- Research Article
- Open Access
Hierarchical Convergence of a Double-Net Algorithm for Equilibrium Problems and Variational Inequality Problems
© Yonghong Yao et al. 2010
- Received: 21 May 2010
- Accepted: 22 December 2010
- Published: 29 December 2010
We consider the following hierarchical equilibrium problem and variational inequality problem (abbreviated as HEVP): find a point such that , for all , where , are two monotone operators and is the solution of the equilibrium problem of finding such that , for all . We note that the problem (HEVP) includes some problems, for example, mathematical program and hierarchical minimization problems as special cases. For solving (HEVP), we propose a double-net algorithm which generates a net . We prove that the net hierarchically converges to the solution of (HEVP); that is, for each fixed , the net converges in norm, as , to a solution of the equilibrium problem, and as , the net converges in norm to the unique solution of (HEVP).
- Variational Inequality
- Equilibrium Problem
- Nonexpansive Mapping
- Real Hilbert Space
- Variational Inequality Problem
for all . It is obvious that any -inverse strongly monotone mapping is monotone and -Lipschitz continuous.
At this point, we wish to point out the link with some monotone variational inequalities and convex programming problems as follows.
a variational inequality studied by Yamada and Ogura .
a problem considered by Luo et al. .
The solution set of equilibrium problems (1.10) and (1.11) are denoted by and , respectively. The equilibrium problem (1.10) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, fixed point problems, minimax problems, Nash equilibrium problem in noncooperative games, and others. We remind the readers to refer to [14–30] and the references therein.
where are two monotone operators. The solution set of (1.12) is denoted by .
It is clear that the hierarchical variational inequality problem and equilibrium problem (1.12) includes the variational inequality problem studied by Yamada and Ogura , mathematical program studied by Luo et al. , hierarchical minimization problem considered by Cabot  and Solodov , as special cases.
For solving (1.12), we propose a double-net algorithm which generates a net . We prove that the net hierarchically converges to the solution of (1.12); that is, for each fixed , the net converges in norm, as , to a solution of the equilibrium problem, and as , the net converges in norm to the unique solution of (1.12).
Let be a real Hilbert space. Throughout this paper, let us assume that a bifunction satisfies the following conditions:
(F1) for all ;
(F2) is monotone, that is, for all ;
(F3) for each , ;
(F4) for each , is convex and lower semicontinuous.
On the equilibrium problems, we have the following important lemma. You can find it in .
Further, if , for all , then the following hold:
(1) is single-valued;
(4) is closed and convex.
Below we gather some basic facts that are needed in the argument of the subsequent sections.
Lemma 2.2 (see ).
In particular, if , then is nonexpansive.
Lemma 2.3 (demiclosedness principle for nonexpansive mappings, see ).
Let be a nonempty closed convex subset of a real Hilbert space and let be a nonexpansive mapping with . If is a sequence in weakly converging to , and if converges strongly to , then ; in particular, if , then .
which is exactly (2.5).
This implies that solves (2.4). The proof is completed.
In this section, we first introduce our double-net algorithm.
Below is our main result of this paper which displays the behavior of the net as and successively.
We divide our detailed proofs into several conclusions as follows. Throughout, we assume all assumptions of Theorem 3.1 are satisfied.
For each fixed , the net is bounded.
It follows that for each fixed , is bounded, so are the nets , and . Note that we use as a positive constant which bounds all bounded terms appearing in the following.
Since is bounded, without loss of generality, we may assume that as , converges weakly to a point . Note that also converges weakly to a point .
This implies that .
Consequently, the weak convergence of to actually implies that strongly. This has proved the relative norm-compactness of the net as .
Notice that (3.31) is equivalent to the fact that . That is, is the unique element in of the contraction . Clearly, this is sufficient to conclude that the entire net converges in norm to as .
The net is bounded.
which implies that is bounded.
The net which solves the variational inequality VI (3.4).
First, we note that the solution of the variational inequality VI (3.4) is unique.
Namely, is a solution of VI (1.12); hence, .
It turns out that solves VI (3.4). By uniqueness, we have . This is sufficient to guarantee that in norm, as . The proof is complete.
By Conclusions 1–4, the proof of Theorem 3.1 is completed.
The solution of (3.45) is denoted by .
Taking in Theorem 3.1, we have the following corollary.
Taking in Theorem 3.1, we have the following corollary.
The work of the second author was partially supported by the Grant NSC 98-2923-E-110-003-MY3 and the work of the third author was partially supported by the Grant NSC 98-2221-E-110-064.
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