Some Iterative Methods for Solving Equilibrium Problems and Optimization Problems
© Huimin He et al. 2010
Received: 3 September 2010
Accepted: 29 October 2010
Published: 31 October 2010
We introduce a new iterative scheme for finding a common element of the set of solutions of the equilibrium problems, the set of solutions of variational inequality for a relaxed cocoercive mapping, and the set of fixed points of a nonexpansive mapping. The results presented in this paper extend and improve some recent results of Ceng and Yao (2008), Yao (2007), S. Takahashi and W. Takahashi (2007), Marino and Xu (2006), Iiduka and Takahashi (2005), Su et al. (2008), and many others.
which is called the variational inequality. For the recent applications, sensitivity analysis, dynamical systems, numerical methods, and physical formulations of the variational inequalities, see [1–24] and the references therein.
for approximating a common element of the set of fixed points of a nonexpansive nonself mapping and the set of solutions of the equilibrium problem and obtained a strong convergence theorem in a real Hilbert space.
and also obtained a strong convergence theorem by viscosity approximation method.
where , , and are the sequences in and is a sequence in . They proved that the sequence defined by (1.17) converges strongly to common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for α-inverse-strongly monotone mappings under some parameters controlling conditions.
where and is also the optimality condition for the minimization problem , where is a potential function for (i.e., for ). The results obtained in this paper improve and extend the recent ones announced by S. Takahashi and W. Takahashi , Iiduka and Takahashi , Marino and Xu , Chen et al. , Y. Yao and J.-C. Yao , Ceng and Yao , Su et al. , and many others.
for , is -strongly monotone. This class of maps are more general than the class of strongly monotone maps. It is easy to see that we have the following implication: -strongly monotonicity relaxed -cocoercivity.
Then is the maximal monotone and if and only if ; see .
Related to the variational inequality problem (1.2), we consider the equilibrium problem, which was introduced by Blum and Oettli  and Noor and Oettli  in 1994. To be more precise, let be a bifunction of into , where is the set of real numbers.
which is known as the equilibrium problem. The set of solutions of (2.12) is denoted by . Given a mapping , let for all . Then if and only if for all , that is, is a solution of the variational inequality. That is to say, the variational inequality problem is included by equilibrium problem, and the variational inequality problem is the special case of equilibrium problem.
where is defined as in Lemma 2.7. Once we have the solutions of the equation (2.19), then it simultaneously solves the fixed points problems, equilibrium points problems, and variational inequalities problems. Therefore, the constrained set is very important and applicable.
Lemma 2.4 (Marino and Xu ).
Lemma 2.5 (see ).
Lemma 2.6 (Blum and Oettli ).
Lemma 2.7 (Combettes and Hirstoaga ).
3. Main Results
That is, (3.47) holds.
This completes the proof.
The computational possibility of the resolvent, , of in Lemma 2.7 and Theorem 3.1 is well defined mathematically, but, in general, the computation of is very difficult in large-scale finite spaces and infinite spaces.
This work is supported by the Fundamental Research Funds for the Central Universities, no. JY10000970006 and National Science Foundation of China, no. 60974082.
- Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Transactions of the American Mathematical Society 1970, 149: 75–88. 10.1090/S0002-9947-1970-0282272-5MathSciNetView ArticleMATHGoogle Scholar
- Noor MA: New approximation schemes for general variational inequalities. Journal of Mathematical Analysis and Applications 2000, 251(1):217–229. 10.1006/jmaa.2000.7042MathSciNetView ArticleMATHGoogle Scholar
- Noor MA: Some developments in general variational inequalities. Applied Mathematics and Computation 2004, 152(1):199–277. 10.1016/S0096-3003(03)00558-7MathSciNetView ArticleMATHGoogle Scholar
- Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 2005, 6(1):117–136.MathSciNetMATHGoogle Scholar
- Flåm SD, Antipin AS: Equilibrium programming using proximal-like algorithms. Mathematical Programming 1997, 78(1):29–41.MathSciNetView ArticleMATHGoogle Scholar
- Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007, 331(1):506–515. 10.1016/j.jmaa.2006.08.036MathSciNetView ArticleMATHGoogle Scholar
- Deutsch F, Yamada I: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numerical Functional Analysis and Optimization 1998, 19(1–2):33–56.MathSciNetView ArticleMATHGoogle Scholar
- Marino G, Xu H-K: A general iterative method for nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2006, 318(1):43–52. 10.1016/j.jmaa.2005.05.028MathSciNetView ArticleMATHGoogle Scholar
- Xu H-K: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society 2002, 66(1):240–256. 10.1112/S0024610702003332MathSciNetView ArticleMATHGoogle Scholar
- Xu HK: An iterative approach to quadratic optimization. Journal of Optimization Theory and Applications 2003, 116(3):659–678. 10.1023/A:1023073621589MathSciNetView ArticleMATHGoogle Scholar
- Yamada I: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), Studies in Computational Mathematics. Volume 8. Edited by: Butnariu D, Censor Y, Reich S. North-Holland, Amsterdam, The Netherlands; 2001:473–504.Google Scholar
- Moudafi A: Viscosity approximation methods for fixed-points problems. Journal of Mathematical Analysis and Applications 2000, 241(1):46–55. 10.1006/jmaa.1999.6615MathSciNetView ArticleMATHGoogle Scholar
- Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications 2003, 118(2):417–428. 10.1023/A:1025407607560MathSciNetView ArticleMATHGoogle Scholar
- Iiduka H, Takahashi W: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Analysis: Theory, Methods & Applications 2005, 61(3):341–350. 10.1016/j.na.2003.07.023MathSciNetView ArticleMATHGoogle Scholar
- Chen J, Zhang L, Fan T: Viscosity approximation methods for nonexpansive mappings and monotone mappings. Journal of Mathematical Analysis and Applications 2007, 334(2):1450–1461. 10.1016/j.jmaa.2006.12.088MathSciNetView ArticleMATHGoogle Scholar
- Yao Y, Yao J-C: On modified iterative method for nonexpansive mappings and monotone mappings. Applied Mathematics and Computation 2007, 186(2):1551–1558. 10.1016/j.amc.2006.08.062MathSciNetView ArticleMATHGoogle Scholar
- Su Y, Shang M, Qin X: An iterative method of solution for equilibrium and optimization problems. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(8):2709–2719. 10.1016/j.na.2007.08.045MathSciNetView ArticleMATHGoogle Scholar
- Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar
- Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994, 63(1–4):123–145.MathSciNetMATHGoogle Scholar
- Noor MA, Oettli W: On general nonlinear complementarity problems and quasi-equilibria. Le Matematiche 1994, 49(2):313–331.MathSciNetMATHGoogle Scholar
- Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 2005, 305(1):227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView ArticleMATHGoogle Scholar
- Ceng L-C, Yao J-C: Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Applied Mathematics and Computation 2008, 198(2):729–741. 10.1016/j.amc.2007.09.011MathSciNetView ArticleMATHGoogle Scholar
- Noor MA, Noor KI, Yaqoob H: On general mixed variational inequalities. Acta Applicandae Mathematicae 2010, 110(1):227–246. 10.1007/s10440-008-9402-4MathSciNetView ArticleMATHGoogle Scholar
- Combettes PL: A block-iterative surrogate constraint splitting method for quadratic signal recovery. IEEE Transactions on Signal Processing 2003, 51(7):1771–1782. 10.1109/TSP.2003.812846MathSciNetView ArticleGoogle Scholar
- 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 Journal on Optimization 2008, 19(4):1881–1893.MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.