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An Implicit Extragradient Method for Hierarchical Variational Inequalities
Fixed Point Theory and Applications volume 2011, Article number: 697248 (2011)
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
The variational inequality problem is to find such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ1_HTML.gif)
The set of solutions of the variational inequality problem is denoted by . It is well known that the variational inequality theory has emerged as an important tool in studying a wide class of obstacle, unilateral, and equilibrium problems; which arise in several branches of pure and applied sciences in a unified and general framework. Several numerical methods have been developed for solving variational inequalities and related optimization problems, see [1–24] and the references therein. In particular, Korpelevich's extragradient method which was introduced by Korpelevič [4] in 1976 generates a sequence
via the recursion
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ2_HTML.gif)
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
.
Recently, hierarchical fixed point problems and hierarchical minimization problems have attracted many authors' attention due to their link with some convex programming problems. See [25–32]. Motivated and inspired by these results in the literature, in this paper we are devoted to solve the following hierarchical variational inequality :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ3_HTML.gif)
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
Let be a real Hilbert space with inner product
and norm
, and let
be a closed convex subset of
. Recall that a mapping
is called
-inverse strongly monotone if there exists a positive real number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ4_HTML.gif)
A mapping is said to be
-contraction if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ5_HTML.gif)
It is well known that, for any , there exists a unique
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ6_HTML.gif)
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
,
.
Such properties of will be crucial in the proof of our main results. Let
be a monotone mapping of
into
. In the context of the variational inequality problem, it is easy to see from property (iii) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ7_HTML.gif)
We need the following lemmas for proving our main result.
Lemma 2.1 (see [13]).
Let be a nonempty closed convex subset of a real Hilbert space
. Let the mapping
be
-inverse strongly monotone, and let
be a constant. Then, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ8_HTML.gif)
In particular, if , then
is nonexpansive.
Lemma 2.2 (see [32]).
Let be a nonempty closed convex subset of a real Hilbert space
. Assume that the mapping
is monotone and weakly continuous along segments, that is,
weakly as
. Then, the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ9_HTML.gif)
is equivalent to the dual variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ10_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ11_HTML.gif)
where is a constant.
Note the fact that is a possible nonself mapping. Hence, if we take
, then (3.1) reduces to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ12_HTML.gif)
Remark 3.1.
We notice that the net defined by (3.1) is well defined. In fact, we can define a self-mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ13_HTML.gif)
From Lemma 2.1, we know that if , the mapping
is nonexpansive.
For any , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ14_HTML.gif)
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.
Take that . Since
, using the relation (2.4), we have
. In particular, if we take
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ15_HTML.gif)
From (3.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ16_HTML.gif)
Noting that is nonexpansive, thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ17_HTML.gif)
That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ18_HTML.gif)
Therefore, is bounded and so are
,
. Since
is
-inverse strongly monotone, it is
-Lipschitz continuous. Consequently,
and
are also bounded.
From (3.6),(2.5), and the convexity of the norm, we deduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ19_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ20_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ21_HTML.gif)
By the property (ii) of the metric projection , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ22_HTML.gif)
where is some appropriate constant. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ23_HTML.gif)
and hence (by (3.7))
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ24_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ25_HTML.gif)
Since , we derive
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ26_HTML.gif)
Next, we show that the net is relatively norm-compact as
. Assume that
is such that
as
. Put
and
.
By the property (ii) of metric projection , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ27_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ28_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ29_HTML.gif)
In particular,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ30_HTML.gif)
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
.
Next we show that . We define a mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ31_HTML.gif)
Then is maximal monotone (see [33]). Let
. Since
and
, we have
. On the other hand, from
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ32_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ33_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ34_HTML.gif)
Noting that ,
, and
is Lipschitz continuous, we obtain
. Since
is maximal monotone, we have
and hence
.
Therefore we can substitute for
in (3.20) to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ35_HTML.gif)
Consequently, the weak convergence of and
to
actually implies that
strongly. This has proved the relative norm-compactness of the net
as
.
Now we return to (3.20) and take the limit as to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ36_HTML.gif)
In particular, solves the following VI
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ37_HTML.gif)
or the equivalent dual VI (see Lemma 2.2)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ38_HTML.gif)
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
.
Finally, if we take that , then VI (3.28) is reduced to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ39_HTML.gif)
Equivalently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ40_HTML.gif)
This clearly implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F697248/MediaObjects/13663_2010_Article_1423_Equ41_HTML.gif)
Therefore, is the minimum-norm solution of
.This completes the proof.
Remark 3.3.
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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
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Yao, Y., Liou, Y. An Implicit Extragradient Method for Hierarchical Variational Inequalities. Fixed Point Theory Appl 2011, 697248 (2011). https://doi.org/10.1155/2011/697248
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DOI: https://doi.org/10.1155/2011/697248