Abstract
In this paper, we propose two new extragradient-proximal algorithms for solving split equilibrium and fixed point problems (SEFPP) in real Hilbert spaces, in which the first equilibrium bifunction is pseudomonotone, the second one is monotone, and the fixed point mappings are nonexpansive. By using the extragradient method incorporated with the proximal point algorithm and cutting techniques, we obtain algorithms for solving (SEFPP). Under certain conditions on parameters, the iteration sequences generated by the proposed algorithms are proved to be weakly and strongly convergent to a solution of (SEFPP). Our results improve and extend the previous results given in the literature.
Similar content being viewed by others
References
Anh, P.N.: A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization 62, 271–283 (2013)
Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)
Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Existence and solution methods for equilibria. Eur. J. Oper. Res. 227, 1–11 (2013)
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)
Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)
Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21, 2071–2084 (2005)
Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algor. 59, 301–323 (2012)
Censor, Y., Segal, A.: The split common fixed point problem for directed operators. J. Convex Anal. 16, 587–600 (2009)
Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algor. 8, 221–239 (1994)
Combettes, P.L., Hirstoaga, A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)
Deepho, J., Kumam, W., Kumam, P.: A new hybrid projection algorithm for solving the split generalized equilibrium problems and the system of variational inequality problems. J. Math. Model. Algor. 13, 405–423 (2014)
Dinh, B.V., Muu, L.D.: A projection algorithm for solving pseudomonotone equilibrium problems and its application to a class of bilevel equilibria. Optimization 64, 559–575 (2015)
Dinh, B.V., Muu, L.D.: On penalty and gap function methods for bilevel equilibrium problems. J. Appl. Math. 2011, 646452 (2011)
Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
He, Z.: The split equilibrium problem and its convergence algorithms. J. Inequal. Appl. 2012, 162 (2012)
Hieu, D.V., Muu, L.D., Anh, P.K.: Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer. Algor. 73, 197–217 (2016)
Iiduka, H., Yamada, I.: A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping. SIAM J. Optim. 19, 1881–1893 (2009)
Iiduka, H., Yamada, I.: A subgradient-type method for the equilibrium problem over the fixed point set and its applications. Optimization 58, 251–261 (2009)
Khatibzadeh, H., Mohebbi, V., Ranjbar, S.: Convergence analysis of the proximal point algorithm for pseudo-monotone equilibrium problems. Optim. Methods Softw. 30, 1146–1163 (2015)
Kraikaew, R., Saejung, S.: On split common fixed point problems. J. Math. Anal. Appl. 415, 513–524 (2014)
Korpelevich, G.M.: An extragradient method for finding saddle points and other problems. Matecon 12, 747–756 (1976)
Maingé, P.-E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47, 1499–1515 (2008)
Maingé, P. -E.: Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints. Eur. J. Oper. Res. 205, 501–506 (2010)
Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)
Mastroeni, G.: On auxiliary principle for equilibrium problems. In: Daniele, P., Giannessi, F., Maugeri, A (eds.) Equilibrium Problems and Variational Models, pp. 289–298. Kluwer Academic Publishers, Dordrecht (2003)
Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150, 275–283 (2011)
Moudafi, A.: The split common fixed-point problem for demicontractive mappings. Inverse Probl. 26, 055007 (2010)
Moudafi, A.: Proximal point algorithm extended to equilibrium problems. J. Nat. Geom. 15, 91–100 (1999)
Muu, L.D., Quoc, T.D.: Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash-Cournot equilibrium model. J. Optim. Theory Appl. 142, 185–204 (2009)
Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. TMA 18, 1159–1166 (1992)
Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)
Tada, A., Takahashi, W.: Weak and strong convergence theorems for nonexpansive mapping and equilibrium problem. J. Optim. Theory Appl. 133, 359–370 (2007)
Takahashi, W., Takeuchi, Y., Kubota, R.: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 341, 276–286 (2008)
Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert space. J. Math. Anal. Appl. 331, 506–515 (2007)
Tam, N.N., Yao, J.C., Yen, N.D.: Solution methods for pseudomonotone variational inequalities. J. Optim. Theory Appl. 138, 253–273 (2008)
Tran, D.Q., Muu, L.D., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)
Acknowledgments
The authors would like to thank the referees very much for their constructive comments and suggestions, especially on the presenting and the structure of the early version of their paper which helped them very much in revising the paper. Their thanks would be addressed to Prof. Le Dung Muu and Prof. Pham Ky Anh for the guidance and discussion. The first author is supported in part by NAFOSTED, under the project 101.01-2014-24, and a grant from Le Quy Don Technical University.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Van Dinh, B., Son, D.X. & Anh, T.V. Extragradient-Proximal Methods for Split Equilibrium and Fixed Point Problems in Hilbert Spaces. Vietnam J. Math. 45, 651–668 (2017). https://doi.org/10.1007/s10013-016-0237-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10013-016-0237-4
Keywords
- Split equilibrium problem
- Split fixed point problem
- Nonexpansive mapping
- Weak and strong convergence
- Pseudomonotonicity