- Research Article
- Open Access
Hierarchical Convergence of a Double-Net Algorithm for Equilibrium Problems and Variational Inequality Problems
© Yonghong Yao et al. 2010
Received: 21 May 2010
Accepted: 22 December 2010
Published: 29 December 2010
We consider the following hierarchical equilibrium problem and variational inequality problem (abbreviated as HEVP): find a point such that , for all , where , are two monotone operators and is the solution of the equilibrium problem of finding such that , for all . We note that the problem (HEVP) includes some problems, for example, mathematical program and hierarchical minimization problems as special cases. For solving (HEVP), we propose a double-net algorithm which generates a net . We prove that the net hierarchically converges to the solution of (HEVP); that is, for each fixed , the net converges in norm, as , to a solution of the equilibrium problem, and as , the net converges in norm to the unique solution of (HEVP).
At this point, we wish to point out the link with some monotone variational inequalities and convex programming problems as follows.
a variational inequality studied by Yamada and Ogura .
a problem considered by Luo et al. .
The solution set of equilibrium problems (1.10) and (1.11) are denoted by and , respectively. The equilibrium problem (1.10) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, fixed point problems, minimax problems, Nash equilibrium problem in noncooperative games, and others. We remind the readers to refer to [14–30] and the references therein.
It is clear that the hierarchical variational inequality problem and equilibrium problem (1.12) includes the variational inequality problem studied by Yamada and Ogura , mathematical program studied by Luo et al. , hierarchical minimization problem considered by Cabot  and Solodov , as special cases.
For solving (1.12), we propose a double-net algorithm which generates a net . We prove that the net hierarchically converges to the solution of (1.12); that is, for each fixed , the net converges in norm, as , to a solution of the equilibrium problem, and as , the net converges in norm to the unique solution of (1.12).
On the equilibrium problems, we have the following important lemma. You can find it in .
Below we gather some basic facts that are needed in the argument of the subsequent sections.
Lemma 2.2 (see ).
Lemma 2.3 (demiclosedness principle for nonexpansive mappings, see ).
Let be a nonempty closed convex subset of a real Hilbert space and let be a nonexpansive mapping with . If is a sequence in weakly converging to , and if converges strongly to , then ; in particular, if , then .
which is exactly (2.5).
3. Main Results
In this section, we first introduce our double-net algorithm.
We divide our detailed proofs into several conclusions as follows. Throughout, we assume all assumptions of Theorem 3.1 are satisfied.
First, we note that the solution of the variational inequality VI (3.4) is unique.
By Conclusions 1–4, the proof of Theorem 3.1 is completed.
The work of the second author was partially supported by the Grant NSC 98-2923-E-110-003-MY3 and the work of the third author was partially supported by the Grant NSC 98-2221-E-110-064.
- Maingé P-E, Moudafi A: Strong convergence of an iterative method for hierarchical fixed-point problems. Pacific Journal of Optimization 2007,3(3):529–538.MathSciNetMATHGoogle Scholar
- Ceng LC, Petruşel A: Krasnoselski-Mann iterations for hierarchical fixed point problems for a finite family of nonself mappings in Banach spaces. Journal of Optimization Theory and Applications 2010,146(3):617–639. 10.1007/s10957-010-9679-0MathSciNetView ArticleMATHGoogle Scholar
- Moudafi A: Krasnoselski-Mann iteration for hierarchical fixed-point problems. Inverse Problems 2007,23(4):1635–1640. 10.1088/0266-5611/23/4/015MathSciNetView ArticleMATHGoogle Scholar
- Moudafi A, Maingé P-E: Towards viscosity approximations of hierarchical fixed-point problems. Fixed Point Theory and Applications 2006, 2006:-10.Google Scholar
- Yao Y, Liou Y-C: Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed point problems. Inverse Problems 2008,24(1):-8.MathSciNetView ArticleMATHGoogle Scholar
- Cianciaruso F, Marino G, Muglia L, Yao Y: On a two-step algorithm for hierarchical fixed point problems and variational inequalities. Journal of Inequalities and Applications 2009, 2009:-13.Google Scholar
- Cianciaruso F, Colao V, Muglia L, Xu H-K: On an implicit hierarchical fixed point approach to variational inequalities. Bulletin of the Australian Mathematical Society 2009,80(1):117–124. 10.1017/S0004972709000082MathSciNetView ArticleMATHGoogle Scholar
- Lu X, Xu H-K, Yin X: Hybrid methods for a class of monotone variational inequalities. Nonlinear Analysis: Theory, Methods & Applications 2009,71(3–4):1032–1041. 10.1016/j.na.2008.11.067MathSciNetView ArticleMATHGoogle Scholar
- Yao Y, Chen R, Xu H-K: Schemes for finding minimum-norm solutions of variational inequalities. Nonlinear Analysis: Theory, Methods & Applications 2010,72(7–8):3447–3456. 10.1016/j.na.2009.12.029MathSciNetView ArticleMATHGoogle Scholar
- Yamada I, Ogura N: Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numerical Functional Analysis and Optimization 2004,25(7–8):619–655.MathSciNetView ArticleMATHGoogle Scholar
- Luo Z-Q, Pang J-S, Ralph D: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge, UK; 1996:xxiv+401.View ArticleMATHGoogle Scholar
- Cabot A: Proximal point algorithm controlled by a slowly vanishing term: applications to hierarchical minimization. SIAM Journal on Optimization 2005,15(2):555–572. 10.1137/S105262340343467XMathSciNetView ArticleMATHGoogle Scholar
- Solodov M: An explicit descent method for bilevel convex optimization. Journal of Convex Analysis 2007,14(2):227–237.MathSciNetMATHGoogle Scholar
- Ceng L-C, Al-Homidan S, Ansari QH, Yao J-C: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. Journal of Computational and Applied Mathematics 2009,223(2):967–974. 10.1016/j.cam.2008.03.032MathSciNetView ArticleMATHGoogle Scholar
- Ceng L-C, Yao J-C: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. Journal of Computational and Applied Mathematics 2008,214(1):186–201. 10.1016/j.cam.2007.02.022MathSciNetView ArticleMATHGoogle Scholar
- Yao Y, Noor MA, Liou Y-C: On iterative methods for equilibrium problems. Nonlinear Analysis: Theory, Methods & Applications 2009,70(1):497–509. 10.1016/j.na.2007.12.021MathSciNetView ArticleMATHGoogle Scholar
- Peng J-W, Yao J-C: A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems. Taiwanese Journal of Mathematics 2008,12(6):1401–1432.MathSciNetMATHGoogle Scholar
- Peng J-W, Yao J-C: Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudo-contractions and monotone mappings. Taiwanese Journal of Mathematics 2009,13(5):1537–1582.MathSciNetMATHGoogle Scholar
- Ceng LC, Petruşel A, Yao JC: Iterative approaches to solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Journal of Optimization Theory and Applications 2009,143(1):37–58. 10.1007/s10957-009-9549-9MathSciNetView ArticleMATHGoogle Scholar
- Yao Y, 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 and Applications 2008, 2008:-15.Google Scholar
- Peng J-W, Liou Y-C, Yao J-C: An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocontractions. Fixed Point Theory and Applications 2009, 2009:-21.Google Scholar
- Ceng LC, Mastroeni G, Yao JC: Hybrid proximal-point methods for common solutions of equilibrium problems and zeros of maximal monotone operators. Journal of Optimization Theory and Applications 2009,142(3):431–449. 10.1007/s10957-009-9538-zMathSciNetView ArticleMATHGoogle Scholar
- 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 and Applications 2007, 2007:-12.Google Scholar
- Yao Y, Noor MA, Zainab S, Liou Y-C: Mixed equilibrium problems and optimization problems. Journal of Mathematical Analysis and Applications 2009,354(1):319–329. 10.1016/j.jmaa.2008.12.055MathSciNetView ArticleMATHGoogle Scholar
- Yao Y, Cho YJ, Chen R: An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems. Nonlinear Analysis: Theory, Methods & Applications 2009,71(7–8):3363–3373. 10.1016/j.na.2009.01.236MathSciNetView ArticleMATHGoogle Scholar
- Marino G, Colao V, Muglia L, Yao Y: Krasnoselski-Mann iteration for hierarchical fixed points and equilibrium problem. Bulletin of the Australian Mathematical Society 2009,79(2):187–200. 10.1017/S000497270800107XMathSciNetView ArticleMATHGoogle Scholar
- Ceng L-C, Yao J-C: A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem. Nonlinear Analysis: Theory, Methods & Applications 2010,72(3–4):1922–1937. 10.1016/j.na.2009.09.033MathSciNetView ArticleMATHGoogle Scholar
- Zeng LC, Lin YC, Yao JC: Iterative schemes for generalized equilibrium problem and two maximal monotone operators. Journal of Inequalities and Applications 2009, 2009:-34.Google Scholar
- Honda T, Takahashi W, Yao J-C: Nonexpansive retractions onto closed convex cones in Banach spaces. Taiwanese Journal of Mathematics 2010,14(3B):1023–1046.MathSciNetMATHGoogle Scholar
- Zeng L-C, Ansari QH, Shyu DS, Yao J-C: Strong and weak convergence theorems for common solutions of generalized equilibrium problems and zeros of maximal monotone operators. Fixed Point Theory and Applications 2010, 2010:-33.Google Scholar
- Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 2005,6(1):117–136.MathSciNetMATHGoogle 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
- Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge, UK; 1990:viii+244.View ArticleMATHGoogle Scholar
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