- Research
- Open access
- Published:
An explicit algorithm for solving the optimize hierarchical problems
Journal of Inequalities and Applications volume 2014, Article number: 405 (2014)
Abstract
In this paper, we consider the variational inequality problem over the generalized mixed equilibrium problem which has a hierarchical structure. Strong convergence of the algorithm to the unique solution is guaranteed under some assumptions.
MSC:47H09, 47H10, 47J20, 49J40, 65J15.
1 Introduction
Let C be a closed convex subset of a real Hilbert space H with the inner product and the norm . We denote weak convergence and strong convergence by the notations ⇀ and →, respectively. Let be a nonlinear mapping and let F be a bifunction of into ℛ, where ℛ is the set of real numbers.
Consider the generalized mixed equilibrium problem which is to find such that
The solution set of (1.1) is denoted by . See [1–4].
If , the problem (1.1) is reduced to the generalized equilibrium problem which is to find such that
The set of solutions of (1.2) is denoted by .
If and , the problem (1.1) is reduced to the equilibrium problem [5] which is to find such that
The solution set of (1.3) is denoted by .
If and , the problem (1.1) is reduced to the Hartmann-Stampacchia variational inequality [6] which is to find such that
The solution set of (1.4) is denoted by .
A mapping is called nonexpansive if for all . If C is bounded closed convex and T is a nonexpansive mapping of C into itself, then is nonempty [7]. A point is a fixed point of T provided . Denote by the set of fixed points of T; that is, .
We discuss the following variational inequality problem over the generalized mixed equilibrium problem, which is called the hierarchical problem over the generalized mixed equilibrium problem, which is to find a point such that
where A, B are two monotone operators. See [8, 9].
A mapping is called α-strongly monotone if there exists a positive real number α such that for all . A mapping is called L-Lipschitz continuous if there exists a positive real number L such that for all . A linear bounded operator A is called strongly positive on H if there exists a constant with the property for all . A mapping is called a ρ-contraction if there exists a constant such that for all .
In 2010, Yao et al. [10] considered the hierarchical problem over the generalized equilibrium problem, being defined by implicit algorithms:
for each . The net hierarchically converges to the unique solution of the problem of the variational inequality which is to find a point such that
where A, B are two monotone operators. The solution set of (1.6) is denoted by Ω. Furthermore, also solves the following variational inequality:
In 2011, Yao et al. [11] studied the hierarchical problem over the fixed point set. Let the sequence be generated by two algorithms as follows.
Implicit Algorithm: , and
Explicit Algorithm: , .
They showed that these two algorithms converge strongly to the unique solution of the problem of the variational inequality which is to find such that
where is a strongly positive linear bounded operator, is a ρ-contraction, and is a nonexpansive mapping.
In this paper, we construct an algorithm and introduce the hierarchical problem over the generalized mixed equilibrium problem. The sequence is generated by the algorithm for ,
where , and satisfy some conditions. Then converges strongly to , which is the unique solution of the variational inequality:
Our results improve the results of Yao et al. [10], Yao et al. [11] and some other authors.
2 Preliminaries
Let C be a nonempty closed convex subset of H. We have the following inequality in an inner product space: , . For every point , there exists a unique nearest point in C, denoted by , such that
is called the metric projection of H onto C. It is well known that is a nonexpansive mapping of H onto C and satisfies
for every . Moreover, is characterized by the following properties: and
for all , . Let B be a monotone mapping of C into H. In the context of the variational inequality problem the characterization of projection (2.1) implies the following:
It is also well known that H satisfies the Opial condition [12], i.e., for any sequence with , the inequality , holds for every with .
For solving the generalized mixed equilibrium problem and the mixed equilibrium problem, let us give the following assumptions for the bifunction F, φ, and the set C:
-
(A1) for all ;
-
(A2) F is monotone, i.e., for all ;
-
(A3) for each , is weakly upper semicontinuous;
-
(A4) for each , is convex;
-
(A5) for each , is lower semicontinuous;
-
(B1) for each and , there exist a bounded subset and such that for any ,
(2.2) -
(B2) C is a bounded set.
Lemma 2.1 [13]
Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from to ℛ satisfying (A1)-(A5) and let be a proper lower semicontinuous and convex function. For and , define a mapping as follows.
for all . Assume that either (B1) or (B2) holds. Then the following results hold:
-
(1)
for each , ;
-
(2)
is single-valued;
-
(3)
is firmly nonexpansive, i.e., for any , ;
-
(4)
;
-
(5)
is closed and convex.
Lemma 2.2 [14]
Let C be a closed convex subset of a real Hilbert space H and let be a nonexpansive mapping. Then is demiclosed at zero, that is, , implies .
Lemma 2.3 [15]
Assume A is a self adjoint and strongly positive linear bounded operator on a Hilbert space H with coefficient and , then .
Lemma 2.4 [16]
Assume is a sequence of nonnegative real numbers such that
where and are sequences in ℛ such that
-
(i)
,
-
(ii)
or .
Then .
3 Strong convergence theorems
In this section, we introduce an explicit algorithm for solving some hierarchical problem over the set of fixed points of a nonexpansive and the generalized mixed equilibrium problem.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H, be a strongly positive linear bounded operator, be ρ-contraction, γ be a positive real number such that . Let be β-inverse-strongly monotone and F be a bifunction from satisfying (A1)-(A5) and let be convex and lower semicontinuous with either (B1) or (B2). Let be a sequence generated by the following algorithm for arbitrary :
where , , and satisfy the following conditions:
-
(C1) ;
-
(C2) ;
-
(C3) , , ;
-
(C4) , .
Then converges strongly to , which is the unique solution of the variational inequality:
Proof We will divide the proof into five steps.
Step 1. We will show is bounded. For any and taking , we note that
From (3.1), we have
It follows by induction that
Therefore is bounded and so are , , and .
Step 2. We show that . Setting and we observe that
where . Setting for all . We observes that
Substituting (3.4) into (3.5) it follows that
where . On the other hand, from and it follows that
and
Substituting into (3.7) and into (3.8), we have
and
From (A2), we have
and then
so
It follows that
Without loss of generality, let us assume that there exists a real number c such that , for all . Then we have
and hence
where . From (3.1), we have
where . Substituting (3.6) and (3.9) into (3.10)
from (C1)-(C4) and the boundedness of , , , , and . Applying Lemma 2.4, we obtain
Step 3. We show that . For each , note that is firmly nonexpansive, then we have
which implies that
From (3.1), we get
By (C3), we have
Setting . It follows that
By using (C3) again, we get
From , we compute
It follows from (3.15) that
Then we get
It follows that
that is,
On the other hand, we note that
Using (3.17), (3.18), (3.19), and (3.20), we note that
Then we have
From (C3), , and (3.12), we obtain
Substituting (3.13) into (3.21), we have
and it follows that
Since we have (C3), (3.12), and (3.22),
By (C4), we obtain
Step 4. Next, we will show that
Indeed, we choose a subsequence of such that
Since is bounded, there exists a subsequence of which converges weakly to . We notice that . Hence, we get . Next, we will show that . Since , we have
From (A2), we also have
and hence
For t with and , let . Since and , we have . So, from (3.25), we have
Since , we have . Further, from the inverse strongly monotonicity of B, we have . So, from (A4), (A5), and the weak lower semicontinuity of φ, and , we have in the limit
as . From (A1), (A4) and (3.26), we also get
Letting , we have, for each ,
This implies that . It is easy to see that is a contraction of H into itself. Hence H is complete, there exists a unique fixed point , such that .
Step 5. Next, we will prove , which solves the variational inequality (1.8). It follows from (3.1) that
which implies that
Since , , and are all bounded, we can choose a constant such that
It follows that
By (C3), we conclude that , as . This completes the proof. □
4 An example
Next, the following example shows that all conditions of Theorem 3.1 are satisfied.
Example 4.1 For instance, let , , , and . Then clearly the sequences , satisfy the following condition:
We will show that the condition (C1) is fulfilled. Indeed, we have
The sequence satisfies the condition (C1) by a p-series.
Next, we will show that the condition (C2) is fulfilled. We compute
The sequence satisfies the condition (C2).
Next, we will show that the condition (C3) is fulfilled. We compute
and
The sequence satisfies the condition (C3).
Finally, we will show that the condition (C4) is fulfilled. We compute
and
The sequence satisfies the condition (C4).
Corollary 4.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a strongly positive linear bounded operator, be ρ-contraction, γ be a positive real number such that and be a nonexpansive mapping with . Let be a sequence generated by the following algorithm for arbitrary :
where satisfy the following conditions:
-
(C1) , ;
-
(C2) , , .
Then converges strongly to , which is the unique solution of the variational inequality:
Proof Setting and T to be a nonexpansive mapping in Theorem 3.1, we obtain the desired conclusion immediately. □
Remark 4.3 Corollary 4.2 generalizes and improves the results of Yao et al. [11].
Corollary 4.4 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a strongly positive linear bounded operator, be a β-inverse-strongly monotone and F be a bifunction from satisfying (A1)-(A5) and let be convex and lower semicontinuous with either (B1) or (B2). Suppose . Let be a sequence by the following algorithm for arbitrary :
where , , and satisfy the following conditions:
-
(C1) ;
-
(C2) ;
-
(C3) , , ;
-
(C4) , .
Then converges strongly to , which is the unique solution of the variational inequality:
Proof Setting T, to be the identity and in Theorem 3.1, we obtain the desired conclusion immediately. □
Corollary 4.5 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a strongly positive linear bounded operator, be ρ-contraction, be β-inverse-strongly monotone and F be a bifunction from satisfying (A1)-(A5) and let be convex and lower semicontinuous with either (B1) or (B2). Let be a sequence generated by the following algorithm for arbitrary :
where , , and satisfy the following conditions:
-
(C1) ;
-
(C2) ;
-
(C3) , , ;
-
(C4) , .
Then converges strongly to , which is the unique solution of the variational inequality
Proof Setting T, to be the identity and in Theorem 3.1, we obtain the desired conclusion immediately. □
References
Al-Mazeooei AE, Latif A, Yao JC: Solving generalized mixed equilibria, variational inequalities, and constrained convex minimization. Abstr. Appl. Anal. 2014. Article ID 587865, 2014:
Ceng LC, Chen CM, Pang CT: Hybrid extragradient-like viscosity methods for generalized mixed equilibrium problems, variational inclusions, and optimization problems. Abstr. Appl. Anal. 2014. Article ID 120172, 2014:
Ceng LC, Chen CM, Wen CF, Pang CT: Relaxed iterative algorithms for generalized mixed equilibrium problems with constraints of variational inequalities and variational inclusions. Abstr. Appl. Anal. 2014. Article ID 345212, 2014:
Ceng LC, Ho JL: Hybrid extragradient method with regularization for convex minimization, generalized mixed equilibrium, variational inequality and fixed point problems. Abstr. Appl. Anal. 2014. Article ID 436069, 2014:
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123-145.
Hartman P, Stampacchia G: On some nonlinear elliptic differential functional equations. Acta Math. 1966, 115: 271-310. 10.1007/BF02392210
Kirk WA: Fixed point theorem for mappings which do not increase distance. Am. Math. Mon. 1965, 72: 1004-1006. 10.2307/2313345
Yao YH, Cho YJ, Yang PX: An iterative algorithm for a hierarchical problem. J. Appl. Math. 2012. Article ID 320421, 2012: Article ID 320421
Yao YH, Kang JI, Cho YJ, Liou YC: Composite schemes for variational inequalities over equilibrium problems and variational inclusions. J. Inequal. Appl. 2013. Article ID 414, 2013:
Yao Y, Liou Y-C, Chen C-P: Hierarchical convergence of a double-net algorithm for equilibrium problems and variational inequality problems. Fixed Point Theory Appl. 2010. Article ID 642584, 2010: 10.1155/2010/642584
Yao Y, Liou Y-C, Kang SM: Algorithms construction for variational inequalities. Fixed Point Theory Appl. 2011. Article ID 794203, 2011: 10.1155/2011/794203
Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 595-597.
Peng JW, Yao JC: A new hybrid-extragradient method for generalized mixed equilibrium problems and fixed point problems and variational inequality problems. Taiwan. J. Math. 2008,12(6):1401-1432.
Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math. 1976, 18: 78-81.
Marino G, Xu HK: A general iterative method for nonexpansive mapping in Hilbert space. J. Math. Anal. Appl. 2006, 318: 43-52. 10.1016/j.jmaa.2005.05.028
Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240-256. 10.1112/S0024610702003332
Acknowledgements
This project was partially supported by Centre of Excellence in Mathematics, the Commission on Higher Education, Ministry of Education, Thailand. The second author and third author were supported by Innovation park, RMUTL Hands-on Researcher Project, Rajamangala University of Technology Lanna Chiangrai under Grant no. 57HRG-10 during the preparation of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Kumam, P., Jitpeera, T. & Yarangkham, W. An explicit algorithm for solving the optimize hierarchical problems. J Inequal Appl 2014, 405 (2014). https://doi.org/10.1186/1029-242X-2014-405
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-405