The smoothing gradient method for a kind of special optimization problem

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  • 1. Business School, University of Shanghai for Science and Technology, Shanghai 200093, China; 2. College of Mathematics and Statistic, Qingdao University, Qingdao 266071, Shangdong, China

Received date: 2017-03-30

  Online published: 2017-06-15

Abstract

In this paper, we study a kind of special nonsmooth optimization problem, which is widely used in the field of compressed sensing and image processing. A smoothing gradient method is proposed and the global convergence is also given. Finally, the related numerical results indicate the efficiency of the given method.

Cite this article

CHEN Yuanyuan, GAO Yan, LIU Zhimin, DU Shouqiang . The smoothing gradient method for a kind of special optimization problem[J]. Operations Research Transactions, 2017 , 21(2) : 119 -125 . DOI: 10.15960/j.cnki.issn.1007-6093.2017.02.013

References

[1] Xiao Y H, Wang Q Y, Hu Q J. Non-smooth equations based method for l_1-norm problems with applications to compressed sensing [J]. Nonlinear Analysis, 2011, 74(11): 3570-3577.
[2] Figueiredo  M A T, Nowak R D, Wright S J. Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems [J]. IEEE Journal of Selected Topics in Signal Processing, 2007, 1(4): 586-597.
[3] Bruckstein A M, Donoho  D L,  Elad M. From sparse solutions of systems of equations to sparse modeling of signals and images [J]. SIAM Review, 2009, 51(1): 34-81.
[4] Alliney  S,  Ruzinsky S A. An algorithm for the minimization of mixed and  norms with application to Bayesian estimation [J]. IEEE Transactions on Signal Processing, 1994, 42(3): 618-627.
[5] Liu Y W, Hu J F. A neural network for l_1-l_2 minimization based on scaled gradient projection: Application to compressed sensing [J]. Neurocomputing, 2016, 173: 988-993.
[6] Donoho D L. Compressed Sensing [J]. IEEE Transactions on Information Theory, 2006, 52(4): 1289-1306.
[7] Lustig M, Donoho D, Pauly J M. Sparse MRI: The application of compressed sensing for rapid MR imaging [J]. Magnetic Resonance in Medicine, 2007, 58(6): 1182-1195.
[8] Mangasarian  O,  Meyer R. Absolute value equations [J]. Linear Algebra and its Applications, 2006, 419(2-3): 359-367.
[9] Mangasarian O. Absolute value equation solution via concave minimization [J]. Optimization Letters, 2007, 1(1): 3-8.
[10] Iqbal J, Iqbal A, Arif M. Levenberg-Marquardt method for solving systems of absolute value equations [J]. Journal of Computational Applied Mathematics, 2015, 282: 134-138.
[11] Nocedal, Wright S J. Numerical Optimization [M]. New York: Springer, 2006.
[12] Chen X. Smoothing methods for nonsmooth, nonconvex minimization [J]. Mathematical Programming, 2012, 134: 71-99.
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