线性方程组l_1范数问题的松弛投影算法及其应用

doi:10.15960/j.cnki.issn.1007-6093.2017.02.007

运筹学学报

线性方程组l_1范数问题的松弛投影算法及其应用

屈彪^1,* 张文伟¹ 于丽超¹

1. 曲阜师范大学管理学院、运筹学研究所, 山东日照 276826

收稿日期:2017-03-24 出版日期:2017-06-15 发布日期:2017-06-15
通讯作者: 屈彪 E-mail: qubiao001@163.com
基金资助:
国家自然科学基金(No. 11271226)

The relaxed projection methods for solving the l_1-norm problem of linear equations and their applications

QU Biao^1,*ZHANG Wenwei¹YU Lichao¹

1. Institute of Operations Research, School of Management, Qufu Normal University, Rizhao 276826, Shandong, China

Received:2017-03-24 Online:2017-06-15 Published:2017-06-15

摘要/Abstract

摘要：

考虑线性方程组l_1范数问题的求解, 在分别将其转化为一个分裂可行问题和凸可行问题的基础上, 设计了几种松弛投影算法, 然后将所设计的求解方法用于信号处理问题的求解上.

关键词: 线性方程组, l_1范数问题, 松弛投影算法

Abstract:

This paper discusses of the methods for solving the l_1-norm problem of linear equations. First, the problem is translated into a split feasibility problem and a convex feasibility problem, respectively. Then，some relaxed projection algorithms are presented. Finally, the new algorithms are applied to solve some signal processing problems.

Key words: linear equations, l_1-norm problem, relaxed projection algorithm

屈彪, 张文伟, 于丽超. 线性方程组l_1范数问题的松弛投影算法及其应用[J]. 运筹学学报, doi: 10.15960/j.cnki.issn.1007-6093.2017.02.007.

QU Biao, ZHANG Wenwei, YU Lichao. The relaxed projection methods for solving the l_1-norm problem of linear equations and their applications[J]. Operations Research Transactions, doi: 10.15960/j.cnki.issn.1007-6093.2017.02.007.

参考文献

[1] Candes E, Romberg J, Tao T. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information [J]. IEEE Transactions on Information Theory, 2006, 52: 489-509.

[2] Shen S S, Donoho D L, Saunders M A. Atomic decompositon by basis pursuit [J]. SIAM Rev, 2001, 43: 129–159.

[3] Carmi A, Censor Y, Gurfil P. Convex feasibility modeling and projection methods for sparse signal recovery [J]. Journal of Computational and Applied Mathematics, 2012, 236: 4318–4335.

[4] Gafni E M, Bertsekas D P. Two-metric projection problems and descent methods for asymmetric variational inequality problems [J]. Math Programming, 1984, 53: 99-110.

[5] He B S. A class of projection and contraction methods for monotone variational inequalities [J]. Applied Mathematics & Optimization, 1997, 35: 69-76.

[6] Solodov M V, Tseng P. Modified projection-type methods for monotone variational inequalities [J]. SIAM J Control and Optimization, 1996, 34(5): 1814-1830.

[7] Xiu N H, Zhang J Z. Some recent advances in projection-type methods for variational inequalities [J]. Journal of Computational and Applied Mathematics, 2003, 152: 559-585.

[8] Fukushima M. A relaxed projection method for variational inequalities [J]. Math Programming, 1986, 35: 58-70.

[9] Qu B, Xiu N H. A note on the CQ algorithm for the split feasibility problem [J]. Inverse Problems, 2005, 21: 1655-1665.

[10] Yang Q Z. The relaxed CQ algorithm solving the split feasibility problem [J]. Inverse Problems, 2004, 20: 1261-1266.

[11] Zarantonello E H. Projections on convex sets in Hilbert space and spectral theory [M]\\Contributions to Nonlinear Functional Analysis, New York: Academic Press, 1971.

[12] Boyd S, Vandenberghe L. Convex optimization [M]. New York: Cambridge University Press, 2009.

[13] Bertsekas D P, Nedi\acute{c} A, Ozdaglar A E. Convex Analysis and Optimization [M]. Nashua: Athena Scientific, 2003.

[14] Rockafellar R T. Convex Analysis [M]. Princeton: Princeton University Press, 1970.

[15] Bauschke H H, Borwein J M. On projection algorithms for solving convex feasibility problems [J]. SIAM Review, 1996, 38: 367-426.

[16] Censor Y, Elfving T. A multiprojection algorithm using Bregman projections in a product space [J]. Numerical Algorithms, 1994, 8: 221-239.

[17] Censor Y, Elfving T, Kopf N, et al. The mltiple-sets split feasibility problem and its applications for inverse problems [J]. Inverse Problems, 2005, 21: 2071-2084.

[18] Liu B H, Qu B, Zheng N. A successive projection algorithm for solving the multiple-sets split feasibility problem [J]. Numerical Functional Analysis and Optimization, 2014, 35: 1459-1466.

[19] Qu B, Liu B H, Zheng N. On the computation of the step-size for the CQ-like algorithms for the split feasibility problem [J]. Applied Mathematics and Computation, 2015, 262: 218–223.

[20] Yu L C, Qu B. A projection algorithm for compressive sensing [J]. Operations Research and Fuzziology, 2015, 5: 1-5.

线性方程组l_1范数问题的松弛投影算法及其应用

The relaxed projection methods for solving the l_1-norm problem of linear equations and their applications

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