Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2017, Article ID 9674528, 15 pages
https://doi.org/10.1155/2017/9674528
Research Article

A Proximal Fully Parallel Splitting Method for Stable Principal Component Pursuit

1School of Mathematics and Statistics, Linyi University, Shandong 276005, China
2School of Data Sciences, Zhejiang University of Finance and Economics, Zhejiang 310018, China
3School of Mathematics and Statistics, Zaozhuang University, Shandong 277160, China
4School of Management, Qufu Normal University, Shandong 276826, China

Correspondence should be addressed to Min Sun; moc.361@uodoaixuoyiz

Received 30 March 2017; Revised 8 June 2017; Accepted 29 August 2017; Published 25 October 2017

Academic Editor: Laurent Bako

Copyright © 2017 Hongchun Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. Wright, Y. Peng, Y. Ma, A. Ganesh, and S. Rao, “Robust principal component analysis: exact recovery of corrupted low-rank matrices by convex optimization,” in Proceedings of the 23rd Annual Conference on Neural Information Processing Systems (NIPS '09), pp. 2080–2088, Vancouver, Canada, December 2009. View at Scopus
  2. E. J. Candes, X. Li, and Y. J. W. Ma, “Robust principle component analysis?” Journal of the ACM, vol. 58, pp. 1–37, 2011. View at Google Scholar
  3. V. Chandrasekaran, S. Sanghavi, P. . Parrilo, and A. S. Willsky, “Rank-sparsity incoherence for matrix decomposition,” SIAM Journal on Optimization, vol. 21, no. 2, pp. 572–596, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. Z. Zhou, X. Li, J. Wright, E. Candès, and Y. Ma, “Stable principal component pursuit,” in Proceedings of the IEEE International Symposium on Information Theory (ISIT '10), pp. 1518–1522, Austin, Tex, USA, June 2010. View at Publisher · View at Google Scholar · View at Scopus
  5. M. Tao and X. Yuan, “Recovering low-rank and sparse components of matrices from incomplete and noisy observations,” SIAM Journal on Optimization, vol. 21, no. 1, pp. 57–81, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. L. Hou, H. He, and J. Yang, “A partially parallel splitting method for multiple-block separable convex programming with applications to robust PCA,” Computational Optimization and Applications, vol. 63, no. 1, pp. 273–303, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  7. J. F. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optimization Methods and Software, vol. 11/12, no. 1-4, pp. 625–653, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  8. R. H. Tutuncu, K. C. Toh, and M. J. Todd, “Solving semidefinite-quadrtic-linear programs using SDPT3,” Math. Program, vol. 95, pp. 189–217, 2003. View at Google Scholar
  9. Y.-H. Xiao and Z.-F. Jin, “An alternating direction method for linear-constrained matrix nuclear norm minimization,” Numerical Linear Algebra with Applications, vol. 19, no. 3, pp. 541–554, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. K. Wang, J. Desai, and H. He, “A proximal partially parallel splitting method for separable convex programs,” Optimization Methods & Software, vol. 32, no. 1, pp. 39–68, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  11. D. Han, X. Yuan, W. Zhang, and X. Cai, “An ADM-based splitting method for separable convex programming,” Computational Optimization and Applications, vol. 54, no. 2, pp. 343–369, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  12. M. Sun, Y. Wang, and J. Liu, Generalized Peaceman-Rachford splitting method for multi-block separable convex programming with applications to robust PCA, Calcolo, vol. 54, no. 1, pp. 77–94, 2017.
  13. B. He, L. Hou, and X. Yuan, “On full Jacobian decomposition of the augmented Lagrangian method for separable convex programming,” SIAM Journal on Optimization, vol. 25, no. 4, pp. 2274–2312, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  14. W. Deng, M.-J. Lai, Z. Peng, and W. Yin, “Parallel multi-block ADMM with o(1/k) convergence,” Journal of Scientific Computing, vol. 71, no. 2, pp. 712–736, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  15. R. Glowinski and A. Marrocco, “Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité, d'une classe de problémes de Dirichlet non-linéaires,” Revue Française D'automatique, Informatique, Recherche Opérationnelle. Analyse Numérique, vol. 9, pp. 41–76, 1975. View at Google Scholar
  16. D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximation,” Computers & Mathematics with Applications, vol. 2, no. 1, pp. 17–40, 1976. View at Publisher · View at Google Scholar · View at Scopus
  17. M. Fortin and R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, North-Holland, Amsterdam, The Netherlands, 1983. View at MathSciNet
  18. J. Eckstein and D. P. Bertsekas, “On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators,” Mathematical Programming, vol. 55, no. 3, Ser. A, pp. 293–318, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. K. J. Arrow, L. Hurwicz, and H. Uzawa, Studies in Linear And Non-Linear Programming, Stanford University Press, Palo Alto, Calif, USA, 1958. View at MathSciNet
  20. C. Chen, B. He, Y. Ye, and X. Yuan, “The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent,” Mathematical Programming, vol. 155, no. 1-2, pp. 57–79, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  21. B. He, M. Tao, and X. Yuan, “Alternating direction method with Gaussian back substitution for separable convex programming,” SIAM Journal on Optimization, vol. 22, no. 2, pp. 313–340, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. B. He, H.-K. Xu, and X. Yuan, “On the proximal Jacobian decomposition of ALM for multiple-block separable convex minimization problems and its relationship to ADMM,” Journal of Scientific Computing, vol. 66, no. 3, pp. 1204–1217, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  23. M. Hong and Z.-Q. Luo, “On the linear convergence of the alternating direction method of multipliers,” Mathematical Programming, vol. 162, no. 1-2, Ser. A, pp. 165–199, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. B. S. He and F. Ma, Convergence study on the proximal alternating direction method with larger step size, Optimization online, 2017.
  25. M. Sun and J. Liu, “The convergence rate of the proximal alternating direction method of multipliers with indefinite proximal regularization,” Journal of Inequalities and Applications, Paper No. 19, 15 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. W. Deng and W. Yin, “On the global and linear convergence of the generalized alternating direction method of multipliers,” Journal of Scientific Computing, vol. 66, no. 3, pp. 889–916, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. K. Guo, D. Han, D. Z. Wang, and T. Wu, “Convergence of ADMM for multi-block nonconvex separable optimization models,” Frontiers of Mathematics in China, vol. 12, no. 5, pp. 1139–1162, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  28. W. Y. Tian and X. M. Yuan, “Faster alternating direction method of multipliers with O(1/n2) convergence rate,” Mathematics of Computation, 2016. View at Google Scholar
  29. M. Fukushima, Fundamentals of Nonlinear Optimization (in Japanese), Asakura Shoten, Tokyo, 2001, Chinese edition: Translated by G.H. Lin, published by Science Press, Beijing, 2011.
  30. B. He, M. Tao, and X. Yuan, “A splitting method for separable convex programming,” IMA Journal of Numerical Analysis (IMAJNA), vol. 35, no. 1, pp. 394–426, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. J.-F. Cai, E. J. Candes, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” SIAM Journal on Optimization, vol. 20, no. 4, pp. 1956–1982, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  32. A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Review, vol. 51, no. 1, pp. 34–81, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. M. H. Xu, “Proximal alternating directions method for structured variational inequalities,” Journal of Optimization Theory and Applications, vol. 134, no. 1, pp. 107–117, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus