# A proximal splitting method for separable convex programming and its application to compressive sensing

Volume 9, Issue 2, pp 392--403 Publication Date: February 21, 2016
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### Authors

Hongchun Sun - School of Sciences, Linyi University, Shandong, 276005, P. R. China. Min Sun - School of Mathematics and Statistics, Zaozhuang University, Shandong, 277160, P. R. China. Houchun Zhou - School of Sciences, Linyi University, Shandong, 276005, P. R. China.

### Abstract

Recently, by taking full exploitation to the special structure of the separable convex programming, some splitting methods have been developed. However, in some practical applications, these methods need to compute the inverse of a matrix, which maybe slow down their convergence rate, especially when the dimension of the matrix is large. To solve this issue, in this paper we shall study the Peaceman-Rachford splitting method (PRSM) by adding a proximal term to its first subproblem and get a new method named proximal Peaceman-Rachford splitting method (PPRSM). Under mild conditions, the global convergence of the PPRSM is established. Finally, the effeciency of the PPRSM is illustrated by testing some applications arising in compressive sensing.

### Keywords

• Proximal Peaceman-Rachford splitting method
• separable convex programming
• compressive sensing.

•  90C25
•  90C30

### References

• [1] D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods , Academic Press, New York-London (1982)

• [2] S. Cao, Y. Xiao, H. Zhu, Linearized alternating directions method for l1-norm inequality constrained $l_1$-norm minimization, Appl. Numer. Math., 85 (2014), 142-153.

• [3] D. R. Han, X. M. Yuan, Convergence analysis of the Peaceman-Rachford splitting method for nonsmooth convex optimization, J. Optim. Theory Appl., (Under-revision)

• [4] B. He, H. Liu, Z. Wang, X. Yuan , A strictly contractive Peaceman-Rachford splitting method for convex programming , SIAM J. Optim., 24 (2014), 1011-1040.

• [5] M. Li, X. Yuan , A strictly contractive Peaceman-Rachford splitting method with logarithmic-quadratic proximal regularization for convex programming, Math. Oper. Res., 2015 , (2015)

• [6] P. L. Lions, B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Num. Anal., 16 (1979), 964-979.

• [7] D. H. Peaceman, H. H. Rachford, The numerical solution of parabolic elliptic differential equations, SIAM J. Appl. Math., 3 (1955), 28-41.

• [8] M. Sun, J. Liu, A proximal Peaceman-Rachford splitting method for compressive sensing, J. Appl. Math. Comput., (2015), 1-15.