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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

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, [118] 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