A general iterative method for addressing mixed equilibrium problems and optimization problems. (English) Zbl 1205.49011
Summary: We introduce a new general iterative method for finding a common element of the set of solutions of a Mixed Equilibrium Problem (MEP), the set of fixed points of an infinite family of nonexpansive mappings \(\{T_n\}^\infty_{n=1}\) and the set of solutions of variational inequalities for a \(\xi \)-inverse-strongly monotone mapping in Hilbert spaces. Furthermore, we establish a strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under some suitable conditions, which solves some optimization problems. Our results extend and improve the recent results of Y. Yao, M. A. Noor, S. Zainab and Y.-C. Liou [J. Math. Anal. Appl. 354, No. 1, 319–329 (2009; Zbl 1160.49013), Y. Yao, M. A. Noor and Y.-C. Liou [On iterative methods for equilibrium problems, Nonlinear Anal. 70, No. 1, 479–509 (2009)] and many others.
MSC:
| 49J30 | Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) |
| 49J40 | Variational inequalities |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47J25 | Iterative procedures involving nonlinear operators |
| 49M05 | Numerical methods based on necessary conditions |
| 90C99 | Mathematical programming |
Keywords:
mixed equilibrium problem; fixed point; variational inequality; nonexpansive mapping; optimization problemsCitations:
Zbl 1160.49013References:
| [1] | Combettes, P. L., Hilbertian convex feasibility problem: convergence of projection methods, Appl. Math. Optim., 35, 311-330 (1997) · Zbl 0872.90069 |
| [2] | Deutsch, F.; Yamada, I., Minimizing certain convex functions over the intersection of the fixed point set of nonexpansive mappings, Numer. Funct. Anal. Optim., 19, 33-56 (1998) · Zbl 0913.47048 |
| [3] | Xu, H. K., An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116, 659-678 (2003) · Zbl 1043.90063 |
| [4] | Yamada, I.; Ogura, N.; Yamashita, Y.; Sakaniwa, K., Quadratic optimization of fixed point of nonexpansive mapping in Hilbert space, Numer. Funct. Anal. Optim., 19, 1-2, 165-190 (1998) · Zbl 0911.47051 |
| [5] | Yao, Y.; Noor, M. A.; Zainab, S.; Liou, Y. C., Mixed equilibrium problems and optimization problems, J. Math. Anal. Appl., 354, 319-329 (2009) · Zbl 1160.49013 |
| [6] | Noor, M. Aslam; Ottli, W., On general nonlinear complementarity problems and quasi equilibria, Le Mathematics (Catania), 49, 313-331 (1994) · Zbl 0839.90124 |
| [7] | Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math. Student, 63, 123-145 (1994) · Zbl 0888.49007 |
| [8] | Changa, S. S.; Lee, H. W.J.; Chan, C. K., A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization, Nonlinear Anal., 70, 3307-3319 (2009) · Zbl 1198.47082 |
| [9] | Gabay, D., Applications of the method of multipliers to variational inequalities, (Fortin, M.; Glowinski, R., Augmented Lagrangian Methods (1983), North-Holland: North-Holland Amsterdam, Holland), 299-331 |
| [10] | Konnov, I. V.; Schaible, S.; Yao, J. C., Combined relaxation method for mixed equilibrium problems, J. Optim. Theory Appl., 126, 309-322 (2005) · Zbl 1110.49028 |
| [11] | Stampacchia, G., Formes bilineaires coercivites sur les ensembles convexes, C. R. Acad. Sci., Paris, 258, 4413-4416 (1964) · Zbl 0124.06401 |
| [12] | Verma, R. U., General convergence analysis for two-step projection methods and application to variational problems, J. Optim. Theory Appl., 18, 11, 1286-1292 (2005) · Zbl 1099.47054 |
| [13] | Combettes, P. L.; Hirstoaga, S. A., Equilibrium programming using proximal-like algorithms, Math. Program., 78, 29-41 (1997) · Zbl 0890.90150 |
| [14] | Ceng, 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) · Zbl 1143.65049 |
| [15] | Jaiboon, C.; Poom, P.; Humphries, U. W., Convergence theorems by the viscosity approximation method for equilibrium problems and variational inequality problems, J. Comput. Math. Optim., 5, 1, 25-56 (2009) · Zbl 1194.47084 |
| [16] | Qin, X.; Shang, M.; Su, Y., A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Anal., 69, 8, 3897-3909 (2008) · Zbl 1170.47044 |
| [17] | Qin, X.; Shang, M.; Su, Y., Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems, Math. Comput. Modelling, 48, 1033-1046 (2008) · Zbl 1187.65058 |
| [18] | 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. (2007), Article ID 64363, 12 pages · Zbl 1153.54024 |
| [19] | Yao, Y. H.; Liou, Y. C.; Yao, J. C., A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems, Fixed Point Theory Appl. (2008), Article ID 417089, 15 pages · Zbl 1203.47087 |
| [20] | Yao, Y.; Noor, M. A.; Liou, Y. C., On iterative methods for equilibrium problems, Nonlinear Anal., 70, 1, 479-509 (2009) · Zbl 1165.49035 |
| [21] | Yao, J. C.; Chadli, O., Pseudomonotone complementarity problems and variational inequalities, (Crouzeix, J. P.; Haddjissas, N.; Schaible, S., Handbook of Generalized Convexity and Monotonicity (2005), Kluwer Academic), 501-558 · Zbl 1106.49020 |
| [22] | Ceng, L. C.; Yao, J. C., Iterative algorithm for generalized set-valued strong nonlinear mixed variational-like inequalities, J. Optim. Theory Appl., 124, 725-738 (2005) · Zbl 1067.49007 |
| [23] | Browder, F. E.; Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20, 197-228 (1967) · Zbl 0153.45701 |
| [24] | Liu, F.; Nashed, M. Z., Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Anal., 6, 313-344 (1998) · Zbl 0924.49009 |
| [25] | Takahashi, W., Nonlinear Functional Analysis (2000), Yokohama Publishers: Yokohama Publishers Yokohama · Zbl 0997.47002 |
| [26] | Rockafellar, R. T., On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149, 75-88 (1970) · Zbl 0222.47017 |
| [27] | Iiduka, H.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal., 61, 341-350 (2005) · Zbl 1093.47058 |
| [28] | Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73, 595-597 (1967) · Zbl 0179.19902 |
| [29] | Marino, G.; Xu, H. K., A general iterative method for nonexpansive mapping in Hilbert spaces, J. Math. Anal. Appl., 318, 43-52 (2006) · Zbl 1095.47038 |
| [30] | Yao, Y., A general iterative method for a finite family of nonexpansive mappings, Nonlinear Anal., 66, 12, 2676-2687 (2007) · Zbl 1129.47058 |
| [31] | Shimoji, K.; Takahashi, W., Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math., 5, 387-404 (2001) · Zbl 0993.47037 |
| [32] | Suzuki, T., Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305, 227-239 (2005) · Zbl 1068.47085 |
| [33] | Xu, H. K., Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298, 279-291 (2004) · Zbl 1061.47060 |
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