Abstract
Recently, He et al. proposed a modified Peaceman–Rachford splitting method (MPRSM) for separable convex programming, which includes compressive sensing (CS) as a special case. In this paper, we further study MPRSM for CS, and regularize its first subproblem by the proximal regularization. Thus the computational load of the subproblem is substantially alleviated. That is, it is easy enough to have a closed-form solution for CS. Convergence of the new method can be guaranteed under the same assumptions as MPRSM. Finally, numerical results, including comparisons with MPPSM are reported to demonstrate the efficiency of the new method.
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References
Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)
Peaceman, D.H., Rachford, H.H.: The numerical solution of parabolic elliptic differential equations. SIAM J. Appl. Math. 3, 28–41 (1955)
Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982)
He, B.S., Liu, H., Wang, Z.R., Yuan, X.M.: A strictly contractive Peaceman–Rachford splitting method for convex programming. SIAM J. Optim 24(3), 1011–1040 (2014)
Yang, J.F., Zhang, Y.: Alternating direction algorithms for \(l_1\)-problems in compressive sensing. SIAM J. Sci. Comput. 33(1), 250–278 (2011)
Xiao, Y.H., Song, H.N.: An inexact alternating directions algorithm for constrained total variation regularized compressive sensing problems. J. Math. Imaging Vis. 44(2), 114–127 (2012)
Cao, S.H., Xiao, Y.H., Zhu, H.: Linearized alternating directions method for \(l_1\)-norm inequality constrained \(l_1\)-norm minimization. Appl. Numer. Math. 85, 142–153 (2014)
He, B.S., Yuan, X.M.: On the O(1/n) convergence rate of Douglas–Rachford alternating direction method. SIAM J. Numer. Anal. 50, 700–709 (2012)
Yang, J.F., Yuan, X.M.: Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization. Math. Comput. 82(281), 301–329 (2013)
He, B.S., Yuan, X.M.: On the direct extension of ADMM for multi-block separable convex program- ming and beyond: from variational inequality perspective. Manuscript, http://www.optimizationonline.org/DB_HTML/2014/03/4293.html. (2014)
Han, D.R., Yuan, X.M., Zhang, W.X., Cai, X.J.: An ADM-based splitting method for separable convex programming. Comput. Optim. Appl. 54(2), 343–369 (2013)
Han, D.R., Yuan, X.M., Zhang, W.X.: An augmented Lagrangian based parallel splitting method for separable convex minimization with applications to image processing. Math. Comput. 83(289), 2263–2291 (2014)
Acknowledgments
The authors gratefully acknowledge the helpful comments and suggestions of the anonymous reviewers. This work was supported by the Foundation of Zaozhuang University Grants No. 2014YB03, Shandong Province Statistical Research Project No. 20143038, and the domestic visiting scholar project funding of Shandong Province outstanding young teachers in higher schools.
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Sun, M., Liu, J. A proximal Peaceman–Rachford splitting method for compressive sensing. J. Appl. Math. Comput. 50, 349–363 (2016). https://doi.org/10.1007/s12190-015-0874-x
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DOI: https://doi.org/10.1007/s12190-015-0874-x