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Operations Research Transactions >

2017 , Vol. 21 >Issue 2: 57 - 65

DOI: https://doi.org/10.15960/j.cnki.issn.1007-6093.2017.02.007

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

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

Received date: 2017-03-24

  Online published: 2017-06-15

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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

Cite this article

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, 2017 , 21(2) : 57 -65 . DOI: 10.15960/j.cnki.issn.1007-6093.2017.02.007

References

[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.
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