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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 907105, 19 pages
http://dx.doi.org/10.1155/2012/907105
Research Article

Hybrid Steepest Descent Viscosity Method for Triple Hierarchical Variational Inequalities

1Department of Mathematics, Shanghai Normal University and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
2Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India
3Center for General Education, Kaohsiung Medical University, Kaohsiung 80708, Taiwan

Received 21 June 2012; Accepted 26 July 2012

Academic Editor: Jen-Chih Yao

Copyright © 2012 L.-C. 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 consider a triple hierarchical variational inequality problem (in short, THVIP). By combining hybrid steepest descent method, viscosity method, and projection method, we propose an approximation method to compute the approximate solution of THVIP. We also study the strong convergence of the sequences generated by the proposed method to a solution of THVIP.

1. Introduction and Formulations

Let be a real Hilbert space whose inner product and norm are denoted by and , respectively. Let be a nonempty closed convex subset of and let be a nonlinear mapping. The variational inequality problem (for short, VIP) is to find such that The inequality (1.1) is called a variational inequality (in short, VI). If the mapping is a monotone operator, then the inequality (1.1) is called a monotone variational inequality. The theory of variational inequalities is well established in the literature because of its applications in science, engineering, social sciences, and so forth. For further detail on variational inequalities and their applications, we refer to [110] and the references therein.

It is well known that the VI (1.1) is equivalent to the fixed point equation where and is the metric projection of onto which assigns to each the only point in , denoted by , such that It is known that the fixed point methods can be implemented to find a solution of the VI (1.1) provided satisfies some conditions and is chosen appropriately. For instance, if is Lipschitzian and strongly monotone (i.e., for some ) and is small enough, then the mapping determined by the right-hand side of (1.2) is a contraction. Hence, the Banach contraction principle guarantees that the sequence of Picard iterates, given by , converges strongly to a unique solution of the VI (1.1).

Furthermore, it is also known that if is inverse strongly monotone (i.e., there is a constant such that ), then the mapping is an averaged mapping (namely, there are and a nonexpansive mapping such that ), then the sequence of Picard iterates, , converges weakly to a solution of the VI (1.1) (if such a solution exists).

In the last decade, the variational inequality problem is considered over the set of fixed points of a nonexpansive mapping; see, for example, [1115] and the reference therein. In particular, Moudafi and Maingé [12] and Xu [14] considered the following VIP over the set of fixed points of a nonexpansive mapping (i.e., ) with , where is another nonexpansive self-mapping on : find such that where we assume that . It is called hierarchical variational inequality problem (in short, HVIP). The HVIP (1.4) is equivalent to the following fixed point problem: Let denote the solution set of the HVIP (1.4). It has been shown in [12] that the HVIP (1.4) contains the HVIP considered in [15], monotone inclusion problem, convex programming problem, minimization problem over a set of fixed points, and so forth, as special cases; see, for example, [12, 14] and the references therein. In the recent past, several kinds of approximation methods for computing the approximate solutions of HVIP are proposed; see, for example, [1115] and the reference therein. Yamada [15] considered the so-called hybrid steepest descent method for solving the VIP over the set of fixed points of a nonexpansive mapping. Moudafi [11] proposed the viscosity approximation method of selecting a particular fixed point of a given nonexpansive mapping which is also a solution of a variational inequality problem. Subsequently, this method was developed by Xu [13]. Moudafi and Maingé [12] and Xu [14] further studied the viscosity method for HVIP.

Very recently, Iiduka [16, 17] considered a variational inequality problem with variational inequality constraint over the set of fixed points of a nonexpansive mapping. Since this problem has a triple structure in contrast with hierarchical constrained optimization problems or hierarchical fixed point problem, it is referred as triple hierarchical-constrained optimization problem (THCOP). He presented some examples of THCOP and developed iterative algorithms to find the solution of such a problem. The convergence analysis of the proposed algorithms is also studied in [16, 17]. Since the original problem is a variational inequality problem, in this paper, we call it the triple hierarchical variational inequality problem (THVIP).

Let be a -Lipschitzian and -strongly monotone operator with constants and , respectively. Let be -Lipschitzian with constant and let be nonexpansive mappings with . Let and , where . We consider the following triple hierarchical variational inequality problem (for short, THVIP): find such that where denotes the solution set of the hierarchical variational inequality problem (1.4) which is assumed to be nonempty.

Recall the function is said to be convex if for all and for all . It is said to be -strongly convex if there exists such that for all and for all , . It is easy to see that if is Fréchet differential and -strongly convex, then the gradient is -strongly monotone.

Now, we illustrate the triple hierarchical variational inequality problem (for short, THVIP) by an example which is closely related to [17, Example 3.1].

Example 1.1. Let be a nonempty closed convex subset of a real Hilbert space and let be -Lipschitz continuous with constant . Suppose that is a convex function with a -Lipschitz continuous gradient, is a convex function with a -Lipschitz continuous gradient, and is an -strongly convex function with an -Lipschitz continuous gradient. Define , and . Then are nonexpansive mappings with and , and is -Lipschitzian and -strongly monotone with and . Assume that . Then for the solution set of the hierarchical variational inequality problem (for short, HVIP), we have When and , where , we have where . In particular, when , we have In this case, when , the following triple hierarchical variational inequality problem (for short, THVIP): find such that reduces to the following THVIP: find such that

In this paper, by combining hybrid steepest descent method, viscosity method, and projection method, we propose an approximation method to compute the approximate solution of THVIP. We also study the strong convergence of the sequences generated by the proposed method to a solution of THVIP. The results of this paper extend and generalize the results given in [12, 14] and several others given in the literature.

2. Preliminaries

Throughout the paper, unless other specified, we assume that is a nonempty closed convex subset of a real Hilbert space . We use and to denote strong and weak convergence to of the sequence , respectively.

Recall that a mapping is called -Lipschitzian on if there exists such that . In particular, if then is called a contraction on ; if then is called a nonexpansive mapping on .

We present some basic facts and results which will be used in the sequel.

Lemma 2.1 (see [18]). Let be a real Hilbert space. Then, for all and .

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

Lemma 2.2. Let be an -Lipschitzian mapping with constant , and let be a -Lipschitzian and -strongly monotone operator with constants and , respectively. Then, for , That is, is strongly monotone with constant .

Lemma 2.3 (see [18, Demiclosedness Principle]). Let be a nonexpansive mapping with . If weakly converges to in and if strongly converges to , then ; in particular, if , then .

In the following lemma, we present some properties of the projection.

Lemma 2.4. Given and . Then (a) if and only if there holds the relation: (b) if and only if there holds the relation: (c) is nonexpansive and monotone, that is,

Lemma 2.5. Let be a real Hilbert space. Then, for all ,

The following lemma plays a key role in proving the main results of this paper.

Lemma 2.6 (see [19, Lemma ]). Let and . Let be an operator on such that, for some constants , is -Lipschitzian and -strongly monotone. Associating with a nonexpansive mapping , define the mapping by Then is a contraction provided , that is, where .

Remark 2.7. If , where is the identity operator of . Then and hence . Also, if , then it is easy to see that In particular, whenever , we have .

3. Approximation Methods and Convergence Results

Let be a -Lipschitzian and -strongly monotone operator with constants and , respectively. Let be a -Lipschitzian mapping with constant and let be a nonexpansive mapping with . Let and , where . We consider the hierarchical variational inequality problem (in short, HVIP) of finding such that We denote by the solution set of the HVIP (3.1).

When , , and are a nonexpansive self-mapping on , the HVIP (3.1) reduces to the following hierarchical variational inequality problem of finding such that It is considered and studied in [12, 14].

We consider a mapping on defined by It is easy to see that is a nonexpansive mapping. Indeed, we have Since , it is known that is nonexpansive on .

Proposition 3.1. Let be a -Lipschitzian and -strongly monotone operator with constants and , respectively. Let be -Lipschitzian with constant and let be nonexpansive with . Let and , where . Let and be a fixed point of the mapping , that is, . Assume remains bounded as , then the following conclusions hold. (a)The solution set of the HVIP (3.1) is nonempty and each weak limit point (as ) of solves the HVIP (3.1).(b)If is strictly monotone, then the net converges weakly to the (unique) solution of the HVIP (3.1).(c)If is strongly monotone (e.g., ), then the net converges strongly to a solution of the HVIP (3.1).

Proof. Let be the set of all weak accumulation points of as ; that is, Then, because is bounded.
To prove (a), we notice that the boundedness of implies that and It thus follows from Lemma 2.3 that . Take a fixed arbitrarily and set Then and Since is the metric projection from onto , utilizing Lemma 2.4, we have Hence, utilizing -Lipschitzian property of , we get It follows that Note that and Since , we have . Thus, utilizing the -strong monotonicity of and -Lipschitzian property of , we know that is monotone because the following inequality holds: Consequently, we have This together with (3.11) implies that Now, if and if is such that , then we obtain from (3.15) and that Replacing by in (3.16), where and , we get Letting yields Consequently, .
To see (b), we assume that is another null sequence in such that . Then and by replacing by in (3.18), we get By interchanging and , we get Adding up (3.19) and (3.20) yields So the strict monotonicity of implies that and converges weakly.
Finally, to prove (c), we observe that the strong monotonicity of and (3.11) implies that where is the strong monotonicity constant of ; that is, A straightforward consequence of (3.22) is that if and if for some null sequence in , then we must have . This shows that is relatively compact in the norm topology, and each of its limit points solves the HVIP (3.1). Finally repeating the argument in the weak convergence case of (b), we see that can have exactly one limit point; hence, converges in norm.

Corollary 3.2 (see [14, Proposition 3.1]). Let be nonexpansive mappings with . Let and be a fixed point of the mapping , that is, . Assume remains bounded as , then the following conclusions hold. (a)The solution set of the HVIP (1.4) is nonempty and each weak limit point (as ) of solves the HVIP (1.4).(b)If is strictly monotone, then the net converges weakly to the solution of the HVIP (1.4).(c)If is strongly monotone (e.g., is a contraction), then the net converges strongly to a solution of the HVIP (1.4).

Now by combining hybrid steepest descent method, viscosity method, and projection method, we define, for each , two mappings and by It is easy to see that is a nonexpansive self-mapping on . Moreover, utilizing Lemma 2.6, we can see that is a -contraction. Indeed, observe that Let be the unique fixed point of . Namely, is the unique solution in to the following:

Theorem 3.3. Let be a -Lipschitzian and -strongly monotone operator with constants and , respectively. Let be -Lipschitzian with constant and let be nonexpansive with . Let and , where . For each , let be the unique solution to (3.26). Assume also that, for each , is nonempty (but not necessarily bounded), and the following assumption holds: Then the strong exists for each . Moreover, the strong exists and solves the THVIP (1.6). Hence, for any null sequence in , there is another null sequence in such that in norm, as .

Proof. Observe that the condition and the fact imply that Therefore, is a strongly monotone operator with constant . Since, for each fixed , the fixed point set of is nonempty, we can apply Proposition 3.1 (c) to get that exists in and solves the following hierarchical variational inequality problem of finding such that Equivalently, , where is the metric projection from onto .
Utilizing the strong monotonicity of , we conclude from (3.29) that for each , Hence, This implies that The inequality (3.32) is yet to imply the boundedness of since may depend on . However, since the solution set of the HVIP (1.4) is nonempty, we can take (an arbitrary) and use assumption to find such that in norm as . Hence, must be bounded (as ). The inequality (3.32) implies and this is sufficient to ensure that is bounded (as closes ).
Now, the boundedness of allows us to apply Corollary 3.2 (a) to conclude that every weak limit point of belongs to the solution set of the HVIP (1.4). Then (3.31) guarantees that every such weak limit point of is also a strong limit point of . Indeed, if is a null sequence in and if , then . By assumption , we get a sequence such that for all and in norm. From (3.31) we derive However, since in norm and weakly, and we find that the right-hand side of (3.34) tends to zero. Hence, in norm.
So to prove the strong convergence of the entire net , it remains to prove that can have only one strong limit point. Let and be two strong limit points of and assume that and both in norm, where and are null sequences in . It remains to verify that .
Since , by assumption , we can find such that in norm as . The HVIP (3.29) implies Taking the limit as yields Similarly, we have Adding up (3.36) and (3.37) gives Utilizing Lemma 2.2, we know that is strongly monotone with constant . Hence, from (3.38) it follows that and so converges in norm to (say) .
Now, for any , since by assumption , we can find such that in norm, (3.29) then implies which in turns implies that is, , the unique fixed point of the contraction . Finally, for any null sequence in , using a diagonalization argument (cf. [1]), we can find another null sequence in such that in norm, as .

Remark 3.4. Theorem 3.3 shows that for any null sequence in , there is another null sequence in such that in norm, as , and is a solution to the HVIP (3.40). Theorem 3.3 is the main result of the present paper in which we improve the result of Moudafi and Maingé [12] by proving that actually converges strongly and also by removing the boundedness of the set . Our proof is different from that of [12]. In the meantime, Theorem 3.3 covers [14, Theorem 3.2] as a special case. For instance, whenever we put , , , and let the -Lipschitzian mapping be a (self-) contraction with coefficient , our Theorem 3.3 reduces to [14, Theorem 3.2].
Now, we present a general result. We show that as long as is taken so that (i.e., ), then in norm, and moreover, solves the HVIP (3.40) on the larger set (i.e., is the unique fixed point in of the contraction ), without the assumption . However, for such a general choice of , this solution may differ from the solution of the HVIP (3.40) on the smaller set (i.e., is the unique fixed point in of the contraction ). We will verify this by taking for simplicity (the argument, however, works for any net in such that ).

Theorem 3.5. Let be a -Lipschitzian and -strongly monotone operator with constants and , respectively. Let be -Lipschitzian with constant and let be nonexpansive with . Let and , where . For each , let be the unique solution in to the following: Then, as , converges in norm to the solution of the HVIP of finding such that equivalently, .

Proof. Write (instead of ) for ; then Take a fixed arbitrarily and put Then from (3.41) we get . Since is the metric projection from onto , we have Also, observe that Utilizing Lemma 2.6, we deduce from (3.41) that It follows that, for any fixed , This implies that In particular, is bounded, and from (3.41), we further get as . Lemma 2.3 ensures that every weak limit point, as , of is a fixed point of . Going back to (3.48), we find that each weak limit point of is actually a strong limit point of because So to prove the strong convergence of , we need only to show the uniqueness of strong limit points of . Assuming and are null sequences in such that and , both in norm. Observing that (3.41) implies where and . Utilizing Lemmas 2.4 and 2.6, we deduce from the monotonicity of that for any fixed , In particular, we have from that for any fixed , So letting yields Repeating the above argument obtains Adding up (3.55) and (3.56) gives us that The strong monotonicity of (Lemma 2.2) then implies . Finally, taking the limit as in (3.53) and letting , we conclude immediately that solves the variational inequality of finding such that Equivalently, . The proof is therefore complete.

Remark 3.6. If and have a common fixed point, then it is not hard to see that for all . Indeed, it suffices to show the inclusion . Let . Then for any fixed we have This implies ; that is . Furthermore, it is clear that . In this case, assumption is reduced to the assumption . Therefore, assumption is equivalent to the assumption .

Corollary 3.7. Let be a -Lipschitzian and -strongly monotone operator with constants and , respectively. Let be -Lipschitzian with constant and let be nonexpansive with . Let and , where . For each , let be the unique solution to (3.26). Then the conclusion of Theorem 3.3 holds. Namely, the strong exists for each fixed , and moreover, the strong exists and solves the THVIP (1.6).

Proof. Since is independent of ; the in both (3.31) and (3.32) does not depend on . Hence, it is immediately clear that is bounded, which then implies via (3.31) that every weak accumulation point of is also a strong accumulation point of . Eventually, converges in norm as shown in the final part of the proof of Theorem 3.5.

Remark 3.8. Theorems 3.3 and 3.5 improve and extend [14, Theorems 3.2 and 3.4], respectively, in the following ways. (a)The contraction mapping in [14, Theorems 3.2 and 3.4] is extended to the case of (possibly nonself) -Lipschitzian mapping from a nonempty closed convex subset to .(b)The convex combination of (self) contraction mapping and nonexpansive mapping in the implicit scheme in [14, Theorem 3.2] is extended to the linear combination of (possibly nonself) -Lipschitzian mapping and hybrid steepest descent method involving . In particular, if , Theorem 3.5 is an extension of [14, Theorem 3.4].(c)In order to guarantee that the net generated by the implicit scheme still lies in , the implicit scheme in [14, Theorem 3.2] is extended to develop our new implicit scheme (3.26) by virtue of the projection method. In particular, if , [14, Theorem 3.4] is extended to the corresponding case in our Theorem 3.5.(d)The new technique of argument is applied to derive our Theorems 3.3 and 3.5. For instance, the characteristic properties (Lemma 2.4) of the metric projection play a key role in proving the strong convergence of the nets and in our Theorems 3.3 and 3.5, respectively.(e)If we put and and let be a contractive self-mapping on with coefficient , then our Theorems 3.3 and 3.5 reduce to [14, Theorems 3.2 and 3.4], respectively. Thus, our Theorems 3.3 and 3.5 cover [14, Theorems 3.2 and 3.4] as special cases, respectively.

Acknowledgments

In this research, the first author was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), Leading Academic Discipline Project of Shanghai Normal University (DZL707), Science and Technology Commission of Shanghai Municipality Grant (075105118), and Shanghai Leading Academic Discipline Project (S30405). The second author was supported by the D.S.T. Research Project no. SR/S4/MS:719/11. Third author was partially supported by Grant NSC 101-2115-M-037-001.

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