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Generalized Peaceman-Rachford splitting method for multiple-block separable convex programming with applications to robust PCA

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Abstract

The Peaceman-Rachford splitting method (PRSM) is an efficient approach for two-block separable convex programming. In this paper we extend this method to the general case where the objective function consists of the sum of multiple convex functions without coupled variables, and present a generalized PRSM. Theoretically, we prove global convergence of the new method and establish the worst-case convergence rate measured by the iteration complexity in the ergodic sense for the new method. Numerically, its efficiency is illustrated by synthetic data about the robust principal component analysis (PCA) model with noisy and incomplete information.

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References

  1. Tao, M., Yuan, X.M.: Recovering low-rank and sparse components of matrices from incomplete and noisy observations. SIAM J. Optim. 21(1), 57–81 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Peng, Y.G., Ganesh, A., Wright, J., Xu, W.L., Ma, Y.: Robust alignment by sparse and low-rank decomposition for linearly correlated images, IEEE Con. Comput. Vision Patt. Recog. (CVPR), 763–770 (2010)

  3. McLACHLAN, G.: Discriminant analysis and statistical pattern recognition, Wiley-Interscience, (2004)

  4. Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximations. Comput. Math. Appl. 2, 16–40 (1976)

    Article  MATH  Google Scholar 

  5. Peaceman, D.H., Rachford, H.H.: The numerical solution of parabolic elliptic differential equations. SIAM J. Appl. Math. 3, 28–41 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Num. Analy. 16, 964–979 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  7. Han, D., Yuan, X.M.: Local linear convergence of the alternating direction method of multipliers for quadratic programs. SIAM J. Numer. Anal. 51(6), 3446–3457 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Sun, M., Liu, J.: A proximal Peaceman-Rachford splitting method for compressive sensing, J. Appl. Math. Comput., (2015) (Accepted)

  10. Li, X.X., Yuan, X.M.: A proximal strictly contractive Peaceman-Rachford splitting method for convex programming with applications to imaging. SIAM J. Imaging Sci. 8(2), 1332–1365 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Duchi, J., Singer, Y.: Efficient online and batch learning using forward backword splitting. J. Mach. Learn. Res. 10, 2899–2934 (2009)

    MathSciNet  MATH  Google Scholar 

  12. Cai, J., Chan, R., Nikolova, M.: Two-phase approach for deblurring images corrupted by impulse plus gaussian noise. Inverse Probl. Imaging 2, 187–204 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Facchinei, F., Pang, J.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Vols I and II. Springer, Berlin (2003)

  14. Hou, L.S., He, H.J., Yang, J.F.: A partially parallel splitting method for multiple-block separable convex programming with applications to robust PCA, Comput. Optim. Appl. (2015) (Accepted)

  15. Candés, E., Li, X., Ma, Y., Wright, J.: Robust principal component analysis? J. ACM 58, 1–37 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fazel, M., Goodman, J.: Approximations for partially coherent optical imaging systems. Stanford University, Tech. Rep. (1998)

  17. He, B.S., Xu, H.K., Yuan, X.M.: On the proximal jacobina decomposition of ALM for multiple-block separable convex minimization problems and its relationship to ADMM, J. Sci. Comput. (2015) (Accepted)

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Acknowledgments

The authors gratefully acknowledge the helpful comments and suggestions of the anonymous reviewers. This work is supported by the National Natural Science Foundation of China (11302188, 11171180), the foundation of Scientific Research Project of Shandong Universities (J15LI11), and the foundation of Zaozhuang University (2014YB03).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zaozhuang University, Shandong, 277160, China

    Min Sun

  2. School of Management, Qufu Normal University, Rizhao, 276826, China

    Yiju Wang

  3. School of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou, 310018, China

    Jing Liu

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  1. Min Sun
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  2. Yiju Wang
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  3. Jing Liu
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Correspondence to Min Sun.

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Sun, M., Wang, Y. & Liu, J. Generalized Peaceman-Rachford splitting method for multiple-block separable convex programming with applications to robust PCA. Calcolo 54, 77–94 (2017). https://doi.org/10.1007/s10092-016-0177-0

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  • DOI: https://doi.org/10.1007/s10092-016-0177-0

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