Abstract and Applied Analysis

Volume 2013 (2013), Article ID 718624, 17 pages

http://dx.doi.org/10.1155/2013/718624

Research Article

Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

Zhao-Rong Kong,¹ Lu-Chuan Ceng,² Qamrul Hasan Ansari,³ and Chin-Tzong Pang⁴

¹Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

²Department of Mathematics, Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China

³Department of Mathematics, Aligarh Muslim University, Aligarh 202 00, India

⁴Department of Information Management, Yuan Ze University, Chung Li 32003, Taiwan

Received 26 December 2012; Accepted 25 January 2013

Academic Editor: Jen-Chih Yao

Copyright © 2013 Zhao-Rong Kong 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 (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI.

1. Introduction and Formulations

Throughout the paper, we will adopt the following terminology and notations. is a real Hilbert space, whose inner product and norm are denoted by and , respectively. The strong (resp., weak) convergence of the sequence to will be denoted by (resp., ). We shall use to denote the weak -limit set of the sequence ; namely,

Throughout the paper, unless otherwise specified, we assume that is a nonempty, closed, and convex subset of a Hilbert space and is a nonlinear mapping. The variational inequality problem (VIP) on is defined as follows:

We denote by the set of solutions of VIP. In particular, if is the set of fixed points of a nonexpansive mapping , denoted by , then (VIP) is called a hierarchical variational inequality problem (HVIP), also known as a hierarchical fixed point problem (HFPP). If we replace the mapping by , where is the identity mapping and is a nonexpansive mapping (not necessarily with fixed points), then the VIP becomes as follows:

This problem, first introduced and studied in [1, 2], is called a hierarchical variational inequality problem, also known as a hierarchical fixed point problem. Observe that if has fixed points, then they are solutions of VIP (3). It is worth mentioning that many practical problems can be written in the form of a hierarchical variational inequality problem; see, for example, [1–18] and the references therein.

If is a -contraction with coefficient (i.e., for some ), then the set of solutions of VIP (3) is a singleton, and it is well known as a viscosity problem, which was first introduced by Moudafi [19] and then developed by Xu [20]. It is not hard to verify that solving VIP (3) is equivalent to finding a fixed point of the nonexpansive mapping , where is the metric projection on the closed and convex set .

Let be -Lipschitzian and -strongly monotone, where , are constants, that is, for all

A mapping is called -strictly pseudocontractive if there exists a constant such that

see [21] for more details. We denote by the fixed point set of ; that is, .

We introduce and consider the following triple hierarchical variational inequality problem (THVIP).

Problem I. Let be -Lipschitzian and -strongly monotone on the nonempty, closed, and convex subset of , where and are positive constants. Let be a -contraction with coefficient , be a nonexpansive mapping, and, for , be -strictly pseudocontractive mapping with . Let and , where . Then the objective is to find such that

where denotes the solution set of the following hierarchical variational inequality problem (HVIP) of finding such that

In particular, whenever a nonexpansive mapping, and an identity mapping, Problem I reduces to the THVIP considered by Ceng et al. [22]. By combining the regularization method, the hybrid steepest-descent method, and the projection method, they proposed an iterative algorithm that generates a sequence via the explicit scheme and studied the convergence analysis of the sequences generated by the proposed method.

We consider and study the following triple hierarchical variational inequality problem.

Problem II. Let be -Lipschitzian and -strongly monotone on the nonempty, closed, and convex subset of , where and are positive constants. Let be a monotone and -Lipschitzian mapping, be a -contraction with coefficient , be a nonexpansive mapping, and be a -strictly pseudocontractive mapping with . Let and , where . Then the objective is to find such that

where denotes the solution set of the following hierarchical variational inequality problem (HVIP) of finding such that

We remark that Problem II is a generalization of Problem I. Indeed, in Problem II we put and , where is a -strictly pseudocontractive mapping for . Then from the definition of strictly pseudocontractive mapping, it follows that

It is clear that the mapping is -inverse strongly monotone. Taking , we know that is a monotone and -Lipschitzian mapping. In this case, . Therefore, Problem II reduces to Problem I.

Motivated and inspired by Korpelevich’s extragradient method [23] and the iterative method proposed in [22], we propose the following multistep hybrid extragradient method for solving Problem II.

Algorithm I. Let be -Lipschitzian and -strongly monotone on the nonempty, closed, and convex subset of , be a monotone and -Lipschitzian mapping, be a -contraction with coefficient , be a nonexpansive mapping, and be a -strictly pseudocontractive mapping. Let , , and , , , , , and , where . The sequence is generated by the following iterative scheme:

where for all . In particular, if , then (11) reduces to the following iterative scheme:

Further, if , then (11) reduces to the following iterative scheme:

moreover, if , then (12) reduces to the following iterative scheme:

We prove that under appropriate conditions the sequence generated by Algorithm I converges strongly to a unique solution of Problem II. Our result improves and extends Theorem  4.1 in [22] in the following aspects.

(a)Problem II generalizes Problem I from the fixed point set of a nonexpansive mapping to the intersection of the fixed point set of a strictly pseudocontractive mapping and the solution set of VIP (2).(b)The Korpelevich extragradient algorithm is extended to develop the multistep hybrid extragradient algorithm (i.e., Algorithm I) for solving Problem II by virtue of the iterative schemes in Theorem  4.1 in [22].(c)The strong convergence of the sequence generated by Algorithm I holds under the lack of the same restrictions as those in Theorem  4.1 in [22].(d)The boundedness requirement of the sequence in Theorem  4.1 in [22] is replaced by the boundedness requirement of the sequence .

We also consider and study the multistep hybrid extragradient algorithm (i.e., Algorithm I) for solving the following system of hierarchical variational inequalities (SHVI).

Problem III. Let be -Lipschitzian and -strongly monotone on the nonempty, closed, and convex subset of , where and are positive constants. Let be a monotone and -Lipschitzian mapping, be a -contraction with coefficient , be a nonexpansive mapping, and be a -strictly pseudocontractive mapping with . Let and , where . Then the objective is to find such that

In particular, if and where is -strictly pseudocontractive for , Problem III reduces to the following Problem IV.

Problem IV. Let be -Lipschitzian and -strongly monotone on the nonempty, closed, and convex subset of , where and are positive constants. Let be a -contraction with coefficient , be a nonexpansive mapping, and, for , be -strictly pseudocontractive mapping with . Let and , where . Then the objective is to find such that

We prove that under very mild conditions the sequence generated by Algorithm I converges strongly to a unique solution of Problem III.

2. Preliminaries

Let be a nonempty, closed, and convex subset of and be a (possibly nonself) -contraction mapping with coefficient ; that is, there exists a constant such that , for all . Now we present some known results and definitions which will be used in the sequel.

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

The following properties of projections are useful and pertinent to our purpose.

Proposition 1 (see [21]). Given any and . One has

(i) ,(ii) , for all ;(iii), for all , which hence implies that is nonexpansive and monotone.

Definition 2. A mapping is said to be

(a)nonexpansive if

(b)firmly nonexpansive if is nonexpansive, or, equivalently,

alternatively, is firmly nonexpansive if and only if can be expressed as

where is nonexpansive; projections are firmly nonexpansive.

Definition 3. Let be a nonlinear operator whose domain is and whose range is .

(a) is said to be monotone if

(b)Given a number , is said to be -strongly monotone if

(c)Given a number , is said to be -inverse strongly monotone (-ism) if

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 cocoercive) operators have been applied widely in solving practical problems in various fields, for instance, in traffic assignment problems; see [24, 25].

Definition 4. 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 are nonexpansive. More precisely, when the last equality holds, we say that is -averaged. Thus firmly nonexpansive mappings (in particular, projections) are -averaged maps.

Proposition 5 (see [26]). Let be a given mapping.

(i) is nonexpansive if and only if the complement is -ism.(ii)If is -ism, then,    is -ism.(iii) is averaged if and only if the complement is -ism for some . Indeed,    is -averaged if and only if is -ism.

Proposition 6 (see [26, 27]). 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 .

On the other hand, it is clear that, in a real Hilbert space , is -strictly pseudocontractive if and only if there holds the following inequality:

This immediately implies that if is a -strictly pseudocontractive mapping, then is -inverse strongly monotone; for further detail, we refer to [21] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings. The so-called demiclosedness principle for strict pseudocontractive mappings in the following lemma will often be used.

Lemma 7 (see [21, Proposition 2.1]). Let be a nonempty closed convex subset of a real Hilbert space and be a mapping.

(i)If is a -strictly pseudocontractive mapping, then satisfies the Lipschitz condition:

(ii)If is a -strictly pseudocontractive mapping, then the mapping is semiclosed at ; that is, if is a sequence in such that and , then .(iii)If is a -(quasi-)strict pseudocontraction, then the fixed point set of is closed and convex so that the projection is well defined.

The following lemma plays a key role in proving strong convergence of the sequences generated by our algorithms.

Lemma 8 (see [28]). Let be a sequence of nonnegative real numbers satisfying the property:

where and are such that

(i);(ii)either or ;(iii) where , for all .

Then, .

The following lemma is not hard to prove.

Lemma 9 (see [20]). Let be a -contraction with and be a nonexpansive mapping. Then

(i) is -strongly monotone:

(ii) is monotone:

The following lemma plays an important role in proving strong convergence of the sequences generated by our algorithm.

Lemma 10 (see [29]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a -strictly pseudo-contractive mapping. Let and be two nonnegative real numbers such that . Then

Lemma 11 (see [30, Lemma 3.1]). Let be a number in and let . 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 , that is,

where . In particular, if   is the identity mapping , then

The following lemma appears implicitly in Reineermann [31].

Lemma 12. Let be a Hilbert space. Then

The following lemma is not difficult to prove.

Lemma 13 (see [32]). Let and be a sequence of nonnegative real numbers and a sequence of real numbers, respectively, such that and . Then .

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 and -Lipschitzian mapping, and let be the normal cone to at , that is, , for all. Define

It is known that in this case is maximal monotone, and if and only if ; see [33].

3. Main Results

We are now in a position to present the convergence analysis of Algorithm I for solving Problem II.

Theorem 14. Let be a -Lipschitzian and -strongly monotone operator with constants , respectively, be a -inverse strongly monotone mapping, be a -contraction with coefficient , be a nonexpansive mapping, and be a -strictly pseudocontractive mapping. Let and , where . Assume that the solution set of the HVIP (9) is nonempty and that the following conditions hold for the sequences and :

(C1) and ;(C2);(C3) and   for all ;(C4) and ;(C5) and ;(C6)there are constants satisfying for each ;(C7).

One has the following.

(a)If is the sequence generated by scheme (11) and is bounded, then converges strongly to the point which is a unique solution of Problem II provided that .(b)If is the sequence generated by the scheme (12) and is bounded, then converges strongly to a unique solution of the following VIP provided that :

Proof. We treat only case (a); that is, the sequence is generated by the scheme (11). Obviously, from the condition it follows that . In addition, in terms of conditions (C2) and (C4), without loss of generality, we may assume that for some , for some .

First of all, we observe (see, e.g., [34]) that and are nonexpansive for all .

Next we divide the remainder of the proof into several steps.

Step  1 ( is bounded). Indeed, take a fixed arbitrarily. Then, we get and for . From (11), it follows that

Put for each . Then, by Proposition 1 (ii), we have

Further, by Proposition 1 (i), we have

So, we obtain

Since , utilizing Lemmas 10 and 12, from (37) and the last inequality, we conclude that

Noticing the boundedness of , we get for some . Moreover, utilizing Lemma 11 we have from (11)

So, calling

we claim that

Indeed, when , it is clear from (42) that (44) is valid, that is,

Assume that (44) is valid for , that is,

Then from (42) and (46) it follows that

This shows that (44) is also valid for . Hence, by induction we derive the claim. Consequently, is bounded (due to ) and so are , and .

Step  2 . Indeed, from (11) and (41), it follows that

where . This together with and implies that

Note that . Hence, taking into account the boundedness of and , we deduce from (49) that

Furthermore, we obtain

which together with (50) implies that

So, from (11) we get

which together with implies that

Note that

This together with (50)–(54) implies that

Step  3 . Indeed, since is -Lipschitz continuous, we have

As is bounded, there is a subsequence of that converges weakly to some . By the same argument as that in [34], we can obtain that from which it follows that

Step  4 . Indeed, we first note that and

It is clear that

Hence, it follows from that is monotone. Putting

and noticing from (11)

we obtain

Set

It can be easily seen from (63) that

This yields that, for all (noticing ),

In (66), the first term is nonnegative due to Proposition 1, and the fourth term is also nonnegative due to the monotonicity of . We, therefore, deduce from (66) that (noticing again )

Note that

Hence it follows from that

Also, since (due to ), and is bounded by Step 1 which implies that is bounded, we obtain from (67) that

This suffices to guarantee that ; namely, every weak limit point of solves the HVIP (9). As a matter of fact, if for some subsequence of , then we deduce from (70) that

that is,

In addition, note that by Step 3. Since is monotone and Lipschitz continuous and is nonempty, closed, and convex, by the Minty lemma [1] the last inequality is equivalent to (9). Thus, we get .

Step  5 ( converges strongly to a unique solution of Problem II.) Indeed, we now take a subsequence of satisfying

Without loss of generality, we may further assume that ; then as we just proved. Since is a solution of the THVIP (8), we get

From (11) and (41), it follows that (noticing that and )

where . It turns out that

However, from and condition (C6) we obtain that

On the other hand, from (41) we have

which, together with , implies that

That is, . Observe that

Since , utilizing Lemma 10 we get

That is, . Taking into account , we have . Furthermore, utilizing Lemma 7 (i), we have

where . Hence, for a big enough constant , from (77), we have

Combining (76)–(83), we get

where and

Now condition (C5) implies that . Moreover, since  +  +   , conditions (C7) and (74) imply that

Therefore, we can apply Lemma 8 to (84) to conclude that . The proof of part (a) is complete. It is easy to see that part (b) now becomes a straightforward consequence of part (a) since, if , THVIP (8) reduces to the VIP in part (b). This completes the proof.

Next we consider a special case of Problem II. In Problem II, put and . In this case, the objective is to find such that

where denotes the solution set of the following hierarchical variational inequality problem (HVIP) of finding such that

Corollary 15. Let be a -inverse strongly monotone mapping, be a -contraction with coefficient , be a nonexpansive mapping, and be a -strictly pseudocontractive mapping. Assume that the solution set of the HVIP (88) is nonempty and that the following conditions hold for the sequences , and :

(C1) and ;(C2);(C3) and for all ;(C4) and ;(C5) and ;(C6)there are constants satisfying for each ;(C7).

One has

(a)If is the sequence generated by the iterative scheme

and is bounded, then converges strongly to the point which is a unique solution of THVIP (87) provided that .(b)If is the sequence generated by the iterative scheme

and is bounded, then converges strongly to a unique solution of the following VIP provided that :

that is, is the minimum-norm solution of HVIP (88).

Furthermore, applying Theorem 14 to Problem I, we get the result as below.

Corollary 16. Let be a -Lipschitzian and -strongly monotone operator with constants , respectively, be a -contraction with coefficient , be a nonexpansive mapping, and be a -strictly pseudocontractive mapping for . Let and , where . Assume that the solution set of the HVIP in Problem I is nonempty and that the following conditions hold for the sequences and :

(C1);(C2) and for all ;(C3) and ;(C4) and ;(C5)there are constants satisfying for each ;(C6).

One has the following.

(a)If is the sequence generated by

such that is bounded, then converges strongly to the point which is a unique solution of Problem I provided that .(b)If is the sequence generated by

such that is bounded, then converges strongly to a unique solution of the following VIP provided that :

Proof. In Theorem 14, we put and where is -strictly pseudocontractive for . Taking and for all , we know that is a -inverse strongly monotone mapping such that . In the scheme (11), we have

Utilizing Theorem 14, we obtain desired result.

On the other hand, we also derive the following strong convergence result of Algorithm I for finding a unique solution of Problem III.

Theorem 17. Let be a -Lipschitzian and -strongly monotone operator with constants , respectively, be a -inverse strongly monotone mapping, be a -contraction with coefficient , be a nonexpansive mapping, and be a -strictly pseudocontractive mapping. Let and , where . Assume that Problem III has a solution and that the following conditions hold for the sequences and :

(C1);(C2);(C3) and   for all ;(C4) and ;(C5);(C6) and .

One has the following.

(a)If is the sequence generated by the scheme (11) and is bounded, then converges strongly to a unique solution of Problem III provided that .(b)If is the sequence generated by the scheme (12) and is bounded, then converges strongly to a unique solution of the following system of variational inequalities provided that :

Proof. We treat only case (a); that is, the sequence is generated by scheme (11). First of all, it is seen easily that and . Hence it follows from the -contractiveness of and that is -strongly monotone and Lipschitz continuous. So, there exists a unique solution of the following VIP:

Consequently, it is easy to see that Problem III has a unique solution . In addition, taking into account condition (C5), without loss of generality we may assume that for some .

Next we divide the rest of the proof into several steps.

Step  1 ( is bounded). Indeed, repeating the same argument as in Step 1 of the proof of Theorem 14 we can derive the claim.

Step  2 . Indeed, repeating the same argument as in Step 2 of the proof of Theorem 14 we can derive the claim.

Step  3 . Indeed, repeating the same argument as in Step 3 of the proof of Theorem 14 we can derive the claim.

Step  4 ( converges strongly to a unique solution of Problem III). Indeed, according to , we can take a subsequence of satisfying

Without loss of generality, we may further assume that ; then due to Step 3. Since is a solution of Problem III, we get

Repeating the same argument as that of (99), we have

Repeating the same argument as that of (76) in the proof of Theorem 14, we obtain

Put and

Then (101) is rewritten as

In terms of conditions (C5) and (C6), we conclude from that

Note that

Consequently, utilizing Lemma 13 we obtain that

So, this together with Lemma 8 leads to . The proof is complete.

Utilizing Theorem 17 we immediately derive the following result.

Corollary 18. Let be a -Lipschitzian and -strongly monotone operator with constants , respectively, be a -inverse strongly monotone mapping, be a -contraction with coefficient , and be a -strictly pseudocontractive mapping such that . Let and , where . Assume that the following conditions hold for the sequences and :

(C1);(C2);(C3) and   for all ;(C4) and ;(C5) and .

One has the following.

(a)If is the sequence generated by the scheme (13) and is bounded, then converges strongly to a unique solution of the following VIP provided that :

(b)If is the sequence generated by the scheme (14), then converges strongly to a unique solution of the following VIP provided that :

Proof. In Theorem 17, putting , we know that the iterative scheme (11) reduces to (13) since there holds for any

In this case, the SVI (15) with VIP constraint is equivalent to the VIP (107). Thus, utilizing Theorem 17 (a) we obtain the desired conclusion (a). As for the conclusion (b), we immediately derive it from and Theorem 17 (b).

In addition, applying Theorem 17 to Problem IV, we derive the result as below.

Corollary 19. Let be a -Lipschitzian and -strongly monotone operator with constants , respectively, be a -contraction with coefficient , be a nonexpansive mapping, and be a -strictly pseudocontractive mapping for . Let and , where . Assume that Problem IV has a solution and that the following conditions hold for the sequences and :

(C1);(C2) and for all ;(C3) and ;(C4);(C5) and .

One has the following.

(a)If is the sequence generated by the scheme in Corollary 16 (a) such that is bounded, then converges strongly to a unique solution of Problem IV provided that .(b)If is the sequence generated by the scheme in Corollary 16 (b) such that is bounded, then converges strongly to a unique solution of the following system of variational inequalities provided that :

Acknowledgments

In this research, first two authors were partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Leading Academic Discipline Project of Shanghai Normal University (DZL707). The third author was partially supported by a research project no. SR/S4/MS: 719/11 of Department of Science and Technology, Govternment of India. The fourth author was supported in part by the National Science Council of the Republic of China.

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