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

- LuChuan Ceng
^{1}, - YeongCheng Liou
^{2}and - Eskandar Naraghirad
^{3}Email author

**2011**:282171

https://doi.org/10.1155/2011/282171

© Lu Chuan Ceng et al. 2011

**Received: **18 August 2010

**Accepted: **23 September 2010

**Published: **11 October 2010

## 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.

## 1. Introduction

where ( ) 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 .

the sequence defined by (1.3) converges weakly to a zero of provided that . 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 be a real sequence in such that

Then converges strongly to a zero of , where .

They also derived the following weak convergence theorem.

Theorem CKZ2.

where for all . Then the sequence converges weakly to a zero of .

respectively, where , , and with . 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, [7–10] and the references therein.

Here, the first iteration step, + + , is to compute the prediction value of approximate zeros of ; the second iteration step, , is to compute the correction value of approximate zeros of . 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

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 [1–5, 7, 11–30]. 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.

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

where , , and satisfy the conditions

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

Lemma 2.4 (see [11]).

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

Lemma 2.5 (see [31]).

It is clear that the following lemma is valid.

Lemma 2.6.

## 3. Main Results

Assume that , where is the set of zeros of . Then for all , where is the set of fixed points of the resolvent .

Theorem 3.1.

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

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

Proof.

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

This shows that the sequence is bounded.

The existence of is guaranteed by Lemma 1 of Bruck [12].

This completes the proof.

Remark 3.2.

we can see that Theorem 3.1 still holds.

Corollary 3.3.

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

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

Proof.

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

where is a fixed point. Then the sequence converges strongly to a zero of , where , if and only if as .

Proof.

In Theorem 3.1, put for all . 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 be a real Hilbert space, a nonempty, closed, and convex subset of , and a maximal monotone operator with . Let be a metric projection from onto . For any given , , and , find conforming to SVME (2.5), where and with . Let , , , and be real sequences in satisfying the following control conditions:

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

Proof.

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

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

where is a bounded sequence in . Then the sequence converges weakly to a fixed point of .

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

Corollary 3.7.

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

## 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 .

Theorem 4.1.

Let , , , and be real sequences in satisfying the following control conditions:

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

Proof.

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

Theorem 4.2.

Let , , , and be real sequences in satisfying the following control conditions:

where is a bounded sequence in . Then the sequence converges weakly to a minimizer of .

Proof.

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

## Declarations

### 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.

## Authors’ Affiliations

## References

- Eckstein J:
**Approximate iterations in Bregman-function-based proximal algorithms.***Mathematical Programming*1998,**83**(1, Ser. A):113–123. 10.1007/BF02680553MathSciNetMATHGoogle Scholar - Rockafellar RT:
**Monotone operators and the proximal point algorithm.***SIAM Journal on Control and Optimization*1976,**14**(5):877–898. 10.1137/0314056MathSciNetView ArticleMATHGoogle Scholar - Güler O:
**On the convergence of the proximal point algorithm for convex minimization.***SIAM Journal on Control and Optimization*1991,**29**(2):403–419. 10.1137/0329022MathSciNetView ArticleMATHGoogle Scholar - Ceng L-C, Wu S-Y, Yao J-C:
**New accuracy criteria for modified approximate proximal point algorithms in Hilbert spaces.***Taiwanese Journal of Mathematics*2008,**12**(7):1691–1705.MathSciNetMATHGoogle Scholar - Cho YJ, Kang SM, Zhou H:
**Approximate proximal point algorithms for finding zeroes of maximal monotone operators in Hilbert spaces.***Journal of Inequalities and Applications*2008,**2008:**-10.Google Scholar - Qin X, Kang SM, Cho YJ:
**Approximating zeros of monotone operators by proximal point algorithms.***Journal of Global Optimization*2010,**46**(1):75–87. 10.1007/s10898-009-9410-6MathSciNetView ArticleMATHGoogle Scholar - Ceng L-C, Yao J-C:
**On the convergence analysis of inexact hybrid extragradient proximal point algorithms for maximal monotone operators.***Journal of Computational and Applied Mathematics*2008,**217**(2):326–338. 10.1016/j.cam.2007.02.010MathSciNetView ArticleMATHGoogle Scholar - Ceng L-C, Yao J-C:
**Generalized implicit hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space.***Taiwanese Journal of Mathematics*2008,**12**(3):753–766.MathSciNetMATHGoogle Scholar - Ceng LC, Petruşel A, Wu SY:
**On hybrid proximal-type algorithms in Banach spaces.***Taiwanese Journal of Mathematics*2008,**12**(8):2009–2029.MathSciNetMATHGoogle Scholar - Ceng L-C, Huang S, Liou Y-C:
**Hybrid proximal point algorithms for solving constrained minimization problems in Banach spaces.***Taiwanese Journal of Mathematics*2009,**13**(2):805–820.MathSciNetMATHGoogle Scholar - Browder FE:
**Nonlinear operators and nonlinear equations of evolution in Banach spaces.***Proceedings of Symposia in Pure Mathematics*1976,**18:**78–81.MathSciNetGoogle Scholar - Bruck, RE Jr.:
**A strongly convergent iterative solution of for a maximal monotone operator in Hilbert space.***Journal of Mathematical Analysis and Applications*1974,**48:**114–126. 10.1016/0022-247X(74)90219-4MathSciNetView ArticleMATHGoogle Scholar - Burachik RS, Iusem AN, Svaiter BF:
**Enlargement of monotone operators with applications to variational inequalities.***Set-Valued Analysis*1997,**5**(2):159–180. 10.1023/A:1008615624787MathSciNetView ArticleMATHGoogle Scholar - Censor Y, Zenios SA:
**Proximal minimization algorithm with -functions.***Journal of Optimization Theory and Applications*1992,**73**(3):451–464. 10.1007/BF00940051MathSciNetView ArticleMATHGoogle Scholar - Cohen G:
**Auxiliary problem principle extended to variational inequalities.***Journal of Optimization Theory and Applications*1988,**59**(2):325–333.MathSciNetMATHGoogle Scholar - Deimling K:
**Zeros of accretive operators.***Manuscripta Mathematica*1974,**13:**365–374. 10.1007/BF01171148MathSciNetView ArticleMATHGoogle Scholar - Dembo RS, Eisenstat SC, Steihaug T:
**Inexact Newton methods.***SIAM Journal on Numerical Analysis*1982,**19**(2):400–408. 10.1137/0719025MathSciNetView ArticleMATHGoogle Scholar - Han D, He B:
**A new accuracy criterion for approximate proximal point algorithms.***Journal of Mathematical Analysis and Applications*2001,**263**(2):343–354. 10.1006/jmaa.2001.7535MathSciNetView ArticleMATHGoogle Scholar - Kamimura S, Takahashi W:
**Approximating solutions of maximal monotone operators in Hilbert spaces.***Journal of Approximation Theory*2000,**106**(2):226–240. 10.1006/jath.2000.3493MathSciNetView ArticleMATHGoogle Scholar - Kamimura S, Takahashi W:
**Strong convergence of a proximal-type algorithm in a Banach space.***SIAM Journal on Optimization*2002,**13**(3):938–945. 10.1137/S105262340139611XMathSciNetView ArticleMATHGoogle Scholar - Lan KQ, Wu JH:
**Convergence of approximants for demicontinuous pseudo-contractive maps in Hilbert spaces.***Nonlinear Analysis*2002,**49**(6):737–746. 10.1016/S0362-546X(01)00130-4MathSciNetView ArticleMATHGoogle Scholar - Nevanlinna O, Reich S:
**Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces.***Israel Journal of Mathematics*1979,**32**(1):44–58. 10.1007/BF02761184MathSciNetView ArticleMATHGoogle Scholar - Opial Z:
**Weak convergence of the sequence of successive approximations for nonexpansive mappings.***Bulletin of the American Mathematical Society*1967,**73:**591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar - Pazy A:
**Remarks on nonlinear ergodic theory in Hilbert space.***Nonlinear Analysis*1979,**3**(6):863–871. 10.1016/0362-546X(79)90053-1MathSciNetView ArticleMATHGoogle Scholar - Qin X, Su Y:
**Approximation of a zero point of accretive operator in Banach spaces.***Journal of Mathematical Analysis and Applications*2007,**329**(1):415–424. 10.1016/j.jmaa.2006.06.067MathSciNetView ArticleMATHGoogle Scholar - Solodov MV, Svaiter BF:
**An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions.***Mathematics of Operations Research*2000,**25**(2):214–230. 10.1287/moor.25.2.214.12222MathSciNetView ArticleMATHGoogle Scholar - Teboulle M:
**Convergence of proximal-like algorithms.***SIAM Journal on Optimization*1997,**7**(4):1069–1083. 10.1137/S1052623495292130MathSciNetView ArticleMATHGoogle Scholar - Tan K-K, Xu HK:
**Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.***Journal of Mathematical Analysis and Applications*1993,**178**(2):301–308. 10.1006/jmaa.1993.1309MathSciNetView ArticleMATHGoogle Scholar - Verma RU:
**Rockafellar's celebrated theorem based on -maximal monotonicity design.***Applied Mathematics Letters*2008,**21**(4):355–360. 10.1016/j.aml.2007.05.004MathSciNetView 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 - Cho YJ, Zhou H, Guo G:
**Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings.***Computers & Mathematics with Applications*2004,**47**(4–5):707–717. 10.1016/S0898-1221(04)90058-2MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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.