# An Implicit Extragradient Method for Hierarchical Variational Inequalities

- Yonghong Yao
^{1}Email author and - YeongCheng Liou
^{2}

**2011**:697248

https://doi.org/10.1155/2011/697248

© Yonghong Yao and Yeong Cheng Liou. 2011

**Received: **20 September 2010

**Accepted: **7 November 2010

**Published: **28 November 2010

## Abstract

As a well-known numerical method, the extragradient method solves numerically the variational inequality of finding such that , for all . In this paper, we devote to solve the following hierarchical variational inequality Find such that , for all . We first suggest and analyze an implicit extragradient method for solving the hierarchical variational inequality . It is shown that the net defined by the suggested implicit extragradient method converges strongly to the unique solution of in Hilbert spaces. As a special case, we obtain the minimum norm solution of the variational inequality .

## 1. Introduction

where is the metric projection from onto , is a monotone operator, and is a constant. Korpelevich [4] proved that the sequence converges strongly to a solution of . Note that the setting of the space is Euclid space .

For this purpose, in this paper, we first suggest and analyze an implicit extragradient method. It is shown that the net defined by this implicit extragradient method converges strongly to the unique solution of in Hilbert spaces. As a special case, we obtain the minimum norm solution of the variational inequality .

## 2. Preliminaries

We denote
by
, where
is called the *metric projection* of
onto
. The metric projection
of
onto
has the following basic properties:

We need the following lemmas for proving our main result.

Lemma 2.1 (see [13]).

In particular, if , then is nonexpansive.

Lemma 2.2 (see [32]).

## 3. Main Result

In this section, we will introduce our implicit extragradient algorithm and show its strong convergence to the unique solution of .

Algorithm 1.

*Let*

*be a closed convex subset of a real Hilbert space*

*. Let*

*be an*

*-inverse strongly monotone mapping. Let*

*be a (nonself) contraction with coefficient*

*. For any*

*, define a net*

*as follows:*

Remark 3.1.

From Lemma 2.1, we know that if , the mapping is nonexpansive.

This shows that the mapping is a contraction. By Banach contractive mapping principle, we immediately deduce that the net (3.1) is well defined.

Theorem 3.2.

Suppose the solution set of is nonempty. Then the net generated by the implicit extragradient method (3.1) converges in norm, as , to the unique solution of the hierarchical variational inequality . In particular, if one takes that , then the net defined by (3.2) converges in norm, as , to the minimum-norm solution of the variational inequality .

Proof.

Therefore, is bounded and so are , . Since is -inverse strongly monotone, it is -Lipschitz continuous. Consequently, and are also bounded.

Next, we show that the net is relatively norm-compact as . Assume that is such that as . Put and .

Since is bounded, without loss of generality, we may assume that converges weakly to a point . Since , we have . Hence, also converges weakly to the same point .

Noting that , , and is Lipschitz continuous, we obtain . Since is maximal monotone, we have and hence .

Consequently, the weak convergence of and to actually implies that strongly. This has proved the relative norm-compactness of the net as .

Therefore, . That is, is the unique solution in of the contraction . Clearly this is sufficient to conclude that the entire net converges in norm to as .

Therefore, is the minimum-norm solution of .This completes the proof.

## Declarations

### Acknowledgments

The authors thank the referees for their comments and suggestions which improved the presentation of this paper. The first author was supported in part by Colleges and Universities, Science and Technology Development Foundation (20091003) of Tianjin and NSFC 11071279. The second author was supported in part by NSC 99-2221-E-230-006

## Authors’ Affiliations

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