Fixed Point Theory and Applications

, 2011:282171 | Cite as

Iterative Approaches to Find Zeros of Maximal Monotone Operators by Hybrid Approximate Proximal Point Methods

  • LuChuan Ceng
  • YeongCheng Liou
  • Eskandar Naraghirad
Open Access
Research Article
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  1. Equilibrium Problems and Fixed Point Theory

Abstract

The purpose of this paper is to introduce and investigate two kinds of iterative algorithms for the problem of finding zeros of maximal monotone operators. Weak and strong convergence theorems are established in a real Hilbert space. As applications, we consider a problem of finding a minimizer of a convex function.

Keywords

Convex Subset Iterative Algorithm Nonexpansive Mapping Maximal Monotone Real Hilbert Space 

1. Introduction

Let Open image in new window be a nonempty, closed, and convex subset of a real Hilbert space Open image in new window . In this paper, we always assume that Open image in new window is a maximal monotone operator. A classical method to solve the following set-valued equation:
is the proximal point method. To be more precise, start with any point Open image in new window , and update Open image in new window iteratively conforming to the following recursion:

where Open image in new window ( Open image in new window ) is a sequence of real numbers. However, as pointed out in [1], the ideal form of the method is often impractical since, in many cases, to solve the problem (1.2) exactly is either impossible or has the same difficulty as the original problem (1.1). Therefore, one of the most interesting and important problems in the theory of maximal monotone operators is to find an efficient iterative algorithm to compute approximate zeros of Open image in new window .

In 1976, Rockafellar [2] gave an inexact variant of the method
where Open image in new window is regarded as an error sequence. This is an inexact proximal point method. It was shown that, if

the sequence Open image in new window defined by (1.3) converges weakly to a zero of Open image in new window provided that Open image in new window . In [3], Güler obtained an example to show that Rockafellar's inexact proximal point method (1.3) does not converge strongly, in general.

Recently, many authors studied the problems of modifying Rockafellar's inexact proximal point method (1.3) in order to strong convergence to be guaranteed. In 2008, Ceng et al. [4] gave new accuracy criteria to modified approximate proximal point algorithms in Hilbert spaces; that is, they established strong and weak convergence theorems for modified approximate proximal point algorithms for finding zeros of maximal monotone operators in Hilbert spaces. In the meantime, Cho et al. [5] proved the following strong convergence result.

Theorem CKZ1.

Let Open image in new window be a real Hilbert space, Open image in new window a nonempty closed convex subset of Open image in new window , and Open image in new window a maximal monotone operator with Open image in new window . Let Open image in new window be the metric projection of Open image in new window onto Open image in new window . Suppose that, for any given Open image in new window , Open image in new window , and Open image in new window , there exists Open image in new window conforming to the following set-valued mapping equation:

Let Open image in new window be a real sequence in Open image in new window such that

(i) Open image in new window as Open image in new window ,

(ii) Open image in new window .

For any fixed Open image in new window , define the sequence Open image in new window iteratively as follows:

Then Open image in new window converges strongly to a zero Open image in new window of Open image in new window , where Open image in new window .

They also derived the following weak convergence theorem.

Theorem CKZ2.

Let Open image in new window be a real Hilbert space, Open image in new window a nonempty closed convex subset of Open image in new window , and Open image in new window a maximal monotone operator with Open image in new window . Let Open image in new window be the metric projection of Open image in new window onto Open image in new window . Suppose that, for any given Open image in new window , Open image in new window , and Open image in new window , there exists Open image in new window conforming to the following set-valued mapping equation:
Let Open image in new window be a real sequence in Open image in new window with Open image in new window , and define a sequence Open image in new window iteratively as follows:

where Open image in new window for all Open image in new window . Then the sequence Open image in new window converges weakly to a zero Open image in new window of Open image in new window .

Very recently, Qin et al. [6] extended (1.7) and (1.10) to the iterative scheme
and the iterative one

respectively, where Open image in new window , Open image in new window , and Open image in new window with Open image in new window . Under appropriate conditions, they derived one strong convergence theorem for (1.11) and another weak convergence theorem for (1.12). In addition, for other recent research works on approximate proximal point methods and their variants for finding zeros of monotone maximal operators, see, for example, [7, 8, 9, 10] and the references therein.

In this paper, motivated by the research work going on in this direction, we continue to consider the problem of finding a zero of the maximal monotone operator Open image in new window . The iterative algorithms (1.7) and (1.10) are extended to develop the following new iterative ones:
respectively, where Open image in new window is any fixed point in Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window with Open image in new window . Under mild conditions, we establish one strong convergence theorem for (1.13) and another weak convergence theorem for (1.14). The results presented in this paper improve the corresponding results announced by many others. It is easy to see that in the case when Open image in new window and Open image in new window for all Open image in new window , the iterative algorithms (1.13) and (1.14) reduce to (1.7) and (1.10), respectively. Moreover, the iterative algorithms (1.13) and (1.14) are very different from (1.11) and (1.12), respectively. Indeed, it is clear that the iterative algorithm (1.13) is equivalent to the following:
Here, the first iteration step Open image in new window , is to compute the prediction value of approximate zeros of Open image in new window ; the second iteration step, Open image in new window , is to compute the correction value of approximate zeros of Open image in new window . Similarly, it is obvious that the iterative algorithm (1.14) is equivalent to the following:

Here, the first iteration step, Open image in new window + Open image in new window + Open image in new window , is to compute the prediction value of approximate zeros of Open image in new window ; the second iteration step, Open image in new window , is to compute the correction value of approximate zeros of Open image in new window . Therefore, there is no doubt that the iterative algorithms (1.13) and (1.14) are very interesting and quite reasonable.

In this paper, we consider the problem of finding zeros of maximal monotone operators by hybrid proximal point method. To be more precise, we introduce two kinds of iterative schemes, that is, (1.13) and (1.14). Weak and strong convergence theorems are established in a real Hilbert space. As applications, we also consider a problem of finding a minimizer of a convex function.

2. Preliminaries

In this section, we give some preliminaries which will be used in the rest of this paper. Let Open image in new window be a real Hilbert space with inner product Open image in new window and norm Open image in new window . Let Open image in new window be a set-valued mapping. The set Open image in new window defined by
is called the effective domain of Open image in new window . The set Open image in new window defined by
is called the range of Open image in new window . The set Open image in new window defined by
is called the graph of Open image in new window . A mapping Open image in new window is said to be monotone if

Open image in new window is said to be maximal monotone if its graph is not properly contained in the one of any other monotone operator.

The class of monotone mappings is one of the most important classes of mappings among nonlinear mappings. Within the past several decades, many authors have been devoted to the study of the existence and iterative algorithms of zeros for maximal monotone mappings; see [1, 2, 3, 4, 5, 7, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. In order to prove our main results, we need the following lemmas. The first lemma can be obtained from Eckstein [1, Lemma 2] immediately.

Lemma 2.1.

Let Open image in new window be a nonempty, closed, and convex subset of a Hilbert space Open image in new window . For any given Open image in new window , Open image in new window , and Open image in new window , there exists Open image in new window conforming to the following set-valued mapping equation (SVME ):
Furthermore, for any Open image in new window , we have

Lemma 2.2 (see [30, Lemma 2.5, page 243]).

Let Open image in new window be a sequence of nonnegative real numbers satisfying the inequality

where Open image in new window , Open image in new window , and Open image in new window satisfy the conditions

(i) Open image in new window , Open image in new window , or equivalently Open image in new window ,

(ii) Open image in new window ,

(iii) Open image in new window , Open image in new window .

Then Open image in new window .

Lemma 2.3 (see [28, Lemma 1, page 303]).

Let Open image in new window and Open image in new window be sequences of nonnegative real numbers satisfying the inequality

If Open image in new window , then Open image in new window exists.

Lemma 2.4 (see [11]).

Let Open image in new window be a uniformly convex Banach space, let Open image in new window be a nonempty closed convex subset of Open image in new window , and let Open image in new window be a nonexpansive mapping. Then Open image in new window is demiclosed at zero.

Lemma 2.5 (see [31]).

Let Open image in new window be a uniformly convex Banach space, and and Open image in new window be a closed ball of Open image in new window . Then there exists a continuous strictly increasing convex function Open image in new window with Open image in new window such that

for all Open image in new window and Open image in new window with Open image in new window .

It is clear that the following lemma is valid.

Lemma 2.6.

Let Open image in new window be a real Hilbert space. Then there holds

3. Main Results

Let Open image in new window be a nonempty, closed, and convex subset of a real Hilbert space Open image in new window . We always assume that Open image in new window is a maximal monotone operator. Then, for each Open image in new window , the resolvent Open image in new window is a single-valued nonexpansive mapping whose domain is all Open image in new window . Recall also that the Yosida approximation of Open image in new window is defined by

Assume that Open image in new window , where Open image in new window is the set of zeros of Open image in new window . Then Open image in new window for all Open image in new window , where Open image in new window is the set of fixed points of the resolvent Open image in new window .

Theorem 3.1.

Let Open image in new window be a real Hilbert space, Open image in new window a nonempty, closed, and convex subset of Open image in new window , and Open image in new window a maximal monotone operator with Open image in new window . Let Open image in new window be a metric projection from Open image in new window onto Open image in new window . For any given Open image in new window , Open image in new window , and Open image in new window , find Open image in new window conforming to SVME (2.5), where Open image in new window with Open image in new window as Open image in new window and Open image in new window with Open image in new window . Let Open image in new window , Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window satisfying the following control conditions:

(i) Open image in new window and Open image in new window ,

(ii) Open image in new window and Open image in new window ,

(iii) Open image in new window and Open image in new window .

Let Open image in new window be a sequence generated by the following manner:

where Open image in new window is a fixed point and Open image in new window is a bounded sequence in Open image in new window . Then the sequence Open image in new window generated by (3.2) converges strongly to a zero Open image in new window of Open image in new window , where Open image in new window , if and only if Open image in new window as Open image in new window .

Proof.

First, let us show the necessity. Assume that Open image in new window as Open image in new window , where Open image in new window . It follows from (2.5) that
and hence

This shows that Open image in new window as Open image in new window .

Next, let us show the sufficiency. The proof is divided into several steps.

Step 1 ( Open image in new window is bounded).

Indeed, from the assumptions Open image in new window and Open image in new window , it follows that
Take an arbitrary Open image in new window . Then it follows from Lemma 2.1 that
and hence
This implies that
we show that Open image in new window for all Open image in new window . It is easy to see that the result holds for Open image in new window . Assume that the result holds for some Open image in new window . Next, we prove that Open image in new window . As a matter of fact, from (3.9), we see that

This shows that the sequence Open image in new window is bounded.

Step 2 ( Open image in new window , where Open image in new window ).

The existence of Open image in new window is guaranteed by Lemma 1 of Bruck [12].

On the other hand, by the nonexpansivity of Open image in new window , we obtain that
From the assumption Open image in new window as Open image in new window and (3.13), we get
From (2.5), we see that
Combining (3.15) with (3.17), we have
In the meantime, from algorithm (3.2) and assumption Open image in new window , it follows that
Thus, from the condition Open image in new window , we have
This together with (3.18) implies that
From Open image in new window and (3.21), we can obtain that

Step 3 ( Open image in new window as Open image in new window ).

Indeed, utilizing (3.8), we deduce from algorithm (3.2) that
Note that Open image in new window and Open image in new window is bounded. Hence it is known that Open image in new window . Since Open image in new window , Open image in new window , and Open image in new window , in terms of Lemma 2.2, we conclude that

This completes the proof.

Remark 3.2.

The maximal monotonicity of Open image in new window is only used to guarantee the existence of solutions of SVME Open image in new window , for any given Open image in new window , Open image in new window , and Open image in new window . If we assume that Open image in new window is monotone (not necessarily maximal) and satisfies the range condition

we can see that Theorem 3.1 still holds.

Corollary 3.3.

Let Open image in new window be a real Hilbert space, Open image in new window a nonempty, closed, and convex subset of Open image in new window , and Open image in new window a demicontinuous pseudocontraction with a fixed point in Open image in new window . Let Open image in new window be a metric projection from Open image in new window onto Open image in new window . For any Open image in new window , Open image in new window , and Open image in new window , find Open image in new window such that

where Open image in new window with Open image in new window as Open image in new window and Open image in new window with Open image in new window . Let Open image in new window , Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window satisfying the following control conditions:

(i) Open image in new window and Open image in new window ,

(ii) Open image in new window and Open image in new window ,

(iii) Open image in new window and Open image in new window .

Let Open image in new window be a sequence generated by the following manner:

where Open image in new window is a fixed point and Open image in new window is a bounded sequence in Open image in new window . If the sequence Open image in new window satisfies the condition Open image in new window as Open image in new window , then the sequence Open image in new window converges strongly to a fixed point Open image in new window of Open image in new window , where Open image in new window .

Proof.

Let Open image in new window . Then Open image in new window is demicontinuous, monotone, and satisfies the range condition:
Then Open image in new window is demicontinuous and strongly pseudocontractive. By the study of Lan and Wu [21, Theorem 2.2], we see that Open image in new window has a unique fixed point Open image in new window ; that is,
that is,

Finally, from the proof of Theorem 3.1, we can derive the desired conclusion immediately.

From Theorem 3.1, we also have the following result immediately.

Corollary 3.4.

Let Open image in new window be a real Hilbert space, Open image in new window a nonempty, closed, and convex subset of Open image in new window , and Open image in new window a maximal monotone operator with Open image in new window . Let Open image in new window be a metric projection from Open image in new window onto Open image in new window . For any Open image in new window , Open image in new window and Open image in new window , find Open image in new window conforming to SVME (2.5), where Open image in new window with Open image in new window as Open image in new window and Open image in new window with Open image in new window . Let Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window satisfying the following control conditions:

(i) Open image in new window ,

(ii) Open image in new window and Open image in new window ,

(iii) Open image in new window .

Let Open image in new window be a sequence generated by the following manner:

where Open image in new window is a fixed point. Then the sequence Open image in new window converges strongly to a zero Open image in new window of Open image in new window , where Open image in new window , if and only if Open image in new window as Open image in new window .

Proof.

In Theorem 3.1, put Open image in new window for all Open image in new window . Then, from Theorem 3.1, we obtain the desired result immediately.

Next, we give a hybrid Mann-type iterative algorithm and study the weak convergence of the algorithm.

Theorem 3.5.

Let Open image in new window be a real Hilbert space, Open image in new window a nonempty, closed, and convex subset of Open image in new window , and Open image in new window a maximal monotone operator with Open image in new window . Let Open image in new window be a metric projection from Open image in new window onto Open image in new window . For any given Open image in new window , Open image in new window , and Open image in new window , find Open image in new window conforming to SVME (2.5), where Open image in new window and Open image in new window with Open image in new window . Let Open image in new window , Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window satisfying the following control conditions:

(i) Open image in new window and Open image in new window ,

(ii) Open image in new window ,

(iii) Open image in new window and Open image in new window .

Let Open image in new window be a sequence generated by the following manner:

where Open image in new window is a bounded sequence in Open image in new window . Then the sequence Open image in new window generated by (3.34) converges weakly to a zero Open image in new window of Open image in new window .

Proof.

Take an arbitrary Open image in new window . Utilizing Lemma 2.1, from the assumption Open image in new window with Open image in new window , we conclude that
It follows from Lemma 2.5 that
Utilizing Lemma 2.3, we know that Open image in new window exists. We, therefore, obtain that the sequence Open image in new window is bounded. It follows from (3.36) that
Note that
In view of (3.38), we obtain that
Also, note that
In view of the assumption Open image in new window and (3.40), we see that

Let Open image in new window be a weakly subsequential limit of Open image in new window such that Open image in new window converges weakly to Open image in new window as Open image in new window . From (3.40), we see that Open image in new window also converges weakly to Open image in new window . Since Open image in new window is nonexpansive, we can obtain that Open image in new window by Lemma 2.4. Opial's condition (see [23]) guarantees that the sequence Open image in new window converges weakly to Open image in new window . This completes the proof.

By the careful analysis of the proof of Corollary 3.3 and Theorem 3.5, it is not hard to derive the following result.

Corollary 3.6.

Let Open image in new window be a real Hilbert space, Open image in new window a nonempty, closed, and convex subset of Open image in new window , and Open image in new window a demicontinuous pseudocontraction with a fixed point in Open image in new window . Let Open image in new window be a metric projection from Open image in new window onto Open image in new window . For any Open image in new window , Open image in new window , and Open image in new window , find Open image in new window such that

where Open image in new window and Open image in new window with Open image in new window . Let Open image in new window , Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window satisfying the following control conditions:

(i) Open image in new window and Open image in new window ,

(ii) Open image in new window ,

(iii) Open image in new window and Open image in new window .

Let Open image in new window be a sequence generated by the following manner:

where Open image in new window is a bounded sequence in Open image in new window . Then the sequence Open image in new window converges weakly to a fixed point Open image in new window of Open image in new window .

Utilizing Theorem 3.5, we also obtain the following result immediately.

Corollary 3.7.

Let Open image in new window be a real Hilbert space, Open image in new window a nonempty, closed, and convex subset of Open image in new window , and Open image in new window a maximal monotone operator with Open image in new window . Let Open image in new window be a metric projection from Open image in new window onto Open image in new window . For any Open image in new window , Open image in new window , and Open image in new window , find Open image in new window conforming to SVME (2.5), where Open image in new window and Open image in new window with Open image in new window . Let Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window satisfying the following control conditions:

(i) Open image in new window ,

(ii) Open image in new window ,

(iii) Open image in new window .

Let Open image in new window be a sequence generated by the following manner:

Then the sequence Open image in new window converges weakly to a zero Open image in new window of Open image in new window .

4. Applications

In this section, as applications of the main Theorems 3.1 and 3.5, we consider the problem of finding a minimizer of a convex function Open image in new window .

Let Open image in new window be a real Hilbert space, and let Open image in new window be a proper convex lower semi-continuous function. Then the subdifferential Open image in new window of Open image in new window is defined as follows:

Theorem 4.1.

Let Open image in new window , Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window satisfying the following control conditions:

(i) Open image in new window and Open image in new window ,

(ii) Open image in new window and Open image in new window ,

(iii) Open image in new window and Open image in new window .

Let Open image in new window be a sequence generated by the following manner:

where Open image in new window is a fixed point and Open image in new window is a bounded sequence in Open image in new window . If the sequence Open image in new window satisfies the condition Open image in new window as Open image in new window , then the sequence Open image in new window converges strongly to a minimizer of Open image in new window nearest to Open image in new window .

Proof.

Since Open image in new window is a proper convex lower semi-continuous function, we have that the subdifferential Open image in new window of Open image in new window is maximal monotone by the study of Rockafellar [2]. Notice that
is equivalent to the following:
It follows that

By using Theorem 3.1, we can obtain the desired result immediately.

Theorem 4.2.

Let Open image in new window be a real Hilbert space and Open image in new window a proper convex lower semi-continuous function such that Open image in new window . Let Open image in new window be a sequence in Open image in new window with Open image in new window and Open image in new window a sequence in Open image in new window such that Open image in new window with Open image in new window . Let Open image in new window be the solution of SVME (2.5) with Open image in new window replaced by Open image in new window ; that is, for any given Open image in new window ,

Let Open image in new window , Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window satisfying the following control conditions:

(i) Open image in new window and Open image in new window ,

(ii) Open image in new window ,

(iii) Open image in new window and Open image in new window .

Let Open image in new window be a sequence generated by the following manner:

where Open image in new window is a bounded sequence in Open image in new window . Then the sequence Open image in new window converges weakly to a minimizer of Open image in new window .

Proof.

We can obtain the desired result readily from the proof of Theorems 3.5 and 4.1.

Notes

Acknowledgment

This research was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and the Dawn Program Foundation in Shanghai.

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© Lu Chuan Ceng et al. 2011

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.

Authors and Affiliations

  • LuChuan Ceng
    • 1
  • YeongCheng Liou
    • 2
  • Eskandar Naraghirad
    • 3
  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiChina
  2. 2.Department of Information ManagementCheng Shiu UniversityKaohsiungTaiwan
  3. 3.Department of MathematicsYasouj UniversityYasoujIran

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