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.
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
3. Further Extension of Hybrid Iterative Algorithm
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
4. Parallel Algorithm and Cyclic Algorithm
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 .
- (1). We prove it by induction. Definitely . Suppose , that is,(4.8)
- (2)By (4.6) and Lemma 2.7, we have(4.12)
- (3)By (4.3) and , we have(4.14)
- (4). Assume that such that , we prove . By the conclusion of step (3), we get(4.19)
- (5)In fact, there exists a subsequence such that(4.23)
- (6)By (4.6), Lemmas 2.1(ii), and 2.7, we obtain(4.25)
This research is supported by the Fundamental Research Funds for the Central Universities (GRANT:ZXH2009D021).
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