- Research Article
- Open Access
Iterative Approaches to Find Zeros of Maximal Monotone Operators by Hybrid Approximate Proximal Point Methods
© Lu Chuan Ceng et al. 2011
- Received: 18 August 2010
- Accepted: 23 September 2010
- Published: 11 October 2010
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.
- Convex Subset
- Iterative Algorithm
- Nonexpansive Mapping
- Maximal Monotone
- Real Hilbert Space
where ( ) is a sequence of real numbers. However, as pointed out in , 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 , 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.  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.  proved the following strong convergence result.
They also derived the following weak convergence theorem.
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.
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 2.2 (see [30, Lemma 2.5, page 243]).
Lemma 2.3 (see [28, Lemma 1, page 303]).
Lemma 2.4 (see ).
Lemma 2.5 (see ).
It is clear that the following lemma is valid.
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:
Next, let us show the sufficiency. The proof is divided into several steps.
The existence of is guaranteed by Lemma 1 of Bruck .
This completes the proof.
we can see that Theorem 3.1 still holds.
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.
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:
Next, we give a hybrid Mann-type iterative algorithm and study the weak convergence of the algorithm.
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:
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 ) 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.
Utilizing Theorem 3.5, we also obtain the following result immediately.
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:
By using Theorem 3.1, we can obtain the desired result immediately.
We can obtain the desired result readily from the proof of Theorems 3.5 and 4.1.
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.
- 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.22.214.171.12422MathSciNetView 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
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.