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).
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.
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).
On the equilibrium problems, we have the following important lemma. You can find it in .
Below we gather some basic facts that are needed in the argument of the subsequent sections.
Lemma 2.2 (see ).
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).
3. Main Results
In this section, we first introduce our double-net algorithm.
We divide our detailed proofs into several conclusions as follows. Throughout, we assume all assumptions of Theorem 3.1 are satisfied.
First, we note that the solution of the variational inequality VI (3.4) is unique.
By Conclusions 1–4, the proof of Theorem 3.1 is completed.
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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