- Research Article
- Open Access
Hybrid Steepest-Descent Methods for Solving Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators
© S. He and X.-L. Liang. 2010
- Received: 30 September 2009
- Accepted: 13 January 2010
- Published: 24 January 2010
Let be a real Hilbert space and let be a boundedly Lipschitzian and strongly monotone operator. We design three hybrid steepest descent algorithms for solving variational inequality of finding a point such that , for all , where is the set of fixed points of a strict pseudocontraction, or the set of common fixed points of finite strict pseudocontractions. Strong convergence of the algorithms is proved.
- Variational Inequality
- Parallel Algorithm
- Nonexpansive Mapping
- Monotone Operator
- Hybrid Algorithm
This is known as the variational inequality problem (i.e., initially introduced and studied by Stampacchia  in 1964. In the recent years, variational inequality problems have been extended to study a large variety of problems arising in structural analysis, economics, optimization, operations research, and engineering sciences; see [1–6] and the references therein.
Yamada  proposed hybrid methods to solve , where is composed of fixed points of a nonexpansive mapping; that is, is of the form
where is a nonexpansive mapping (i.e., for all ), is Lipschitzian and strongly monotone.
He and Xu  proved that has a unique solution and iterative algorithms can be devised to approximate this solution if is a boundedly Lipschitzian and strongly monotone operator and is a closed convex subset of . In the case where is the set of fixed points of a nonexpansive mapping, they invented a hybrid iterative algorithm to approximate the unique solution of and this extended the Yamada's results.
The main purpose of this paper is to continue our research in . We assume that is a boundedly Lipschitzian and strongly monotone operator as in , but is the set of fixed points of a strict pseudo-contraction , or the set of common fixed points of finite strict pseudo-contractions . For the two cases of , we will design the hybrid iterative algorithms for solving and prove their strong convergence, respectively. Relative definitions are stated as below.
Let be a nonempty closed and convex subset of a real Hilbert space , and , then
Obviously, the nonexpansive mapping class is a proper subclass of the strict pseudo-contraction class and the Lipschitzian operator class is a proper subclass of the boundedly Lipschitzian operator class, respectively.
We will use the following notations:
(i) for weak convergence and for strong convergence,
(ii) denotes the weak -limit set of
(iii) denotes a closed ball with center and radius .
We need some facts and tools which are listed as lemmas below.
Let be a real Hilbert space. The following expressions hold:
Lemma 2.2 (see ).
If and satisfy the following conditions:
Lemma 2.3 (see ).
Let be a nonempty closed convex subset of a real Hilbert space and is a nonexpansive mapping. If a one has sequence in such that and then
Lemma 2.4 (see ).
Let be a nonempty closed convex subset of a real Hilbert space , if is a -strict pseudo-contraction, then the mapping is demiclosed at 0. That is, if is a sequence in such that and then
Lemma 2.5 (see ).
Assume that is a nonempty closed convex subset of a real Hilbert space , if is boundedly Lipschitzian and -strongly monotone, then variational inequality (1.1) has a unique solution.
then is nonexpansive and
so is nonexpansive. is obvious.
where the constants and are such that and , respectively, and is defined as in Lemma 2.6 above. Then restricted to is a contraction.
Therefore, restricted to that is a contraction with coefficient , where
Lemma 2.8 (see ).
Assume is a closed convex subset of a Hilbert space .
Let , and be given as in (i) above. Suppose that , then
Suppose that the conclusion holds for , we prove that
Yamada got the following result.
Theorem 3.1 (see ).
Assume that is a real Hilbert space, is nonexpansive such that and is -strongly monotone and -Lipschitzian. Fix a constant . Assume also that the sequence satisfies the following conditions:
(iii) , or .
then converges strongly to the unique solution of .
He and Xu  proved that has a unique solution if is a boundedly Lipschitzian and strongly monotone operator and is a closed convex subset of . Using this result, they were able to relax the global Lipschitz condition on in Theorem 3.1 to the weaker bounded Lipschitz condition and invented a hybrid iterative algorithm to approximate the unique solution of . Their result extended the Yamada's above theorem.
We have the following result.
If the sequences and satisfy the following conditions:
(iii) , , or ,
then generated by (3.2) converges strongly to the unique solution of .
It proves that . Therefore, for all . Thus is bounded. It is not difficult to verify that the sequences and are all bounded.
By (3.2) and Lemma 2.7, we have
Finally conditions (i)–(iii) and (3.14) allow us to apply Lemma 2.2 to the relation (3.13) to conclude that
In this section, we discuss the parallel algorithm and the cyclic algorithm, respectively, for solving the variational inequality over the set of the common fixed points of finite strict pseudo-contractions.
Since is a nonempty closed convex subset of , (4.1) has a unique solution. Throughout this section, is an arbitrary fixed point, , is the Lipschitz constant of on , the fixed constant satisfies , and the sequence belongs to .
Firstly we consider the parallel algorithm. Take a positive sequence such that and let
Using Lemma 2.8 and Thorem 3.2, the following conclusion can be deduced directly.
Suppose that and satisfy the same conditions as in Theorem 3.2. Then the sequence generated by the parallel algorithm (4.3) converges strongly to the unique solution of (4.1).
For each let
where the constant such that . Then we turn to defining the cyclic algorithm as follows:
We get the following result
If satisfies the following conditions:
(iii) , or
then the sequence generated by (4.6) converges strongly to the unique solution of .
This research is supported by the Fundamental Research Funds for the Central Universities (GRANT:ZXH2009D021).
- Stampacchia G: Formes bilinéaires coercitives sur les ensembles convexes. Comptes Rendus de l'Académie des Sciences 1964, 258: 4413–4416.MathSciNetMATHGoogle Scholar
- Cottle RW, Giannessi F, Lions JL: Variational Inequalities and Complementarity Problems: Theory and Applications. John Wiley & Sons, New York, NY, USA; 1980.Google Scholar
- Fukushima M: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Mathematical Programming 1992,53(1):99–110. 10.1007/BF01585696MathSciNetView ArticleMATHGoogle Scholar
- Geobel K, Reich S: Uniform Convexity, Nonexpansive Mappings, and Hyperbolic Geometry. Dekker, New York, NY, USA; 1984.Google Scholar
- Glowinski R, Lions J-L, Trémolières R: Numerical Analysis of Variational Inequalities, Studies in Mathematics and Its Applications. Volume 8. North-Holland, Amsterdam, The Netherlands; 1981:xxix+776.Google Scholar
- He BS: A class of implicit methods for monotone variational inequalities. In Reports of the Institute of Mathematics. Nanjing University, China; 1995.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, Studies in Computational Mathematics. Volume 8. Edited by: Butnariu D, Censor Y, Reich S. North-Holland, Amsterdam, The Netherlands; 2001:473–504.Google Scholar
- He SN, Xu HK: Variational inequalities governed by boundedly Lipschitzian and strongly monotone operators. Fixed Point Theory 2009,10(2):245–258.MathSciNetMATHGoogle Scholar
- Xu HK: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society 2002,66(1):240–256. 10.1112/S0024610702003332MathSciNetView ArticleMATHGoogle Scholar
- Geobel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge, UK; 1990:viii+244.View ArticleGoogle Scholar
- Acedo GL, Xu H-K: Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2007,67(7):2258–2271. 10.1016/j.na.2006.08.036MathSciNetView ArticleMATHGoogle 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.