- Research Article
- Open Access
An Implicit Extragradient Method for Hierarchical Variational Inequalities
© Yonghong Yao and Yeong Cheng Liou. 2011
- Received: 20 September 2010
- Accepted: 7 November 2010
- Published: 28 November 2010
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 .
- Variational Inequality
- Maximal Monotone
- Real Hilbert Space
- Variational Inequality Problem
- Nonempty Closed Convex Subset
where is the metric projection from onto , is a monotone operator, and is a constant. Korpelevich  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 .
We denote by , where is called the metric projection of onto . The metric projection of onto has the following basic properties:
(i) for all ;
(ii) for every ;
(iii) for all , ;
(iv) for all , .
We need the following lemmas for proving our main result.
Lemma 2.1 (see ).
In particular, if , then is nonexpansive.
Lemma 2.2 (see ).
In this section, we will introduce our implicit extragradient algorithm and show its strong convergence to the unique solution of .
where is a constant.
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.
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 .
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.
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
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