On a Two-Step Algorithm for Hierarchical Fixed Point Problems and Variational Inequalities
© Filomena Cianciaruso et al. 2009
Received: 5 May 2009
Accepted: 12 September 2009
Published: 28 September 2009
A common method in solving ill-posed problems is to substitute the original problem by a family of well-posed (i.e., with a unique solution) regularized problems. We will use this idea to define and study a two-step algorithm to solve hierarchical fixed point problems under different conditions on involved parameters.
1. Introduction and Preliminar Results
A common method in solving ill-posed problems is to substitute the original problem by a family of well-posed (i.e., with a unique solution) regularized problems. We will use this idea to define and study a two-step algorithm to solve hierarchical fixed point problems under different conditions on involved parameters. We will see that choosing appropriate hypotheses on the parameters, we will obtain convergence to the solution of well-posed problems. Changing these assumptions, we will obtain convergence to one of the solutions of a ill-posed problem. The results are situaded on the lines of research of Byrne , Yang and Zhao , Moudafi , and Yao and Liou .
In this paper, we consider variational inequalities of the form
where are nonexpansive mappings such that the fixed points set of ( ) is nonempty and is a nonempty closed convex subset of a Hilbert space . If we denote with the set of solutions of (1.1), it is evident that .
Variational inequalities of (1.1) cover several topics recently investigated in literature as monotone inclusion ( and the references therein), convex optimization , quadratic minimization over fixed point set (see, e.g., [5, 7–10] and the references therein).
There are in literature many papers in which iterative methods are defined in order to solve (1.1).
Recently, in  Moudafi defined the following explicit iterative algorithm
where and are two sequences in and he proved a weak-convergence's result. In order to obtain a strong-convergence result, Maingé and Moudafi in  introduced and studied the following iterative algorithm
and give some conditions which assure that the method converges to a solution which solves some variational inequality.
We will confront the two methods (1.3) and (1.4) later.
We recall some general results of the Hilbert spaces theory and of the monotone operators theory.
Lemma 1.3 (see ).
Finally, we conclude this section with a lemma due to Xu on real sequences which has a fundamental role in the sequel.
Lemma 1.4 (see ).
2. Convergence of the Two-Step Iterative Algorithm
Suppose that (H1), (H3) hold. Also, assume that either (H4) and (H5) hold, or (H6) and (H7) hold. Then
So, if (H4) and (H5) hold, we obtain the asymptotic regularity by Lemma 1.4.
If, instead, (H6) and (H7) hold, from (H1) we can write
so, the asymptotic regularity follows by Lemma 1.4 also.
In order to prove (2), we can observe that
Suppose that the hypotheses of Proposition 2.2 hold. Then
Moreover, noting that
It is clear that our algorithm (1.4) is different from (1.3). At the same time, our algorithm (1.4) includes some algorithms in the literature as special cases. For instance, if we take in (1.4), then we get which is well-known as the viscosity method studied by Moudafi  and Xu .
We do not know the rate of convergence of our method. Nevertheless, the rates of convergence of our method (1.4) that generates the sequence and the Mainge-Moudafi method (1.3), seem not comparable. To see this, we consider three examples. In such examples we take , , , , .
Reassuming, we cannot affirm that our method is more convenient or better than the Mainge-Moudafi method, but only that seems to us that it is the first time that it is introduced a two-step iterative approach to the VIP (1.1). In some case, our method approximates the solution more rapidly than Mainge-Moudafi method, in some other case it happens the contrary and in some other cases, both methods give the same sequence.
The authors are extremely grateful to the anonymous referees for their useful comments and suggestions. This work was supported in part by Ministero dell'Universitá e della Ricerca of Italy.
- Byrne C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems 2004,20(1):103–120. 10.1088/0266-5611/20/1/006MathSciNetView ArticleMATHGoogle Scholar
- Yang Q, Zhao J: Generalized KM theorems and their applications. Inverse Problems 2006,22(3):833–844. 10.1088/0266-5611/22/3/006MathSciNetView ArticleMATHGoogle Scholar
- Moudafi A: Krasnoselski-Mann iteration for hierarchical fixed-point problems. Inverse Problems 2007,23(4):1635–1640. 10.1088/0266-5611/23/4/015MathSciNetView ArticleMATHGoogle Scholar
- Yao Y, Liou Y-C: Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed point problems. Inverse Problems 2008,24(1):-8.Google Scholar
- Yamada I: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), Studies in Computational Mathematics. Volume 8. North-Holland, Amsterdam, The Netherlands; 2001:473–504.Google Scholar
- Solodov M: An explicit descent method for bilevel convex optimization. Journal of Convex Analysis 2007,14(2):227–237.MathSciNetMATHGoogle Scholar
- Marino G, Xu H-K: A general iterative method for nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2006,318(1):43–52. 10.1016/j.jmaa.2005.05.028MathSciNetView ArticleMATHGoogle Scholar
- Moudafi A: Viscosity approximation methods for fixed-points problems. Journal of Mathematical Analysis and Applications 2000,241(1):46–55. 10.1006/jmaa.1999.6615MathSciNetView ArticleMATHGoogle Scholar
- Xu H-K: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society 2002,66(1):240–256. 10.1112/S0024610702003332MathSciNetView ArticleMATHGoogle Scholar
- Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleMATHGoogle Scholar
- Maingé P-E, Moudafi A: Strong convergence of an iterative method for hierarchical fixed-point problems. Pacific Journal of Optimization 2007,3(3):529–538.MathSciNetMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.