A Douglas-Rachford splitting method for solving equilibrium problems

Nonlinear Analysis, Theory, Methods and Applications

Volumn 75, Issue 16, 2012, Pages 6053-6059

  Briceño Arias, Luis M ^a  

^a UNIVERSITY OF CHILE   (Chile)

Author keywords

Douglas Rachford splitting; Equilibrium problem; Maximally monotone operator; Monotone inclusion

Indexed keywords

BIFUNCTIONS; DOUGLAS-RACHFORD SPLITTING; EQUILIBRIUM PROBLEM; MONOTONE OPERATORS; SPLITTING METHOD; STANDARD CONDITIONS;

MATHEMATICAL TECHNIQUES; NONLINEAR ANALYSIS;

PROBLEM SOLVING;

EID: 84864677657     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2012.06.014     Document Type: Article

References (37)

1. G. Allen Variational inequalities, complementarity problems, and duality theorems J. Math. Anal. Appl. 58 1977 1 10
2. M. Bianchi, and S. Schaible Generalized monotone bifunctions and equilibrium problems J. Optim. Theory Appl. 90 1996 31 43 (Pubitemid 126411261)
3. E. Blum, and W. Oettli From optimization and variational inequalities to equilibrium problems Math. Student 63 1994 123 145
4. P.L. Combettes, and S.A. Hirstoaga Equilibrium programming in Hilbert spaces J. Nonlinear Convex Anal. 6 2005 117 136
5. S.D. Flåm, and A. Antipin Equilibrium programming using proximal-like algorithms Math. Program. 78 1997 29 41 (Pubitemid 127372132)
6. A.N. Iusem, and W. Sosa New existence results for equilibrium problems Nonlinear Anal. 52 2003 621 635 (Pubitemid 35276306)
7. A.N. Iusem, and W. Sosa Iterative algorithms for equilibrium problems Optimization 52 2003 301 316 (Pubitemid 36989417)
8. I.V. Konnov Application of the proximal point method to nonmonotone equilibrium problems J. Optim. Theory Appl. 119 2003 317 333
9. A. Moudafi Mixed equilibrium problems: sensitivity analysis and algorithmic aspect Comput. Math. Appl. 44 2002 1099 1108 (Pubitemid 35250561)
10. W. Oettli A remark on vector-valued equilibria and generalized monotonicity Acta Math. Vietnam. 22 1997 215 221
11. A. Moudafi Proximal methods for a class of bilevel monotone equilibrium problems J. Global Optim. 47 2010 287 292
12. A. Moudafi Second-order differential proximal methods for equilibrium problems J. Inequal. Pure Appl. Math. 4 2003 7 (Article 18)
13. A. Moudafi, and M. Thera Proximal and dynamical approaches to equilibrium problems M. Thera, R. Tichatschke, Ill-Posed Variational Problems and Regularization Techniques Lecture Notes in Economics and Mathematical Systems vol. 477 1999 Springer-Verlag New York 187 201
14. B. Martinet Régularisation d'inéquations variationnelles par approximations successives Rev. Française Informat. Recherche Opérationnelle 4 1970 154 158
15. R.T. Rockafellar Monotone operators and the proximal point algorithm SIAM J. Control Optim. 14 1976 877 898
16. G. Kassay, S. Reich, and S. Sabach Iterative methods for solving systems of variational inequalities in reflexive Banach spaces SIAM J. Optim. 21 2011 1319 1344
17. L.M. Briceño-Arias Outer approximation method for constrained composite fixed point problems involving Lipschitz pseudo contractive operators Numer. Funct. Anal. Optim. 32 2011 1099 1115
18. L.-C. Ceng, and J.-C. Yao A hybrid iterative scheme for mixed equilibrium problems and fixed point problems J. Comput. Appl. Math. 214 2008 186 201 (Pubitemid 351220932)
19. L.-C. Ceng, and J.-C. Yao A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem Nonlinear Anal. 72 2010 1922 1937
20. I.V. Konnov, S. Schaible, and J.C. Yao Combined relaxation method for mixed equilibrium problems J. Optim. Theory Appl. 126 2005 309 322 (Pubitemid 41152024)
21. J.W. Peng Iterative algorithms for mixed equilibrium problems, strict pseudocontractions and monotone mappings J. Optim. Theory Appl. 144 2010 107 119
22. J.-W. Peng, and J.-C. Yao Two extragradient methods for generalized mixed equilibrium problems, nonexpansive mappings and monotone mappings Comput. Math. Appl. 58 2009 1287 1301
23. J.-W. Peng, and J.-C. Yao Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems Math. Comput. Modelling 49 2009 1816 1828
24. Y. Yao, Y.J. Cho, and R. Chen An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems Nonlinear Anal. 71 2009 3363 3373
25. A. Moudafi On the convergence of splitting proximal methods for equilibrium problems in Hilbert spaces J. Math. Anal. Appl. 359 2009 508 513
26. P.-L. Lions, and B. Mercier Splitting algorithms for the sum of two nonlinear operators SIAM J. Numer. Anal. 16 1979 964 979
27. B.F. Svaiter On weak convergence of the Douglas-Rachford method SIAM J. Control Optim. 49 2011 280 287
28. K. Aoyama, Y. Kimura, and W. Takahashi Maximal monotone operators and maximal monotone functions for equilibrium problems J. Convex Anal. 15 2008 395 409
29. S. Sabach Products of finitely many resolvents of maximal monotone mappings in reflexive Banach spaces SIAM J. Optim. 21 2011 1289 1308
30. H.H. Bauschke, and P.L. Combettes Convex Analysis and Monotone Operator Theory in Hilbert Spaces 2011 Springer New York
31. P.L. Combettes Solving monotone inclusions via compositions of nonexpansive averaged operators Optimization 53 2004 475 504 (Pubitemid 40190083)
32. I.V. Konnov Combined Relaxation Methods for Variational Inequalities 2001 Springer-Verlag Berlin
33. J. Douglas, and H.H. Rachford On the numerical solution of heat conduction problems in two and three space variables Trans. Amer. Math. Soc. 82 1956 421 439
34. 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 293 318
35. J. Eckstein, and B.F. Svaiter General projective splitting methods for sums of maximal monotone operators SIAM J. Control Optim. 48 2009 787 811
36. L.M. Briceño-Arias, and P.L. Combettes Monotone operator methods for Nash equilibria in non-potential games D. Bailey, H.H. Bauschke, P. Borwein, F. Garvan, M. Théra, J. Vanderwerff, H. Wolkowicz, Computational and Analytical Mathematics 2013 Springer New York (in press)
37. P.L. Combettes, and V.R. Wajs Signal recovery by proximal forward-backward splitting Multiscale Model. Simul. 4 2005 1168 1200 (Pubitemid 44932287)