Hybrid Extragradient-Like Viscosity Methods for Generalized Mixed Equilibrium Problems, Variational Inclusions, and Optimization Problems

Ceng, Lu-Chuan; Chen, Chi-Ming; Pang, Chin-Tzong

doi:https://doi.org/10.1155/2014/120172

Abstract and Applied Analysis

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Advances on Multivalued Operators and Related Fixed Point Problems

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Volume 2014 | Article ID 120172 | https://doi.org/10.1155/2014/120172

Hybrid Extragradient-Like Viscosity Methods for Generalized Mixed Equilibrium Problems, Variational Inclusions, and Optimization Problems

Lu-Chuan Ceng,¹Chi-Ming Chen,²and Chin-Tzong Pang³

Academic Editor: Erdal Karapinar

Received20 Nov 2013

Accepted20 Dec 2013

Published09 Feb 2014

Abstract

We introduce and analyze a new hybrid extragradient-like viscosity iterative algorithm for finding a common solution of a generalized mixed equilibrium problem, a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings, and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some mild conditions, we prove the strong convergence of the sequence generated by the proposed algorithm to a common solution of these three problems which also solves an optimization problem.

1. Introduction

Let be a real Hilbert space with inner product and norm , be a nonempty closed convex subset of , and 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 strongly positive on if there exists a constant such that 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 . Recall that the classical variational inequality problem (VIP) is to find a point such that The solution set of VIP (3) is denoted by . The VIP (3) was first discussed by Lions [1] and has been extensively studied since then. See, for example, [2–5].

In 1976, Korpelevič [6] proposed an iterative algorithm for solving the VIP (3) in Euclidean space : with as a given number, which is known as the extragradient method. The literature on the VIP is vast and Korpelevich's extragradient method has received great attention given by many authors, see, for example, [7–23] and the references therein. Let be a real-valued function, be a nonlinear mapping, and be a bifunction. In 2008, Peng and Yao [14] introduced the following generalized mixed equilibrium problem (GMEP) of finding such that We denote the set of solutions of GMEP (5) by . The GMEP (5) 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, [13, 16, 24–28].

We present some special cases of GMEP (5) as follows.

If , then GMEP (5) reduces to the generalized equilibrium problem (GEP) which is to find such that This problem was introduced and studied by S. Takahashi and W. Takahashi [29]. The set of solutions of GEP is denoted by .

If , then GMEP (5) reduces to the mixed equilibrium problem (MEP) which is to find such that It is considered and studied in [30, 31]. The set of solutions of MEP is denoted by .

If ,, then GMEP (5) reduces to the equilibrium problem (EP) which is to find such that This was considered and studied in [32, 33]. The set of solutions of EP is denoted by . It is worth mentioning that the EP is a unified model of several problems, namely, variational inequality problems, optimization problems, saddle point problems, complementarity problems, fixed point problems, Nash equilibrium problems, and so forth.

For the bifunction and real-valued function in the GMEP (5), as in [14], we assume that is a bifunction satisfying conditions (H1)–(H4) and is a lower semicontinuous and convex function with restriction (H5), where(H1) for all ;(H2) is monotone, that is; for any ;(H3) is upper-hemicontinuous, that is; for each , (H4) is convex and lower semicontinuous for each ;(H5)for each and , there exists a bounded subset and such that for any ,

A differentiable function is called(i)convex, if where is the Frechet derivative of at ;(ii)strongly convex, if there exists a constant such that

It is easy to see that if is a differentiable strongly convex function with constant , then is strongly monotone with constant .

Given a positive number , let be the solution set of the auxiliary mixed equilibrium problem; that is, for each , In particular, whenever ,, is rewritten as .

Let be two bifunctions and be two nonlinear mappings. Consider the following system of generalized equilibrium problems (SGEP): find such that where and are two constants. It is introduced and studied in [15], that the SGEP reduces to a system of variational inequalities whenever . It is worth mentioning that the system of variational inequalities is a tool to solve the Nash equilibrium problem for noncooperative games.

In 2010, Ceng and Yao [15] transformed the SGEP into a fixed point problem in the following way.

Proposition CY (see [15]). Let be two bifunctions satisfying conditions (H1)–(H4) and let be -inverse-strongly monotone for . Let for . Then, is a solution of SGEP if and only if is a fixed point of the mapping defined by , where . Here, one denotes the fixed point set of by .

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

In 2011, for the case where, letbe a contraction, be differentiable and strongly convex, ,andbe given. Yao et al. [25] proposed the hybrid iterative algorithm for finding a common element of the setand the fixed setof an infinite family of nonexpansive mappingsonas follows: They proved the strong convergence of the sequence generated by the hybrid iterative algorithm (16) to a point under some appropriate conditions. This point also solves the following optimization problem: where is the potential function of .

Let be a contraction and be a strongly positive bounded linear operator on . Assume that is a lower semicontinuous and convex functional, that satisfy conditions (H1)–(H4), and that are inverse-strongly monotone. Very recently, Ceng et al. [16] introduced the following hybrid extragradient-like iterative algorithm: for finding a common solution of GMEP (5), SGEP (14), and the fixed point problem of an infinite family of nonexpansive mappings on , where ,, and are given. The authors proved the strong convergence of the sequence generated by the hybrid iterative algorithm (17) to a point under some suitable conditions. This point also solves the following optimization problem: where is the potential function of .

On the other hand, let be a single-valued mapping of into and be a set-valued mapping with . Consider the following variational inclusion: find a point such that We denote by the solution set of the variational inclusion (18). In particular, if , then . If , then problem (18) becomes the inclusion problem introduced by Rockafellar [34]. It is known that problem (18) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria and game theory, etc. Let a set-valued mapping be maximal monotone. We define the resolvent operator associated with and as follows: where is a positive number.

In 1998, Huang [35] studied problem (18) in the case where is maximal monotone and is strongly monotone and Lipschitz continuous with . Subsequently, Zeng et al. [36] further studied problem (18) in the case which is more general than Huang's [35]. Moreover, the authors [36] obtained the same strong convergence conclusion as in Huang's result [35]. In addition, the authors also gave the geometric convergence rate estimate for approximate solutions. Also, various types of iterative algorithms for solving variational inclusions have been further studied and developed; for more details, refer to [21, 26, 37, 38] and the references therein.

Motivated and inspired by the above facts, we, in this paper, introduce and analyze a new iterative algorithm by a hybrid extragradient-like viscosity method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions of a finite family of variational inclusions for maximal monotone and inverse strong monotone mappings, and the set of fixed points of a countable family of nonexpansive mappings in a real Hilbert space. Under some appropriate conditions, we prove the strong convergence of the sequence generated by the proposed algorithm to a common solution of these three problems. Such a solution also solves an optimization problem. Several special cases are also discussed. The results presented in this paper are the supplement, extension, improvement, and generalization of the previously known results in this area.

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 easy to see that the projection is -ism. Inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields.

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 1. For given and :(i),;(ii),;(iii),. (This implies thatis nonexpansive and monotone.)

By using the technique of [31], we can readily obtain the following elementary result.

Proposition 2 (see [16, Lemma 1 and Proposition 1]). Let be a nonempty closed convex subset of a real Hilbert space and let be a lower semicontinuous and convex function. Let be a bifunction satisfying the conditions (H1)–(H4). Assume that(i)is strongly convex with constantand the functionis weakly upper semicontinuous for each;(ii)for each and , there exists a bounded subset and such that for any ,

Then the following hold:(a)for each ,;(b) is single-valued;(c) is nonexpansive if is Lipschitz continuous with constant and where for ;(d)for all and (e);(f) is closed and convex. In particular, whenever is a bifunction satisfying the conditions (H1)–(H4) and ,, then, that is, for any , ( is firmly nonexpansive) and In this case, is rewritten as . If, in addition, , then is rewritten as (see [15, Lemma 2.1] for more details).

Remark 3. Suppose is strongly convex with constant and is Lipschitz continuous with constant . Then is -strongly monotone and -Lipschitz continuous with positive constants . Utilizing Proposition 2(d) we obtain that for all and which immediately implies that
We need some facts and tools in a real Hilbert space which are listed as lemmas below.

Lemma 4. Let be a real inner product space. Then there holds the following inequality:

Lemma 5. 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

We have the following crucial lemmas concerning the -mappings defined by (15).

Lemma 6 (see [39, Lemma 3.2]). 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 by (15).

Remark 7 (see [40, Remark 3.1]). It can be known from Lemma 6 that if is a nonempty bounded subset of , then for there exists such that for all

Remark 8 (see [40, Remark 3.2]). Utilizing Lemma 6, we define a mapping as follows: Such a is called the -mapping generated by and . Since is nonexpansive, is also nonexpansive. Indeed, observe that for each If is a bounded sequence in , then we put . Hence, it is clear from Remark 3 that for an arbitrary there exists such that for all This implies that

Lemma 9 (see [39, Lemma 3.3]). Let be a sequence of nonexpansive self-mappings on such that and let be a sequence in for some . Then, .

Lemma 10 (see [41, Theorem 10.4 (Demiclosedness Principle)]). Let be a nonempty closed convex subset of a real Hilbert space . Let be nonexpansive. Then is demiclosed on . 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 .

Lemma 11. Let be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 1(i)) implies

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

Lemma 12. Let be an -Lipschitzian mapping with constant and be a -Lipschitzian and -strongly monotone operator with positive constants . Then for ,
That is, is strongly monotone with 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 conditions: for all .

Lemma 13 (see [42, Lemma 3.1]). is a contraction provided ; that is, where .

Recall that a set-valued mapping is called monotone if for all ,, and imply A set-valued mapping is called maximal monotone if is monotone and for each , where is the identity mapping of . We denote by the graph of . It is known that a monotone mapping is maximal if and only if, for , for every implies that . Next we provide an example to illustrate the concept of maximal monotone mapping.

Let be a monotone, -Lipschitz-continuous mapping and let be the normal cone to at ; that is, Define Then, is maximal monotone and if and only if ; see [34].

Assume that is a maximal monotone mapping. Let . In terms of Huang [35] (see also [36]), there holds the following property for the resolvent operator .

Lemma 14. is single-valued and firmly nonexpansive; that is, Consequently, is nonexpansive and monotone.

Lemma 15 (see [21]). Let be a maximal monotone mapping with . Then for any given , is a solution of problem (17) if and only if satisfies

Lemma 16 (see [36]). Let be a maximal monotone mapping with and let be a strongly monotone, continuous, and single-valued mapping. Then for each , the equation has a unique solution for .

Lemma 17 (see [21]). Let be a maximal monotone mapping with and be a monotone, continuous, and single-valued mapping. Then for each . In this case, is maximal monotone.

Lemma 18 (see [30]). Let be a nonempty closed convex subset of a real Hilbert space , and be a proper lower semicontinuous differentiable convex function. If is a solution the minimization problem then, In particular, if solves (OP), then

Lemma 19 (see [43]). Let be a sequence of nonnegative real numbers satisfying where ,, and are three real sequences satisfying the conditions:(i),;(ii);(iii),.Then .

3. Main Results

We introduce and analyze a new iterative algorithm by hybrid extragradient-like viscosity method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions of a finite family of variational inclusions for maximal monotone and inverse strong monotone mappings, and the set of fixed points of a countable family of nonexpansive mappings in a real Hilbert space. Under appropriate conditions imposed on the parameter sequences we will prove a strong convergence of the proposed algorithm.

Theorem 20. Let be a nonempty closed convex subset of a real Hilbert space . Let be an integer. Let be a bifunction from to satisfying (H1)–(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone and -inverse strongly monotone, respectively, where . Let be a sequence of nonexpansive self-mappings on and be a sequence in for some . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let and . Let be an -Lipschitzian mapping with . Let be the -mapping defined by (15) and be a -strongly positive bounded linear operator with . Assume that is nonempty. Suppose ,, and are three sequences in . Assume that(i) is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each ;(ii)for each , there exist a bounded subset and such that for any , (iii),, and (iv),, and satisfies (v). Given arbitrarily, then the sequence generated iteratively by converges strongly to provided is firmly nonexpansive, where is a unique solution of the VIP:

Proof. Since and , we may assume, without loss of generality, that . Since is a -strongly positive bounded linear operator on , we know that Observe that that is, is positive. It follows that Put for all and , where is the identity mapping on . Then we have that .
We divide the rest of the proof into several steps.
Step 1. We show that is bounded. Indeed, take arbitrarily. Since , is -inverse strongly monotone and , we have, for any , Since , , and is -inverse strongly monotone, where ,, by Lemma 14 we deduce that for each Combining (62) and (63), we have Set . Then from (56) and (64), we obtain Therefore, by Lemma 13 we have By induction, we get Therefore, is bounded and so are the sequences ,,,,,, and .
Step 2. We show that as .
Indeed, define Then from the definition of , we obtain It follows that From (15), since ,, and are all nonexpansive, we have where is a constant such that On the other hand, we estimate . Taking into consideration that and , we may assume, without loss of generality, that and . Utilizing Remark 3 and Lemma 14, we have where for some .
Note that Hence, from (70), (71), (74), and (75) it follows that where for some . So, utilizing Lemma 13 we obtain from (72) and (77) that where for some . Applying Lemma 19 to (78), we deduce from conditions (iii) and (v) that
Step 3. as . Indeed, utilizing Lemmas 4 and 5(b) we obtain from (56) and (64) that Utilizing Lemmas 4 and 13, we conclude from (56) and (80) that which leads to Since , , and , we deduce from the boundedness of ,, and that Note that So, it follows from (83) and that
Step 4. , and as . Indeed, for , we find that From (56), (63), and (86), we obtain which immediately implies that From and , we have Furthermore, from the firm nonexpansivity of , we have which implies that From (87) and (91), we have It follows that So, from (85), (89), and , we obtain Next we show that ,. Observe that Combining (87) and (95), we have which leads to Since , and , we obtain from (85) that By Lemmas 5(a) and (14), we obtain which implies Combining (87) and (100), we have which yields From (85), (98), and , we get From (103) we get By (94) and (104), we have together with (85), yields Note that Hence from (83) and (106) we have Also, observe that From (108), Remark 8, and the boundedness of we immediately get
Step 5. We show that , where is a unique solution of the VIP:
Indeed, it is clear that is -Lipschitzian. Note that It is clear that Hence by Lemma 12 we deduce from that is -strongly monotone. In the meantime, it is easy to see that is -Lipschitzian with the constant . Thus, there exists a unique solution in to the VIP (111). Equivalently, has a unique fixed point ; that is, .
First, we observe that there exists a subsequence of such that Since is bounded, there exists a subsequence of which converges weakly to some . Without loss of generality, we may assume that . From (94) and (103)–(105), we have that , and , where . By (110) we have that as . Then, by Lemma 10 we obtain (due to Lemma 9). Next, we prove that . As a matter of fact, since is -inverse strongly monotone, is a monotone and Lipschitz continuous mapping. It follows from Lemma 17 that is maximal monotone. Let ; that is, . Again, since ,,, we have that is, In terms of the monotonicity of , we get and hence In particular, Since (due to (103)) and (due to the Lipschitz continuity of ), we conclude from and,that It follows from the maximal monotonicity of that ; that is, . Therefore, .
Next, we show that . In fact, from , we know that From (H2) it follows that Replacing by , we have Put for all and . Then, from (123) we have Since as , we deduce from the Lipschitz continuity of and that and as . Further, from the monotonicity of , we have . So, from (H4), the weakly lower semicontinuity of and , we have From (H1), (H4), and (125) we also have and hence Letting , we have, for each , This implies that . Therefore, . This shows that . Consequently, from (111) and (114) we have
Step 6. Finally, we show that , where .
Indeed, in terms of Lemma 4 we have Utilizing Lemmas 4 and 13, we conclude from (130) that Note that . Hence, leads to . In addition, since and , we get from (129) Applying Lemma 19 to (131), we have that as . This completes the proof.

Corollary 21. Let be a nonempty closed convex subset of a real Hilbert space . Let be an integer. Let be a bifunction from to satisfying (H1)–(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone and -inverse strongly monotone, respectively, where . Let be a sequence of nonexpansive self-mappings on and be a sequence in for some . Let be a -strongly positive bounded linear operator. Given and , let . Suppose and . Let be an -Lipschitzian mapping with . Let be the -mapping defined by (15). Assume that is nonempty. Let ,, and be three sequences in . Assume that the conditions (i)–(v) of Theorem 20 hold. If is firmly nonexpansive, then for a given arbitrary , the sequence generated iteratively by converges strongly to , which solves the following optimization problem: where is the potential function of .

Proof. Given and , let . Then is a -Lipschitzian and -strongly monotone operator with positive constants and . Suppose and . In this case, it is easy from (56) to see that Then, for , all conditions of Theorem 20 are satisfied. Therefore, utilizing Theorem 20 we infer that converges strongly to , where is a unique solution of the VIP: Utilizing Lemma 18, we know that solves the following optimization problem: where is the potential function of .

Corollary 22. Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from to satisfying (H1)–(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone and -inverse strongly monotone, respectively, for . Let be a sequence of nonexpansive self-mappings on and be a sequence in for some . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let and . Let be an -Lipschitzian mapping with . Let be the -mapping defined by (15) and be a -strongly positive bounded linear operator with . Assume that is nonempty. Let ,, and be three sequences in . Assume that the conditions (i)–(v) of Theorem 20 hold. If is firmly nonexpansive, then for a given arbitrary , the sequence generated iteratively by converges strongly to , where is a unique solution of the VIP:

Corollary 23. Let be a nonempty closed convex subset of a real Hilbert space . Let be an integer. Let be a bifunction from to satisfying (H1)–(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let be -inverse strongly monotone, respectively, where . Let be a sequence of nonexpansive self-mappings on and be a sequence in for some . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let and . Let be an -Lipschitzian mapping with . Let be the -mapping defined by (15) and be a -strongly positive bounded linear operator with . Assume that is nonempty. Let ,, and be three sequences in . Assume that the conditions (i)–(v) of Theorem 20 hold, where is a bounded sequence such that for some . If is firmly nonexpansive, then for a given arbitrary , the sequence generated iteratively by converges strongly to , where is a unique solution of the VIP:

Proof. In Theorem 20, for all , is equivalent to Put . Then it follows that Observe that for all So, whenever for some , we obtain the desired result by using Theorem 20.

Let be a -strictly pseudocontractive mapping. For recent convergence result for strictly pseudocontractive mappings, we refer to [44]. Putting , we know that for all Note that Hence, we have for all Consequently, if is a -strictly pseudocontractive mapping, then the mapping is -inverse strongly monotone.

Corollary 24. Let be a nonempty closed convex subset of a real Hilbert space . Let be an integer. Let be a bifunction from to satisfying (H1)–(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be a -strictly pseudocontractive mapping and an -inverse strongly monotone mapping, respectively, where . Let be a sequence of nonexpansive self-mappings on and be a sequence in for some . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let and . Let be an -Lipschitzian mapping with . Let be the -mapping defined by (15) and be a -strongly positive bounded linear operator with . Assume that is nonempty, where . Let ,, and be three sequences in . Assume that the conditions (i)–(v) of Theorem 20 hold where . If is firmly nonexpansive, then for a given arbitrary , the sequence generated iteratively by converges strongly to , where is a unique solution of the VIP:

Proof. Since is a -strictly pseudocontractive mapping, the mapping is -inverse strongly monotone. In this case, put . Moreover, we obtain that So, from Theorem 20, we obtain the desired result.

Corollary 25. Let be a nonempty closed convex subset of a real Hilbert space . Let be an integer. Let be a bifunction from to satisfying (H1)–(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone and -inverse strongly monotone, respectively, where . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let and . Let be an -Lipschitzian mapping with . Let be a -strongly positive bounded linear operator with . Assume that is nonempty. Let ,, and be three sequences in . Assume that conditions (i)–(v) of Theorem 20 hold. Given arbitrarily, let the sequence be generated iteratively by If is firmly nonexpansive, then converges strongly to , where is a unique solution of the VIP:

Proof. Put for all integers and all . Then, the desired result follows from Theorem 20.

Corollary 26. Let be a nonempty closed convex subset of a real Hilbert space . Let be an integer. Let be a maximal monotone mapping and let be -inverse strongly monotone, where . Let be a sequence of nonexpansive self-mappings on and be a sequence in for some . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let and . Let be an -Lipschitzian mapping with . Let be the -mapping defined by (15) and be a -strongly positive bounded linear operator with . Assume that is nonempty. Let ,, and be three sequences in . Assume that,(i),, and (ii),;(iii). For a given arbitrary , let the sequence be generated iteratively by Then the sequence converges strongly to , where is a unique solution of the VIP:

Proof. Put , for all , for all and . Take for all . Then we get in Theorem 20 and the conclusion follows.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Ph.D. Program Foundation of Ministry of Education of China (20123127110002). This work was supported partly by the National Science Council of the Republic of China.

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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.

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