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Computational Optimization and Applications

, Volume 63, Issue 1, pp 273–303 | Cite as

A partially parallel splitting method for multiple-block separable convex programming with applications to robust PCA

  • Liusheng Hou
  • Hongjin He
  • Junfeng Yang
Article

Abstract

We consider a multiple-block separable convex programming problem, where the objective function is the sum of m individual convex functions without overlapping variables, and the constraints are linear, aside from side constraints. Based on the combination of the classical Gauss–Seidel and the Jacobian decompositions of the augmented Lagrangian function, we propose a partially parallel splitting method, which differs from existing augmented Lagrangian based splitting methods in the sense that such an approach simplifies the iterative scheme significantly by removing the potentially expensive correction step. Furthermore, a relaxation step, whose computational cost is negligible, can be incorporated into the proposed method to improve its practical performance. Theoretically, we establish global convergence of the new method in the framework of proximal point algorithm and worst-case nonasymptotic \({\mathcal {O}}(1/t)\) convergence rate results in both ergodic and nonergodic senses, where t counts the iteration. The efficiency of the proposed method is further demonstrated through numerical results on robust PCA, i.e., factorizing from incomplete information of an unknown matrix into its low-rank and sparse components, with both synthetic and real data of extracting the background from a corrupted surveillance video.

Keywords

Augmented Lagrangian method Multiple-block Convex programming Partially parallel splitting method Proximal point algorithm 

Notes

Acknowledgments

The authors are grateful to Prof. B. S. He, Prof. X. M. Yuan and Prof. C. H. Chen for their valuable comments on the paper. The work of L. S. Hou was supported by 2013NXY43 and the Natural Science Foundation of China (NSFC-11471156). The work of H. J. He was supported by the Natural Science Foundation of China (NSFC-11301123) and the Zhejiang Provincial NSFC Grant (LZ14A010003). The work of J. F. Yang was supported by the Natural Science Foundation of China (NSFC-11371192), the Fundamental Research Funds for the Central Universities (20620140574), the Program for New Century Excellent Talents in University (NCET-12-0252), and the Key Laboratory for Numerical Simulation of Large Scale Complex Systems of Jiangsu Province.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Mathematics and Information Technology, Key Laboratory of Trust Cloud Computing and Big Data AnalysisNanjing Xiaozhuang UniversityNanjingChina
  2. 2.Department of Mathematics, School of ScienceHangzhou Dianzi UniversityHangzhouChina
  3. 3.Department of MathematicsNanjing UniversityNanjingChina

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