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Journal of Applied Mathematics

Volume 2014, Article ID 129379, 25 pages

http://dx.doi.org/10.1155/2014/129379
Research Article

Triple Hierarchical Variational Inequalities with Constraints of Mixed Equilibria, Variational Inequalities, Convex Minimization, and Hierarchical Fixed Point Problems

1Department of Mathematics, Shanghai Normal University and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China

2Department of Food and Beverage Management, Vanung University, Chung-Li 320061, Taiwan

3Department of Information Management, and Innovation Center for Big Data and Digital Convergence, Yuan Ze University, Chung-Li 32003, Taiwan

4Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Received 7 April 2014; Accepted 27 April 2014; Published 26 May 2014

Academic Editor: Jen-Chih Yao

Copyright © 2014 Lu-Chuan Ceng et al. 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.

Abstract

We introduce and analyze a hybrid iterative algorithm by virtue of Korpelevich's extragradient method, viscosity approximation method, hybrid steepest-descent method, and averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inequality problems (VIPs), the solution set of general system of variational inequalities (GSVI), and the set of minimizers of convex minimization problem (CMP), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others.

1. Introduction

Let be a nonempty closed convex subset of a real Hilbert space and let be the metric projection of onto . Let be a nonlinear mapping on . We denote by the set of fixed points of and by the set of all real numbers. A mapping is called -Lipschitz continuous if there exists a constant such that In particular, if , then is called a nonexpansive mapping; if , then is called a contraction.

Let be a nonlinear mapping on . We consider the following variational inequality problem (VIP): find a point such that The solution set of VIP (2) is denoted by .

The VIP (2) was first discussed by Lions [1]. There are many applications of VIP (2) in various fields; see, for example, [25]. It is well known that if is a strongly monotone and Lipschitz-continuous mapping on , then VIP (2) has a unique solution. In 1976, Korpelevich [6] proposed an iterative algorithm for solving the VIP (2) in Euclidean space : with a given number, which is known as the extragradient method. The literature on the VIP is vast, among which, Korpelevich's extragradient method has received great attention in various applications and undergone improvements in many ways; see, for example, [720] and references therein, to name but a few.

Let be a real-valued function, let be a nonlinear mapping, and let be a bifunction. In 2008, Peng and Yao [8] introduced the following generalized mixed equilibrium problem (GMEP) of finding such that We denote the set of solutions of GMEP (4) by . The (4) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games, and others. The GMEP is further considered and studied; see, for example, [10, 16, 18, 19, 2123]. In particular, if , then GMEP (4) is reduced to the generalized equilibrium problem (GEP) which is to find such that It was introduced and studied by S. Takahashi and W. Takahashi [24]. The set of solutions of GEP is denoted by .

If , then GMEP (4) is reduced to the mixed equilibrium problem (MEP), which is to find such that It was considered and studied in [25]. The set of solutions of MEP is denoted by .

If and , then GMEP (4) is reduced to the equilibrium problem (EP), which is to find such that It was considered and studied in [26, 27]. The set of solutions of EP is denoted by . It is worth mentioning that the EP is a unified model of several problems, namely, the variational inequality problems, the optimization problems, the saddle point problems, the complementarity problems, the fixed point problems, the Nash equilibrium problems, and so forth.

It was assumed in [8] that is a bifunction satisfying conditions (A1)–(A4) and is a lower semicontinuous and convex function with a restriction (B1) or (B2), where(A1) for all ;(A2) is monotone; that is, for any ;(A3) is upper-hemicontinuous; that is, for each , (A4) is convex and lower semicontinuous for each ;(B1)for each and , there exists a bounded subset and such that for any , (B2) is a bounded set.

Given a positive number . Let be the solution set of the auxiliary mixed equilibrium problem; that is, for each ,

Let be two mappings. Consider the following general system of variational inequalities (GSVI) of finding such that where and are two constants, which was considered and studied in [9, 11, 28]. In particular, if , then the GSVI (11) is reduced to the following problem of finding such that which is defined by Verma [29] and called as a new system of variational inequalities (NSVI). Furthermore, if , then the NSVI reduces to the classical VIP (2). In 2008, Ceng et al. [9] transformed the GSVI (11) into a fixed point problem as follows.

Proposition CWY (see [9]). For given is a solution of the GSVI (11) if and only if is a fixed point of the mapping defined by where .

In particular, if the mapping is -inverse-strongly monotone for , then the mapping is nonexpansive for all , . We denote by the fixed point set of the mapping .

Let be a convex and continuously Fréchet differentiable functional. Consider the convex minimization problem (CMP) of minimizing over the constraint set as considered and studied in [13, 14, 3032]. We denote by the set of minimizers of CMP (14). The gradient-projection algorithm (GPA) generates a sequence determined by the gradient and the metric projection : or more generally where, in both (15) and (16), the initial guess is taken from arbitrarily and the parameters or are positive real numbers. The convergence of algorithms (15) and (16) depends on the behavior of the gradient . As a matter of fact, it is known that if is -strongly monotone and -Lipschitz continuous, then, for , the operator is a contraction; hence, the sequence defined by the GPA (15) converges in a norm to the unique solution of CMP (14). More generally, if is chosen to satisfy the property then the sequence defined by the GPA (16) converges in a norm to the unique minimizer of CMP (14). If the gradient is only assumed to be Lipschitz continuous, then can only be weakly convergent if is infinite-dimensional (a counterexample is given in Section 5 of Xu [31]). Recently, Xu [31] used averaged mappings to study the convergence analysis of the GPA, which is hence an operator-oriented approach.

Very recently, Ceng and Al-Homidan [23] introduced and analyzed the following iterative algorithm by hybrid steepest-descent viscosity method and derived its strong convergence under appropriate conditions.

Theorem CA (see [23, Theorem 21]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be two integers. Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function, where . Let and be -inverse-strongly monotone and -inverse-strongly monotone, respectively, where and . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let be an -Lipschitzian mapping with a constant . Let and , where . Assume that and that either (B1) or (B2) holds. For arbitrarily given , let be a sequence generated by where here is nonexpansive and for each . Assume that the following conditions hold:(i) for each , and ( );(ii) and ;(iii) and for all ;(iv) and for all .Then converges strongly as to a point , which is a unique solution in to the VIP: Equivalently, .

In 2009, Yao et al. [33] considered the following hierarchical fixed point problem (HFPP): find hierarchically a fixed point of a nonexpansive mapping with respect to another nonexpansive mapping ; namely, find such that The solution set of HFPP (20) is denoted by . It is obvious to see that solving HFPP (20) is equivalent to the fixed point problem of the composite mapping ; that is, find such that . The authors [33] introduced and analyzed the following iterative algorithm for solving HFPP (20):

Theorem YLM (see [33, Theorem 3.2]). Let be a nonempty closed convex subset of a real Hilbert space . Let and be two nonexpansive mappings of into itself. Let be a fixed contraction with . Let and be two sequences in . For any given , let be the sequence generated by (21). Assume that the sequence is bounded and that(i) ;(ii) , ;(iii) , and ;(iv) ;(v) there exists a constant such that for each , where . Then converges strongly to which solves the VIP ,  for all .

Very recently, Iiduka [34, 35] considered a variational inequality with a variational inequality constraint over the set of fixed points of a nonexpansive mapping. Since this problem has a triple structure in contrast with the bilevel programming problems or the hierarchical constrained optimization problems or the hierarchical fixed point problem, it is referred to as the triple hierarchical constrained optimization problem (THCOP). He presented some examples of THCOP and developed iterative algorithms to find the solution of such problem. The convergence analysis of the proposed algorithms is also studied in [34, 35]. Since the original problem is of a variational inequality, in this paper, we call it triple hierarchical variational inequality (THVI). Subsequently, Ceng et al. [36] introduced and considered the following triple hierarchical variational inequality (THVI).

Problem 1. Let be two nonexpansive mappings with , let be a -contractive mapping with a constant , and let be a -Lipschitzian and -strongly monotone mapping with constants . Let and , where . Consider the following THVI: find such that which denotes the solution set of the following hierarchical variational inequality (HVI): find such that where the solution set is assumed to be nonempty.

The authors [36] proposed both implicit and explicit iterative methods and studied the convergence analysis of the sequences generated by the proposed methods. In this paper, we introduce and study the following triple hierarchical variational inequality (THVI) with constraints of mixed equilibria, variational inequalities, and convex minimization problem.

Problem 2. Let be two integers. Let be a convex functional with -Lipschitz continuous gradient . Let be a bifunction from to and let be a proper lower semicontinuous and convex function, where . Let and be inverse-strongly monotone mappings, where , , and . Let be a nonexpansive mapping and let be a sequence of nonexpansive mappings on . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let be an -Lipschitzian mapping with a constant . Let , , and , where . Consider the following triple hierarchical variational inequality (THVI): find such that where denotes the solution set of the following hierarchical variational inequality (HVI): find such that where the solution set is assumed to be nonempty.

Motivated and inspired by the above facts, we introduce and analyze a hybrid iterative algorithm by the virtue of Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, and the averaged mapping approach to the GPA. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings , the solution set of finitely many GMEPs, the solution set of finitely many VIPs, the solution set of GSVI (11), and the set of minimizers of CMP (14), which is just a unique solution of the THVI (24). In addition, we also consider the application of the proposed algorithm to solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI (11), and CMP (14). That is, under very mild conditions, it is proven that the proposed algorithm converges strongly to a unique solution of the VIP: , for all ; equivalently, . The results obtained in this paper improve and extend the corresponding results announced by many others.

2. Preliminaries

Throughout this paper, we assume that is a real Hilbert space whose inner product and norm are denoted by and , respectively. Let be a nonempty closed convex subset of . We write to indicate that the sequence converges weakly to and to indicate that the sequence converges strongly to . Moreover, we use to denote the weak -limit set of the sequence ; that is,

Recall that a mapping is called(i)monotone if (ii) -strongly monotone if there exists a constant such that (iii) -inverse-strongly monotone if there exists a constant such that

It is obvious that if is -inverse-strongly monotone, then is monotone and -Lipschitz continuous. Moreover, we also have that, for all and , So, if , then is a nonexpansive mapping from to .

The metric projection from onto is the mapping which assigns to each point the unique point satisfying the property

Some important properties of projections are gathered in the following proposition.

Proposition 1. For given and :(i) , for all ;(ii) , for all ;(iii) , for all .Consequently, is nonexpansive and monotone.

Definition 2. A mapping is said to be(a)nonexpansive if (b)firmly nonexpansive if is nonexpansive or equivalently if is -inverse-strongly monotone ( -ism), then alternatively, is firmly nonexpansive if and only if can be expressed as where is nonexpansive; projections are firmly nonexpansive.

It can be easily seen that if is nonexpansive, then is monotone and a projection is -ism. The inverse-strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields.

Definition 3. A mapping is said to be an averaged mapping if it can be written as the average of the identity and a nonexpansive mapping; that is, where and is nonexpansive. More precisely, when the last equality holds, we say that is -averaged. Thus firmly nonexpansive mappings (in particular, projections) are -averaged mappings.

Proposition 4 (see [37]). Let be a given mapping.(i) is nonexpansive if and only if the complement is -ism.(ii) If is -ism, then for , is -ism.(iii) is averaged if and only if the complement is -ism for some . Indeed, for is -averaged if and only if is -ism.

Proposition 5 (see [37, 38]). Let be given operators.(i) If for some and if is averaged and is nonexpansive, then is averaged.(ii) is firmly nonexpansive if and only if the complement is firmly nonexpansive.(iii) If for some and if is firmly nonexpansive and is nonexpansive, then is averaged.(iv) The composite of finitely many averaged mappings is averaged. That is, if each of the mappings is averaged, then so is the composite . In particular, if is -averaged and is -averaged, where , then the composite is -averaged, where .(v) If the mappings are averaged and have a common fixed point, then The notation denotes the set of all fixed points of the mapping ; that is, .

Next we list some elementary conclusions for the MEP.

Proposition 6 (see [25]). Assume that satisfies (A1)–(A4) and let be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For and , define a mapping as follows: for all . Then the following hold:(i) for each , is nonempty and single-valued;(ii) is firmly nonexpansive; that is, for any , (iii) ;(iv) is closed and convex;(v) , for all and .

We need some facts and tools in a real Hilbert space which are listed as lemmas below.

Lemma 7. Let be a real inner product space. Then there holds the following inequality:

Lemma 8. Let be a monotone mapping. In the context of the variational inequality problem, the characterization of the projection (see Proposition 1(i)) implies that

Lemma 9 (see [39, Demiclosedness principle]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive self-mapping on . Then is demiclosed. That is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that . Here is the identity operator of .

Let be an infinite family of nonexpansive mappings on and let be a sequence of nonnegative numbers in . For any , define a mapping on as follows: Such a mapping is called the W-mapping generated by and .

Lemma 10 (see [40, Lemma 3.2]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive self-mappings on such that and let be a sequence in for some . Then, for every and , the limit exists where is defined as in (41).

Remark 11 (see [41, Remark 3.1]). It can be known from Lemma 10 that if is a nonempty bounded subset of , then for there exists such that for all

Remark 12 (see [41, Remark 3.2]). Utilizing Lemma 10, we define a mapping as follows: Such a is called the W-mapping generated by   and . Since is nonexpansive, is also nonexpansive. If is a bounded sequence in , then it is clear from Remark 11 that

Lemma 13 (see [40, Lemma 3.3]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive self-mappings on such that , and let be a sequence in for some . Then, .

The following lemma can be easily proven, and therefore, we omit the proof.

Lemma 14. Let be an -Lipschitzian mapping with constant and let be a -Lipschitzian and -strongly monotone operator with positive constants . Then for , That is, is strongly monotone with the constant .

Let be a nonempty closed convex subset of a real Hilbert space . We introduce some notations. Let be a number in and let . Associating with a nonexpansive mapping , we define the mapping by where is an operator such that, for some positive constants , is -Lipschitzian and -strongly monotone on ; that is, satisfies the following conditions: for all .

Lemma 15 (see [42, Lemma 3.1]). is a contraction provided that ; that is, where .

Lemma 16 (see [42]). Let be a sequence of nonnegative numbers satisfying the conditions where and are sequences of real numbers such that(i) and or equivalently (ii) , or .Then .

Lemma 17 (see [39]). Let be a real Hilbert space. Then the following hold:(a) for all ;(b) for all and with ;(c)if is a sequence in such that , it follows that

A set-valued mapping is called monotone if for all , and imply that . A monotone set-valued mapping is called maximal if its graph is not properly contained in the graph of any other monotone set-valued mapping. It is known that a monotone set-valued mapping is maximal if and only if for , for every implies that . Let be a monotone and Lipschitz continuous mapping and let be the normal cone to at , that is, Define

Lemma 18 (see [43]). Let be a monotone mapping. Then there hold the following statements:(i) is maximal monotone;(ii) .

3. Strong Convergence Criteria for the THVI and HFPP

In this section, we will introduce and analyze an iterative algorithm for finding a solution of the THVI (24) with constraints of several problems: the finitely many GMEPs, the finitely many VIPs, GSVI (11), and CMP (14) in a real Hilbert space. This algorithm is based on the Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, and the averaged mapping approach to the GPA. We prove the strong convergence of the proposed algorithm to a unique solution of THVI (24) under suitable conditions. In addition, we also consider the application of the proposed algorithm to solve a hierarchical fixed point problem with the same constraints.

Theorem 19. Let be a nonempty closed convex subset of a real Hilbert space . Let be two integers. Let be a convex functional with -Lipschitz continuous gradient . Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function, where . Let and be -inverse-strongly monotone, -inverse-strongly monotone, and -inverse-strongly monotone, respectively, where , , and . Let be a nonexpansive mapping, let be a sequence of nonexpansive mappings on , and let be a sequence in for some . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let be an -Lipschitzian mapping with the constant . Let , , and , where . Assume that either (B1) or (B2) holds. For arbitrarily given , let be a sequence generated by where here is nonexpansive, for each , , for , and is the W-mapping defined by (41). Suppose that the following conditions are satisfied:(H1) for each , and    ;(H2) and ;(H3) , , and ;(H4) ;(H5) and for all ;(H6) and for all . Then there hold the following:(i) ;(ii) provided ;(iii) provided additionally.

Proof. Since is -Lipschitzian, it follows that is -ism; see [44] (see also [31]). By Proposition 4(ii) we know that for , is -ism. So by Proposition 4(iii) we deduce that is -averaged. Now since the projection is -averaged, it is easy to see from Proposition 5(iv) that the composite is -averaged for . Hence we obtain that for each , is -averaged for each . Therefore, we can write where is nonexpansive and for each . It is clear that Put for all and , for all , and , where is the identity mapping on . Then we have and .

We divide the rest of the proof into several steps.

Step 1. Let us show that is bounded. Indeed, taking into account the assumption in Problem 2, we know that . By (H4), we may assume, without loss of generality, that for all . Taking arbitrarily. Then from (30) and Proposition 6(ii) we have Similarly, we have Combining (59) and (60), we have Since is -inverse-strongly monotone for , and for , we deduce that, for any , Utilizing Lemma 15 and the relation , from (54), (61), and (62), we obtain that and hence By induction, we get Hence is bounded and so are the sequences .

Step 2. Let us show that .

Indeed, taking into account the -inverse-strong monotonicity of , we know that is nonexpansive for . Hence it follows that for any given , This together with the boundedness of implies that is bounded. Also, observe that <