. 核范数和谱范数下广义Sylvester方程最小二乘问题的一类改进算法[J]. 计算数学, 2018, 40(4): 387-401.
. AN IMPROVED ALGORITHM FOR LEAST SQUARES PROBLEM OF GENERALIZED SYLVESTER EQUATION UNDER NUCLEAR NORM AND SPECTRAL NORM[J]. Mathematica Numerica Sinica, 2018, 40(4): 387-401.
 
[1] Lei Y, Liao A P. A minimal residual algorithm for the inconsistent matrix equation AXB=C over symmetrices[J]. Applied Mathematricx and Computation, 2007, 188:499-513.
[2] Michael NG K, Wang F, Yuan X M. Inexact alternating direction methods for image recovery[J]. SIAM Journal on Scientific Computing, 2011, 33:1643-1668.
[3] Peng Y X, Hu X Y, Zhang L. An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C[J]. Applied Mathematics and Computation, 2005, 160:763-777.
[4] Yang J F. Yuan X M. Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization[J]. Mathematrics of Computation, 2012, 82:301-329.
[5] Li J F, Hu X Y, Zhang L. Dykstra's algorithm for constrained least-squares doubly symmetric matrix problems[J]. Theoretical Computer Science, 2010, 411:2818-2826.
[6] Han D R, Yuan X M. Local linear convergence of the alternating direction method of multipliers for quadratic programs[J]. SIAM Journal on Numerical Analysis, 2013, 51:3446-3457.
[7] Li Q N. Alternating direction method for a class of constrained matrix approximation problems[J]. Pacific Journal of Optimization, 2012, 8:765-778.
[8] Brigin E G, Martinezand J M, Raydan M. Inexact Spectral Projected Gradient methods on convex sets[J]. SIMA Journal on Numerical Analysis, 2003, 23:539-559.
[9] Ding F, Chen T W. On iterative solutions of general coupled matrix equations[J]. SIAM Journal on Control and Optimization, 2006, 44:2269-2284.
[10] 李姣芬, 宋丹丹, 李涛, 等. 核范数和谱范数下广义Sylvester方程最小二乘问题的有效算法[J]. 计算数学, 2017, 39(2):129-150. 浏览
[11] Bouhamidi A, Jbilou K, Raydan M. Convex constrained optimization for large-scale generalized Sylvester equations[J]. Computational Optimization and Applications, 2011, 48:233-253.
[12] Boyd S, Parikh N, Chu E, Peleato B, Eckstein J. Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers[J]. Foundations and Trends in Machine Learning[J]. 2011, 3:1-122.
[13] He B S, Tao M and Yuan X M. Alternating direction method with Gaussian back substitution for separable convex programming[J]. SIAM Journal Optimization[J]. 2012, 22:313-340.
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