A convergent 3-block semiproximal alternating direction method of multipliers for conic programming with 4-type constraints

SIAM Journal on Optimization

Volumn 25, Issue 2, 2015, Pages 882-915

  Sun, Defeng ^a   Toh, Kim Chuan ^a   Yang, Liuqin ^a  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Conic programming; Convergence; Multiblock ADMM; SDP; Semiproximal ADMM

Indexed keywords

CONSTRAINT THEORY; CONVEX OPTIMIZATION; ITERATIVE METHODS;

CONIC PROGRAMMING; CONVERGENCE; MULTI BLOCKS; SDP; SEMIPROXIMAL ADMM;

NUMERICAL METHODS;

EID: 84940394631     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/140964357     Document Type: Article

References (44)

1. J. M. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization: Theory and Examples, Springer, New York, 2006.
2. S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs, Math. Program., 120 (2009), pp. 479-495.
3. S. Burer, R. D. Monteiro, and Y. Zhang, A computational study of a gradient-based log-barrier algorithm for a class of large-scale SDPS, Math. Program., 95 (2003), pp. 359-379.
4. C. Chen, B. He, Y. Ye, and X. Yuan, The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent, Math. Program., Ser. A, to appear.
5. W. Deng and W. Yin, On the Global and Linear Convergence of the Generalized Alternating Direction Method of Multipliers, Rice CAAM Technical report 12-14, 2014.
6. J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Program., 55 (1992), pp. 293-318.
7. J. Eckstein and W. Yao, Understanding the convergence of the alternating direction method of multipliers: Theoretical and computational perspectives, Pac. J. Optim., to appear.
8. A. Eisenblätter, M. Grötschel, and A. M. Koster, Frequency planning and ramifications of coloring, Discuss. Math. Graph Theory, 22 (2002), pp. 51-88.
9. M. Fazel, T. K. Pong, D. F. Sun, and P. Tseng, Hankel matrix rank minimization with applications in system identification and realization, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 946-977.
10. M. Fortin and R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary Value Problems, Stud. Math. Appl. 15, Elsevier, Amsterdam, 1983.
11. D. Gabay, Applications of the method of multipliers to variational inequalities, in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, M. Fortin and R. Glowinski, eds., Stud. Math. Appl. 15, Elsevier, Amsterdam, 1983, pp. 299-331.
12. D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Comput. Math. Appl., 2 (1976), pp. 17-40.
13. R. Glowinski, Lectures on Numerical Methods for Nonlinear Variational Problems, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Tata Inst. Fund. Res., Bombay, 1980.
14. R. Glowinski and A. Marrocco, Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité, d'une classe de problèmes de dirichlet non linéares, Rev. Francaise Automat., Inform. Recherche Opér., 9 (1975), pp. 41-76.
15. P. Hahn and M. Anjos, QAPLIB: A Quadratic Assignment Problem Library, http://www.seas.upenn.edu/qaplib.
16. D. Han, X. Yuan, W. Zhang, and X. Cai, An ADM-based splitting method for separable convex programming, Comput. Optim. Appl., 54 (2013), pp. 343-369.
17. B. He, M. Tao, M. Xu, and X. Yuan, An alternating direction-based contraction method for linearly constrained separable convex programming problems, Optimization, 62 (2013), pp. 573-596.
18. B. He, M. Tao, and X. Yuan, Alternating direction method with gaussian back substitution for separable convex programming, SIAM J. Optim., 22 (2012), pp. 313-340.
19. B. He, M. Tao, and X. Yuan, A splitting method for separable convex programming, IMA J. Numer. Anal., 20 (2014), pp. 1-33.
20. B. He, H. Yang, and S. Wang, Alternating direction method with self-adaptive penalty parameters for monotone variational inequalities, J. Optim. Theory Appl., 106 (2000), pp. 337-356.
21. B. He and X. Yuan, Linearized alternating direction method of multipliers with Gaussian back substitution for separable convex programming, Numer. Algebra Control Optim., 3 (2013), pp. 247-260.
22. M. R. Hestenes, Multiplier and gradient methods, J. Optim. Theory Appl., 4 (1969), pp. 303-320.
23. M. Y. Hong and Z. Q. Luo, On the Linear Convergence of the Alternating Direction Method of Multipliers, preprint, arXiv:1208.3922, 2012.
24. X. D. Li, D. F. Sun, and K.-C. Toh, A Schur complement based semi-proximal ADMM for convex quadratic conic programming and extensions, Math. Program., Ser. A, to appear.
25. R. D. Monteiro, C. Ortiz, and B. F. Svaiter, A first-order block-decomposition method for solving two-easy-block structured semidefinite programs, Math. Program. Comput., 6 (2014), pp. 103-150.
26. J. J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cones mutuellement polaires, C. R. Acad. Sci., 255 (1962), pp. 238-240.
27. J. Nie and L. Wang, Regularization methods for SDP relaxations in large-scale polynomial optimization, SIAM J. Optim., 22 (2012), pp. 408-428.
28. J. Nie and L. Wang, Semidefinite Relaxations for Best Rank-1 Tensor Approximations, preprint, arXiv:1308.6562, 2013.
29. J. Peng and Y. Wei, Approximating k-means-type clustering via semidefinite programming, SIAM J. Optim., 18 (2007), pp. 186-205.
30. J. Povh and F. Rendl, Copositive and semidefinite relaxations of the quadratic assignment problem, Discrete Optim., 6 (2009), pp. 231-241.
31. J. Povh, F. Rendl, and A. Wiegele, A boundary point method to solve semidefinite programs, Computing, 78 (2006), pp. 277-286.
32. M. Powell, A method for nonlinear constraints in minimization problems, in Optimization, R. Fletcher, ed., Academic Press, New York, 1969, pp. 283-298.
33. R. T. Rockafellar, Convex Analysis, Princeton Math. Ser. 28, Princeton University Press, Princeton, NJ, 1970.
34. R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res., 1 (1976), pp. 97-116.
35. R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Grundlehren Math. Wiss. 317, Springer-Verlag, Berlin, 1998.
36. N. Sloane, Challenge Problems: Independent Sets in Graphs, http://www.research.att.com/njas/doc/graphs.html (2005).
37. M. Tao and X. Yuan, Recovering low-rank and sparse components of matrices from incomplete and noisy observations, SIAM J. Optim., 21 (2011), pp. 57-81.
38. K. C. Toh, Solving large scale semidefinite programs via an iterative solver on the augmented systems, SIAM J. Optim., 14 (2004), pp. 670-698.
39. M. Trick, V. Chvatal, B. Cook, D. Johnson, C. McGeoch, and R. Tarjan, The Second Dimacs Implementation Challenge-NP Hard Problems: Maximum Clique, Graph Coloring, and Satisfiability http://dimacs.rutgers.edu/Challenges (1992).
40. P. Tseng, Convergence of a block coordinate descent method for nondifferentiable minimization, J. Optim. Theory Appl., 109 (2001), pp. 475-494.
41. S. Wang and L. Liao, Decomposition method with a variable parameter for a class of monotone variational inequality problems, J. Optim. Theory Appl., 109 (2001), pp. 415-429.
42. X. Wang, M. Hong, S. Ma, and Z.-Q. Luo, Solving Multiple-Block Separable Convex Minimization Problems Using Two-Block Alternating Direction Method of Multipliers, preprint, arXiv:1308.5294, 2013.
43. Z. Wen, D. Goldfarb, and W. Yin, Alternating direction augmented Lagrangian methods for semidefinite programming, Math. Program. Comput., 2 (2010), pp. 203-230.
44. X. Y. Zhao, D. F. Sun, and K.-C. Toh, A Newton-CG augmented Lagrangian method for semidefinite programming, SIAM J. Optim., 20 (2010), pp. 1737-1765.