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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 396753, 9 pages
http://dx.doi.org/10.1155/2014/396753
Research Article

An Implementable First-Order Primal-Dual Algorithm for Structured Convex Optimization

College of Communications Engineering, PLA University of Science and Technology, Nanjing 210007, China

Received 2 December 2013; Accepted 17 February 2014; Published 30 March 2014

Academic Editor: Guanglu Zhou

Copyright © 2014 Feng Ma 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. S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning, vol. 3, no. 1, pp. 1–122, 2010. View at Publisher · View at Google Scholar · View at Scopus
  2. E. J. Candès, X. Li, Y. Ma, and J. Wright, “Robust principal component analysis?” Journal of the ACM, vol. 58, no. 3, Article 11, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  3. T. Goldstein and S. Osher, “The split Bregman method for L1-regularized problems,” SIAM Journal on Imaging Sciences, vol. 2, no. 2, pp. 323–343, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  4. B. He, M. Xu, and X. Yuan, “Solving large-scale least squares semidefinite programming by alternating direction methods,” SIAM Journal on Matrix Analysis and Applications, vol. 32, no. 1, pp. 136–152, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. L. Xue, S. Ma, and H. Zou, “Positive-definite 1-penalized estimation of large covariance matrices,” Journal of the American Statistical Association, vol. 107, no. 500, pp. 1480–1491, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  6. S. Ma, L. Xue, and H. Zou, “Alternating direction methods for latent variable Gaussian graphical model selection,” Neural Computation, vol. 25, no. 8, pp. 2172–2198, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  7. Z. Wen, D. Goldfarb, and W. Yin, “Alternating direction augmented Lagrangian methods for semidefinite programming,” Mathematical Programming Computation, vol. 2, no. 3-4, pp. 203–230, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J. Yang and Y. Zhang, “Alternating direction algorithms for 1-problems in compressive sensing,” SIAM Journal on Scientific Computing, vol. 33, no. 1, pp. 250–278, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  9. J. Yang, Y. Zhang, and W. Yin, “A fast alternating direction method for TVL1-L2 signal reconstruction from partial fourier data,” IEEE Journal of Selected Topics in Signal Processing, vol. 4, no. 2, pp. 288–297, 2010. View at Publisher · View at Google Scholar
  10. R. T. Rockafellar, “Augmented Lagrangians and applications of the proximal point algorithm in convex programming,” Mathematics of Operations Research, vol. 1, no. 2, pp. 97–116, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, vol. 9, Society for Industrial and Applied Mathematics, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  12. 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. 1–3, pp. 293–318, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. B. He and X. Yuan, “On the O(1/n) convergence rate of the Douglas-Rachford alternating direction method,” SIAM Journal on Numerical Analysis, vol. 50, no. 2, pp. 700–709, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. Hong and Z. Q. Luo, “On the linear convergence of the alternating direction method of multipliers,” In press, http://arxiv.org/abs/1208.3922.
  15. W. Deng and W. Yin, “On the global and linear convergence of the generalized alternating direction method of multipliers,” Preprint. In press, http://www.optimization-online.org/DB_FILE/2012/08/3578.pdf.
  16. D. Han and X. Yuan, “A note on the alternating direction method of multipliers,” Journal of Optimization Theory and Applications, vol. 155, no. 1, pp. 227–238, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. C. Chen, B. He, Y. Ye, and X. Yuan, “The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent,” Preprint. In press, http://www.optimization-online.org/DB_FILE/2013/09/4059.pdf.
  18. B. He and X. Yuan, “The unified framework of some proximal-based decomposition methods for monotone variational inequalities with separable structures,” Pacific Journal of Optimization, vol. 8, no. 4, pp. 817–844, 2012.
  19. S. Ma, “Alternating proximal gradient method for convex minimization.,” Preprint. In press, http://www.optimization-online.org/DB_FILE/2012/09/3608.pdf.
  20. B. Martinet, “Régularisation d'inéquations variationnelles par approximations successives,” Revue Française d'Informatique et de Recherche Opérationnelle, vol. 4, no. 3, pp. 154–158, 1970. View at Zentralblatt MATH · View at MathSciNet
  21. B. He, X. Yuan, and W. Zhang, “A customized proximal point algorithm for convex minimization with linear constraints,” Computational Optimization and Applications, vol. 56, no. 3, pp. 559–572, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. X. Cai, G. Gu, B. He, and X. Yuan, “A relaxed customized proximal point algorithm for separable convex programming,” Preprint. In press, http://www.optimization-online.org/DB_FILE/2011/08/3141.pdf.
  23. G. Gu, B. He, and X. Yuan, “Customized proximal point algorithms for linearly constrained convex minimization and saddle-point problems: a unified approach,” Computational Optimization and Applications, 2013. View at Publisher · View at Google Scholar
  24. X. Cai, G. Gu, B. He, and X. Yuan, “A proximal point algorithm revisit on the alternating direction method of multipliers,” Science China Mathematics, vol. 56, no. 10, pp. 2179–2186, 2013. View at Publisher · View at Google Scholar
  25. J. Yang and X. Yuan, “Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization,” Mathematics of Computation, vol. 82, no. 281, pp. 301–329, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. N. Parikh and S. Boyd, Proximal Algorithms. Foundations and Trends in Optimization, 2013.