Hybrid Steepest-Descent Methods for Solving Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators

Abstract

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.

1. Introduction

Let be a real Hilbert space with the inner product and the norm , let be a nonempty closed convex subset of and let be a nonlinear operator. We consider the problem of finding a point such that

(1.1)

This is known as the variational inequality problem (i.e., initially introduced and studied by Stampacchia [1] 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 [7] proposed hybrid methods to solve , where is composed of fixed points of a nonexpansive mapping; that is, is of the form

(1.2)

where is a nonexpansive mapping (i.e., for all ), is Lipschitzian and strongly monotone.

He and Xu [8] 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 [8]. We assume that is a boundedly Lipschitzian and strongly monotone operator as in [8], 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

(1) is called Lipschitzian on , if there there exists a positive constant such that

(1.3)

(2) is called boundedly Lipschitzian on , if for each nonempty bounded subset of , there exists a positive constant depending only on the set such that

(1.4)

(3) is said to be -strongly monotone on , if there exists a positive constant such that

(1.5)

(4) is said to be a -strict pseudo-contraction if there exists a constant such that

(1.6)

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 .

2. Preliminaries

We need some facts and tools which are listed as lemmas below.

Lemma 2.1.

Let be a real Hilbert space. The following expressions hold:

(i)

(ii)

Lemma 2.2 (see [9]).

Assume that is a sequence of nonnegtive real numbers satisfying the property

(2.1)

If and satisfy the following conditions:

(i)

(ii)

(iii)

then

Lemma 2.3 (see [10]).

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 [11]).

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 [8]).

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.

Lemma 2.6.

Assume that is a -strict pseudo-contraction, and the constant satisfies Let

(2.2)

then is nonexpansive and

Proof.

Using Lemma 2.1(i) and the conception of -strict pseudo-contraction, we get

(2.3)

so is nonexpansive. is obvious.

Lemma 2.7.

Assume that is a real Hilbert space, is a -strict pseudo-contraction such that and is a boundedly Lipschitzian and -strongly monotone operator. Take arbitrarily and set . Denote by the Lipschitz constant of on and let

(2.4)

where the constants and are such that and , respectively, and is defined as in Lemma 2.6 above. Then restricted to is a contraction.

Proof.

If , that is, , by Lemma 2.6, we have

(2.5)

It suggests that . Since is Lipschitzian and -strongly monotone on , using Lemma 2.6, we obtain

(2.6)

Therefore, restricted to that is a contraction with coefficient , where

Lemma 2.8 (see [11]).

Assume is a closed convex subset of a Hilbert space .

(i)Given an integer , assume that for each , is a -strict pseudo-contraction for some . Assume is a positive sequence such that . Then is a -strict pseudo-contraction, with

• (ii) Let , and be given as in (i) above. Suppose that , then

(2.7)

Lemma 2.9.

Assume that is a -strict pseudo-contraction for some let if , then

(2.8)

Proof.

We prove it by induction. For , set , . Obviously

(2.9)

Now we prove

(2.10)

, if , then the conclusion holds. In fact, we can claim that . From Lemma 2.6, we know that is nonexpansive and Take , then

(2.11)

Since , we get

(2.12)

Namely, that is,

(2.13)

Suppose that the conclusion holds for , we prove that

(2.14)

It suffices to verify

(2.15)

, . Using Lemma 2.6 again, take ,

(2.16)

Since , we have

(2.17)

this implies that

(2.18)

Namely,

(2.19)

From (2.19) and inductive assumption, we get

(2.20)

therefore

(2.21)

Substituting it into (2.19), we obtain Thus we assert that

(2.22)

3. Further Extension of Hybrid Iterative Algorithm

Yamada got the following result.

Theorem 3.1 (see [7]).

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:

(i);

(ii)

(iii), or .

Take arbitrarily and define by

(3.1)

then converges strongly to the unique solution of .

He and Xu [8] 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.

In this section, we mainly focus on further extension of our hybrid algorithm in [8]. Consider , where is composed of fixed points of a -strict pseudo-contraction such that and is still -strongly monotone and boundedly Lipschitzian. Fix a point arbitrarily, set . Denote by the Lipschitz constant of on . Fix the constant satisfying . Assume also that the sequences and satisfy for a constant and , respectively. Let and , define by the scheme:

(3.2)

We have the following result.

Theorem 3.2.

If the sequences and satisfy the following conditions:

(i);

(ii)

(iii), , or ,

then generated by (3.2) converges strongly to the unique solution of.

Proof.

We prove that for all by induction. It is trivial that . Suppose we have proved , that is,

(3.3)

Using Lemma 2.7, We then derive from (3.2) and (3.3) that

(3.4)

However, since and we get

(3.5)

This together with (3.4) implies that

(3.6)

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

(3.7)

where . By Lemma 2.2 and conditions (i)–(iii), we conclude that

(3.8)

Since , it is straitforward from (3.2) that

(3.9)

On the other hand

(3.10)

By the condition and (3.8)–(3.10), we obtain

(3.11)

By Lemma 2.4 and (3.11), we obtain

(3.12)

Lemma 2.5 asserts that has a unique solution . Now we prove that . By Lemma 2.1(ii), (3.2), and Lemma 2.7, we have

(3.13)

Let us show that

(3.14)

In fact, there exists a subsequence such that

(3.15)

Without loss of generality, we may further assume that . Since is the unique solution of , we obtain

(3.16)

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.

Let be a real Hilbert space and a -strongly monotone and boundedly Lipschitzian operator. Let be a positive integer and a -strict pseudo-contraction for some such that We consider the problem of finding such that

(4.1)

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

(4.2)

By using Lemma 2.8, we assert that is a -strict pseudo-contraction with and holds. Thus VI(4.1) is equivalent to VI and we can use scheme (3.2) to solve VI(4.1). In fact, taking in the scheme (3.2), we get the so-called parallel algorithm

(4.3)

Using Lemma 2.8 and Thorem 3.2, the following conclusion can be deduced directly.

Theorem 4.1.

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

(4.4)

where the constant such that . Then we turn to defining the cyclic algorithm as follows:

(4.5)

Indeed, the algorithm above can be rewritten as

(4.6)

where , namely, is one of circularly. For convenience, we denote (4.6) as

(4.7)

We get the following result

Theorem 4.2.

If satisfies the following conditions:

(i);

(ii);

(iii), or

then the sequence generated by (4.6) converges strongly to the unique solution of .

Proof.

We break the proof process into six steps.

• (1) . We prove it by induction. Definitely . Suppose , that is,

  

  (4.8)

We have from , (4.8), and Lemma 2.7 that

(4.9)

where Observing , we get

(4.10)

This together with (4.9) implies that

(4.11)

It suggests that . Therefore, for all . We can also prove that the sequences , , are all bounded.

• (2) By (4.6) and Lemma 2.7, we have

  

  (4.12)

where Since satisfies (i)–(iii), using Lemma 2.2, we get

(4.13)

• (3) By (4.3) and , we have

  

  (4.14)

Recursively,

(4.15)

By Lemma 2.6, is nonexpansive, we obtain

(4.16)

Adding all the expressions above, we get

(4.17)

Using this together with the conclusion of step (2), we obtain

(4.18)

• (4) . Assume that such that , we prove . By the conclusion of step (3), we get

  

  (4.19)

Observe that, for each is some permutation of the mappings , since are finite, all the full permutation are , there must be some permutation that appears infinite times. Without loss of generality, suppose that this permutation is , we can take a subsequence such that

(4.20)

It is easy to prove that is nonexpansive. By Lemma 2.3, we get

(4.21)

Using Lemmas 2.6 and 2.9, we obtain

(4.22)

• (5) In fact, there exists a subsequence such that

  

  (4.23)

Without loss of generality, we may further assume that Since is the solution of , we obtain

(4.24)

• (6) By (4.6), Lemmas 2.1(ii), and 2.7, we obtain

  

  (4.25)

From the conclusion of step (5) and Lemma 2.2, we get

(4.26)