Abstract
The purpose of this paper is to construct two superimposed optimization methods for solving the mixed equilibrium problem and variational inclusion. We show that the proposed superimposed methods converge strongly to a solution of some optimization problem. Note that our methods do not involve any projection.
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Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
Ceng L.C., Al-Homidan S., Ansari Q.H., Yao J.C.: An iterative scheme for equilibrium problems and fixed point problems of strict pseudocontraction mappings. J. Comput. Appl. Math. 223, 967–974 (2009)
Cianciaruso F., Marino G., Muglia L., Yao Y.: A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem. Fixed Point Theory Appl. 2010(383740), 19 (2010)
Chadli O., Wong N.C., Yao J.C.: Equilibrium problems with applications to eigenvalue problems. J. Optim. Theory Appl. 117, 245–266 (2003)
Colao V., Marino G.: Strong convergence for a minimization problem on points of equilibrium and common fixed points of an infinite family of nonexpansive mappings. Nonlinear Anal. 73, 3513–3524 (2010)
Combettes P.L., Hirstoaga S.A.: Equilibrium programming using proximal-like algorithms. Math. Program. 78, 29–41 (1997)
Combettes P.L., Hirstoaga A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)
Giannessi F., Maugeri A., Pardalos P.M.: Equilibrium Problems and Variational Models. Kluwer, Dordrecht (2001)
Giannessi F., Pardalos P.M., Rapcsak T.: New Trends in Equilibrium Systems. Kluwer, Dordrecht (2001)
Gilbert R.P., Panagiotopoulos P.D., Pardalos P.M.: From Convexity to Nonconvexity. Kluwer, Dordrecht (2001)
Konnov I.V., Schaible S., Yao J.C.: Combined relaxation method for mixed equilibrium problems. J. Optim. Theory Appl. 126, 309–322 (2005)
Moudafi A.: Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9, 37–43 (2008)
Moudafi, A., Théra, M.: Proximal and dynamical approaches to equilibrium problems. In: Lecture Notes in Economics and Mathematical Systems, vol. 477, Springer, pp. 187–201 (1999)
Nadezhkina N., Takahashi W.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128, 191–201 (2006)
Noor M.A., Oettli W.: On general nonlinear complementarity problems and quasi equilibria. Mathematiche (Catania) 49, 313–331 (1994)
Peng J.W., Wang Y., Shyu D.S., Yao J.C.: Common solutions of an iterative scheme for variational inclusions, equilibrium problems and fixed point problems. J. Inequal. Appl. 2008(720371), 15 (2008)
Peng J.W., Yao J.C.: A new hybrid-extragradient method for generalized mixed equilibrium problems and fixed point problems and variational inequality problems. Taiwan. J. Math. 12, 1401–1433 (2008)
Qin X., Cho Y.J., Kang S.M.: Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 72, 99–112 (2010)
Robinson S.M.: Generalized equation and their solutions, part I: basic theory. Math Program. Study 10, 128–141 (1979)
Rockafella R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Saewan S., Kumam P.: A modified hybrid projection method for solving generalized mixed equilibrium problems and fixed point problems in Banach spaces. Comput. Math. Appl. 62, 1723–1735 (2011)
Shehu Y.: Strong convergence theorems for nonlinear mappings, variational inequality problems and system of generalized mixed equilibrium problems. Math. Comput. Model. 54, 2259–2276 (2011)
Takahashi S., Takahashi W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506–515 (2007)
Takahashi S., Takahashi W.: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 69, 1025–1033 (2008)
Takahashi S., Takahashi W., Toyoda M.: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 147, 27–41 (2010)
Xu H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)
Yao Y., Cho Y.J., Chen R.: An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems. Nonlinear Anal. 71, 3363–3373 (2009)
Yao Y., Cho Y.J., Liou Y.C.: Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems. Eur. J. Oper. Res. 212, 242–250 (2011)
Yao, Y., Liou, Y.C.: Composite algorithms for minimization over the solutions of equilibrium problems and fixed point problems. Abstr. Appl. Anal. Article ID 763506, 19 pp. (2010). doi:10.1155/2010/763506.
Yao, Y., Liou, Y.C., Yao, J.C.: Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory Appl. Article ID 64363, 12 pp. (2007)
Yao Y., Liou Y.C., Yao J.C.: New relaxed hybrid extragradient method for fixed point problems, a general system of variational inequality problems and generalized mixed equilibrium problems. Optimization 60(3), 395–412 (2011)
Yao Y., Noor M.A., Liou Y.C.: On iterative methods for equilibrium problems. Nonlinear Anal. 70, 497–507 (2009)
Yao Y., Noor M.A., Liou Y.C., Kang S.M.: Some new algorithms for solving mixed equilibrium problems. Comput. Math. Appl. 60, 1351–1359 (2010)
Yao Y., Noor M.A., Zainab S., Liou Y.C.: Mixed equilibrium problems and optimization problems. J. Math. Anal. Appl. 354, 319–329 (2009)
Zeng L.C., Yao J.C.: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 214, 186–201 (2008)
Zhang S.S., Lee J.H.W., Chan C.K.: Algorithms of common solutions for quasi variational inclusion and fixed point problems. Appl. Math. Mech. Engl. Ed. 29, 571–581 (2008)
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Yao, Y., Liou, YC. & Wong, NC. Superimposed optimization methods for the mixed equilibrium problem and variational inclusion. J Glob Optim 57, 935–950 (2013). https://doi.org/10.1007/s10898-012-9982-4
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DOI: https://doi.org/10.1007/s10898-012-9982-4
Keywords
- Mixed equilibrium problem
- Variational inclusion
- Monotone operator
- Superimposed optimization method
- Resolvent