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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 436069, 27 pages
http://dx.doi.org/10.1155/2014/436069
Research Article

Hybrid Extragradient Method with Regularization for Convex Minimization, Generalized Mixed Equilibrium, Variational Inequality and Fixed Point Problems

1Department of Mathematics, Shanghai Normal University and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
2Department of Finance, Tainan University of Technology, No. 529, Zhongzheng Road, YongKang District, Tainan 71002, Taiwan

Received 15 October 2013; Revised 25 November 2013; Accepted 25 November 2013; Published 4 February 2014

Academic Editor: Erdal Karapınar

Copyright © 2014 Lu-Chuan Ceng and Juei-Ling Ho. 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 two iterative algorithms by the hybrid extragradient method with regularization for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings and the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under mild conditions.

1. Introduction

Throughout this paper, we assume that is a real Hilbert space with inner product and norm ; let be a nonempty closed convex subset of and let be the metric projection of onto . Let be a self-mapping on . We denote by the set of fixed points of and by the set of all real numbers.

Let be a real-valued function, let be a nonlinear mapping, and let be a bifunction. In 2008, Peng and Yao [1] introduced the following generalized mixed equilibrium problem (GMEP) of finding such that We denote the set of solutions of GMEP (1) by . The GMEP (1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, and Nash equilibrium problems in noncooperative games. The GMEP is further considered and studied in [25].

Let be a convex and continuously Fréchet differentiable functional. Consider the convex minimization problem (CMP) of minimizing over the constraint set (assuming the existence of minimizers). We denote by the set of minimizers of CMP (2). The gradient-projection algorithm (GPA) generates a sequence determined by the gradient and the metric projection as follows: or more generally, where, in both (3) and , the initial guess is taken from arbitrarily and the parameters or are positive real numbers. The convergence of algorithms (3) and depends on the behavior of the gradient .

Since the Lipschitz continuity of the gradient implies that it is actually -inverse strongly monotone (ism) [6], its complement can be an averaged mapping (i.e., it can be expressed as a proper convex combination of the identity mapping and a nonexpansive mapping). Consequently, the GPA can be rewritten as the composite of a projection and an averaged mapping, which is again an averaged mapping. This shows that averaged mappings play an important role in the GPA. Recently, Xu [7] used averaged mappings to study the convergence analysis of the GPA, which is hence an operator-oriented approach.

Assume that the CMP (2) is consistent and the gradient is -Lipschitz continuous with . Let be a -contraction with . Xu [7] introduced the following hybrid GPA: where and . It was proven that under appropriate conditions the sequence converges in norm to a minimizer of CMP (2); see [7, Theorem 5.2].

It is worth emphasizing that the regularization, in particular the traditional Tikhonov regularization, is usually used to solve ill-posed optimization problems. Consider the regularized minimization problem where is the regularization parameter and again is convex with -Lipschitz continuous gradient . In [7], Xu introduced another hybrid GPA with regularization where (i) for all ; (ii) (and ) as ; (iii) ; and (iv) as . It was proven that converges strongly to the minimum-norm solution of CMP (2); see [7, Theorem 6.1]. Very recently, the hybrid GPA with regularization is extended to develop new extragradient methods with regularization in Ceng et al. [8, 9] for finding a common solution of the split feasibility problem (SFP) and the fixed point problem of a nonexpansive mapping in a real Hilbert space.

On the other hand, consider the following variational inequality problem (VIP): find a such that The solution set of VIP (7) is denoted by .

The VIP (7) was first discussed by Lions [10] and now is well known; there are a lot of different approaches towards solving VIP (7) in finite-dimensional and infinite-dimensional spaces, and the research is intensively continued. The VIP (7) has many applications in computational mathematics, mathematical physics, operations research, mathematical economics, optimization theory, and other fields; see, for example, [1114]. It is well known that if is a strongly monotone and Lipschitz-continuous mapping on , then VIP (7) has a unique solution. Not only the existence and uniqueness of solutions are important topics in the study of VIP (7), but also how to actually find a solution of VIP (7) is important.

Motivated by the idea of Korpelevič's extragradient method [15], Nadezhkina and Takahashi [16] introduced an extragradient iterative scheme: where is a monotone, -Lipschitz continuous mapping, is a nonexpansive mapping, and for some and for some . They proved the weak convergence of to an element of . Recently, inspired by Nadezhkina and Takahashi's iterative scheme [16], Zeng and Yao [17] introduced another iterative scheme for finding an element of and derived the weak convergence result. Furthermore, by combining the CQ method and extragradient method, Nadezhkina and Takahashi [18] introduced an iterative process: They proved the strong convergence of to an element of under appropriate conditions. Later on, Ceng and Yao [19] introduced an extragradient-like approximation method which is based on the above extragradient method and viscosity approximation method and derived a strong convergence result as well. Next, recall some concepts. 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.

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.

Definition 1. Let be a nonempty subset of a normed space and let be a self-mapping on .(i) is asymptotically nonexpansive (cf. [20]) if there exists a sequence of positive numbers satisfying the property and (ii) is asymptotically nonexpansive in the intermediate sense [21] provided is uniformly continuous and (iii) is uniformly Lipschitzian if there exists a constant such that

It is clear that every nonexpansive mapping is asymptotically nonexpansive and every asymptotically nonexpansive mapping is uniformly Lipschitzian.

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [20] as an important generalization of the class of nonexpansive mappings. The existence of fixed points of asymptotically nonexpansive mappings was proved by Goebel and Kirk [20] as follows.

Theorem GK (see [20, Theorem ]). If is a nonempty closed convex bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive mapping has a fixed point in .

The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Bruck et al. [21]. Recently, Kim and Xu [22] introduced the concept of asymptotically -strict pseudocontractive mappings in a Hilbert space as follows.

Definition 2. Let be a nonempty subset of a Hilbert space . A mapping is said to be an asymptotically -strict pseudocontractive mapping with sequence if there exists a constant and a sequence in with such that

They studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptotically -strict pseudocontractive mapping with sequence is a uniformly -Lipschitzian mapping with .

Recently, Sahu et al. [23] considered the concept of asymptotically -strict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian.

Definition 3. Let be a nonempty subset of a Hilbert space . A mapping is said to be an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence if there exist a constant and a sequence in with such that

Put . Then    , and (17) reduces to the relation

Whenever for all in (18), then is an asymptotically -strict pseudocontractive mapping with sequence . In 2009, Sahu et al. [23] derived the weak and strong convergence of the modified Mann iteration process for the class of asymptotically -strictly pseudocontractive mappings in the intermediate sense with sequence . More precisely, they established the following theorems.

Theorem SXY1. Let be a nonempty closed convex subset of a real Hilbert space and let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence such that and . Assume that is a sequence in such that and . Let be a sequence in generated by the modified Mann iteration process: Then converges weakly to an element of .

Theorem SXY2. Let be a nonempty closed convex subset of a real Hilbert space and let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence such that is nonempty and bounded. Let be a sequence in such that for all . Let be the sequence in generated by the following (CQ) algorithm: where and . Then converges strongly to .

Subsequently, the iterative algorithms in Theorems SXY1 and SXY2 are extended to develop new iterative algorithms for finding a common solution of the VIP and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space; see, for example, [24, 25].

On the other hand, Yao et al. [26] introduced two iterative algorithms for finding a common element of the set of fixed points of an asymptotically -strict pseudocontraction and the set of solutions of a mixed equilibrium problem in a real Hilbert space. Then they obtained some weak and strong convergence theorems for the proposed iterative algorithms. Very recently, motivated by Yao et al. [26], Cai and Bu [3] introduced two iterative algorithms for finding a common element of the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings, and the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. Then they proved some strong and weak convergence theorems for the proposed iterative algorithms under appropriate conditions.

In this paper, inspired by the above facts, we introduce two iterative algorithms by hybrid extragradient method with regularization for finding a common element of the set of solutions of the CMP (2) for a convex functional with -Lipschitz continuous gradient , the set of solutions of finite GMEPs, the set of solutions of finite VIPs for inverse strong monotone mappings, and the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. Then we prove some strong and weak convergence theorems for the proposed iterative algorithms under mild conditions. For recent related results, see, for example, [7, 24, 2731] and ther references therein.

2. Preliminaries

Let be 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,

The metric (or nearest point) 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 4. For given and :(i) ,   for all   ;(ii) ,  for all   ;(iii) ,  for all   .

Consequently, is nonexpansive and monotone.

If is an -inverse-strongly monotone mapping of into , then it is obvious that is -Lipschitz continuous. We also have that, for all and , So if , then is a nonexpansive mapping from to .

Definition 5. A mapping is said to be(a)nonexpansive if (b)firmly nonexpansive if is nonexpansive, or equivalently, if is -inverse strongly monotone ( -ism), 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. It is also easy to see that a projection is -ism. Inverse strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields.

Definition 6. 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 maps.

Proposition 7 (see [32]). 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 8 (see [32, 33]). 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, = .

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

Lemma 9. Let be a real inner product space. Then there holds the following inequality:

Lemma 10. Let be a bounded sequence on a reflexive Banach space . If , then .

Lemma 11. Let be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 4 (i)) implies

Lemma 12. Let be a real Hilbert space. Then the following hold:(i) for all ;(ii) for all and for all ;(iii)If is a sequence in such that , it follows that

Lemma 13 ([23, Lemma 2.5]). Let be a real Hilbert space. Given a nonempty closed convex subset of and points and given also a real number , the set is convex (and closed).

Lemma 14 ([23, Lemma 2.6]). Let be a nonempty subset of a Hilbert space and let be an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Then for all and .

Lemma 15 ([23, Lemma 2.7]). Let be a nonempty subset of a Hilbert space and let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Let be a sequence in such that and as . Then as .

Lemma 16 (demiclosedness principle [23, Proposition 3.1]). Let be a nonempty closed convex subset of a Hilbert space and let be a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Then is demiclosed at zero in the sense that if is a sequence in such that and , then .

Lemma 17 ([23, Proposition 3.2]). Let be a nonempty closed convex subset of a Hilbert space and let be a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence such that . Then is closed and convex.

Remark 18. Lemmas 16 and 17 give some basic properties of an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Moreover, Lemma 16 extends the demiclosedness principles studied for certain classes of nonlinear mappings in Kim and Xu [22], Górnicki [34], Marino and Xu [35], and Xu [36].

To prove a weak convergence theorem by a modified extragradient method with regularization for the CMP (2) and the fixed point problem of an asymptotically -strict pseudocontractive mapping in the intermediate sense, we need the following lemma due to Osilike et al. [37].

Lemma 19 (see [37, page 80]). Let and be sequences of nonnegative real numbers satisfying the inequality If and , then exists. If, in addition, has a subsequence which converges to zero, then .

Corollary 20 (see [38, page 303]). Let and be two sequences of nonnegative real numbers satisfying the inequality If converges, then exists.

Recall that a Banach space is said to satisfy the Opial condition [39] if for any given sequence which converges weakly to an element , there holds the inequality It is well known in [39] that every Hilbert space satisfies the Opial condition.

Lemma 21 (see [24, Proposition 3.1]). Let be a nonempty closed convex subset of a real Hilbert space and let be a sequence in . Suppose that where and are sequences of nonnegative real numbers such that and . Then converges strongly in .

A set-valued mapping is called monotone if for all , and imply . A monotone mapping is maximal if its graph is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for for all implies . Let be a monotone, -Lipschitz continuous mapping, and let be the normal cone to at ; that is, . Define It is known that in this case is maximal monotone, and if and only if ; see [40].

For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions:(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.

Lemma 22 (see [41]). 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:(1)for each ;(2) is single-valued;(3) is firmly nonexpansive; that is, for any , (4) ;(5) is closed and convex.

Lemma 23 (see [42]). Let be a closed convex subset of a real Hilbert space . Let be a sequence in and . Let . If is such that and satisfies the condition then as .

3. Strong Convergence Theorem

In this section, we prove a strong convergence theorem for a hybrid extragradient iterative algorithm with regularization for finding a common element of the set of solutions of the CMP (2) for a convex functional with -Lipschitz continuous gradient , the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings, and the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. This iterative algorithm with regularization is based on the extragradient method, shrinking projection method, Mann-type iterative method, and hybrid gradient projection algorithm (GPA) with regularization.

Theorem 24. 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 , . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Assume that is nonempty and bounded. Let be a sequence in and let be sequences in such that and . Pick any and set , . Let be a sequence generated by the following algorithm: where , and . Assume that the following conditions hold:(i) ;(ii) for some ;(iii) ,for all ;(iv) ,for all .Then converge strongly to provided either (B1) or (B2) holds.

Proof. First of all, one can show that is -averaged for each , where which shows that is nonexpansive. Furthermore, for with , we have Without loss of generality, we may assume that Consequently, it follows that for each integer , is -averaged with This immediately implies that is nonexpansive for all .
We divide the proof into several steps.
Step 1. We show that is well defined. It is obvious that is closed and convex. As the defining inequality in is equivalent to the inequality by Lemma 13 we know that is convex for every .
Next we show that for all . Put for all and , for all and , and , where is the identity mapping on . Then we have that and . Suppose that for some . Take arbitrarily. Then from (23) and Lemma 22 we have Similarly, we have Combining (52) and (53), we have Also, it follows from (44) that Note that for every . Then, by Proposition 4(ii), we have Further, by Proposition 4(i), we have So from (54) and (55), we obtain By Lemma 12 and (58), we have It follows from (59) and that Hence . This implies that for all . Therefore, is well defined.
Step 2. We prove that as .
Indeed, let . From and , we obtain This implies that is bounded and hence , and are also bounded. Since and , we have Therefore, exists. From , we obtain which implies It follows from that and hence From (64) and , we have Note that Since and (66), we obtain
Step 3. We prove that , and as .
Indeed, from (58) and (59) it follows that Next we prove that For , it follows from (23) that By (52), (53), (58), (69), and (71), we obtain which implies that Since , , , , and , we conclude from (68) and the boundedness of that By Lemmas 12 and 22, we have <