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# Convergence of a proximal point algorithm for maximal monotone operators in Hilbert spaces

Zhiqiang Wei1 and Guohong Shi2*

Author Affiliations

1 School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

2 College of Science, Hebei University of Engineering, Handan 056038, China

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Journal of Inequalities and Applications 2012, 2012:137 doi:10.1186/1029-242X-2012-137

 Received: 9 February 2012 Accepted: 13 June 2012 Published: 13 June 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this article, we consider the proximal point algorithm for the problem of approximating zeros of maximal monotone mappings. Strong convergence theorems for zero points of maximal monotone mappings are established in the framework of Hilbert spaces.

2000 AMS Subject Classification: 47H05; 47H09; 47J25.

##### Keywords:
fixed point; nonexpansive mapping; maximal monotone mapping; zero

### 1. Introduction

The theory of maximal monotone operators has emerged as an effective and powerful tool for studying many real world problems arising in various branches of social, physical, engineering, pure and applied sciences in unified and general framework. Recently, much attention has been payed to develop efficient and implementable numerical methods including the projection method and its variant forms, auxiliary problem principle, proximal-point algorithm and descent framework for solving variational inequalities and related optimization problems (see [1-32] and the references therein). The proximal point algorithm, can be traced back to Martinet [33] in the context of convex minimization and Rockafellar [34] in the general setting of maximal monotone operators, has been extended and generalized in different directions by using novel and innovative techniques and ideas.

In this article, we investigate the problem of approximating a zero of the maximal monotone mapping based on a proximal point algorithm in the framework of Hilbert spaces. Strong convergence of the iterative algorithm is obtained.

### 2. Preliminaries

Throughout this article, we assume that H is a real Hilbert space, whose inner product and norm are denoted by 〈·,·〉 and ǀǀ · ǀǀ, respectively. Let T be a set-valued mapping.

(a) The set D(T) defined by

is called the effective domain of T.

(b) The set R(T) defined by

is called the range of T.

(c) The set G(T) defined by

is said to be the graph of T.

Recall the following definitions.

(c) T is said to be monotone if

(d) T is said to be maximal monotone if it is not properly contained in any other monotone operator.

For a maximal monotone T : D(T) → 2H, we can defined the resolvent of T by

(2.1)

It is well known that Jt: H D(T) is nonexpansive, and F(Jt) = T-1(0), where F(Jt) denotes the set of fixed points of Jt. The Yosida approximation Tt is defined by

It is well known that Ttx T Jtx, ∀x H and ǁTtxǁ ≤ ǀTxǀ, where

for all x D(T).

Let C be a nonempty, closed and convex subset of H. Next, we always assume that T: C → 2H is a maximal monotone mapping with , where T-1(0) denotes the set of zeros of T.

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 devoting to the studies on the existence and convergence of zero points for maximal monotone mappings. A classical method to solve the following set-valued equation

(2.2)

is the proximal point algorithm. To be more precise, start with any point x0 H, and update xn+1 iteratively conforming to the following recursion

(2.3)

where {βn} ⊂ [β, ∞), (β > 0) is a sequence of real numbers. However, as pointed in [15], the ideal form of the method is often impractical since, in many cases, to solve the problem (2.3) exactly is either impossible or the same difficult as the original problem (2.2). 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 approximately zeroes of T.

In 1976, Rockafellar [35] gave an inexact variant of the method

(2.4)

where {en} is regarded as an error sequence. This is an inexact proximal point algorithm. It was shown that, if

(A)

then the sequence {xn} defined by (1.4) converges weakly to a zero of T provided that . In [16], Güller obtained an example to show that Rockafellar's proximal point algorithm (1.4) does not converge strongly, in general.

Recently, many authors studied the problems of modifying Rockafellar's proximal point algorithm so that strong convergence is guaranteed. Cho et al. [13] proved the following result.

Theorem CKZ. Let H be a real Hilbert space, a nonempty closed convex subset of H, and T: Ω → 2H a maximal monotone operator with . Let Pbe the metric projection of H onto Ω. Suppose that, for any given xn H, βn > 0 and en H, there exists conforming to the SVME (2.4), where {βn} ⊂ (0, + ∞) with βn → ∞ as n → ∞ and

(B)

Let {αn} be a real sequence in [0, 1] such that

(i) αn → 0 as n → ∞,

(ii)

for any fixed u ∈ Ω, define the sequence {xn} iteratively as follows:

Then {xn} converges strongly to a fixed point z of T, where z = limt→ ∞ Jtu.

In this article, motivated by Theorem CKZ, we continue to consider the problem of approximating a zero of the maximal monotone mapping T. Strong convergence theorems are established under mild restrictions imposed on the error sequence {en} comparing with the restriction (B). The results which include Cho et al. [13] as a special case also improve the corresponding results announced by many others.

In order to prove our main result, we need the following lemmas.

Lemma 2.1. (Bruck [[35], Lemma 1]). Let H be a Hilbert space and C a nonempty, closed and convex subset H. For all u C, limt→ ∞ Jtu exists and it is the point of T-1(0) nearest u.

Lemma 2.2 (Eckstein [[15], Lemma 2]). For any given xn C, λn > 0, and en H, there exists conforming to the following set-valued mapping equation (in short, SVME):

(2.5)

Furthermore, for any p T-1(0), we have

and

Lemma 2.3 (Liu [36]). Assume that {αn} is a sequence of nonnegative real numbers such that

where n} is a sequence in (0,1) and {δn} is a sequence such that

(i) ;

(ii) lim supn→∞ δn/γn ≤ 0 or .

Then limn→∞ αn = 0.

### 3. Main results

Theorem 3.1. Let H be a real Hilbert space, C a nonempty, closed and convex subset of H and T: C → 2H a maximal monotone operator with . Let PC be a metric projection from H onto C. For any xn H and λn > 0, find and en H conforming to the SVME (2.5), where {λn} ⊂ (0, ∞) with λn → ∞ as n → ∞ and

(C)

with supn≥0 ηn = η < 1. Let {αn} and {βn} be real sequences in [0, 1] satisfying αn + βn < 1 and the following control conditions:

Let {xn} be a sequence generated by the following manner:

(3.1)

where u C is a fixed element. Then the sequence {xn} generated by (3.1) strongly converges to a zero point z of T, where z = limt→∞ Jtu, if and only if en → 0 as n → ∞.

Proof. First, show that the necessity. Assume that xn z as n → ∞, where z T-1(0). It follows from (2.5) that

This implies that

It follows that as n → ∞. Note that

This shows that en → 0 as n → ∞.

Next, we show the sufficiency. The proof is divided into three steps.

Step 1. Show that {xn} is bounded.

From the assumption (C), we see that

For any p T-1 (0), it follows from Lemma 2.2 that

That is,

(3.2)

It follows from (3.2) that

(3.3)

Putting

we show that ǀǀxnǀǀ ≤ M for all n ≥ 0. It is easy to see that the result holds for n = 0. Assume that the result holds for some n ≥ 0. That is, ǀǀxn - pǀǀ ≤ M. Next, we prove that ǀǀxn+1 - pǀǀ ≤ M. Indeed, we see from (3.3) that

This shows that the sequence {xn} is bounded.

Step 2. Show that lim supn→∞u - z, xn+1 -z〉 ≤ 0, where z = limt→∞ Jtu.

From Lemma 2.1, we see that limt→∞ Jtu exists, which is the point of T-1(0) nearest to u. Since T is maximal monotone, Ttu TJtu and Tλn xn TJλn xn, we see

Since λn → ∞ as n → ∞, for any t > 0, we have

(3.4)

On the other hand, by the nonexpansivity of Jλn, we obtain

From the assumption en → 0 as n → ∞ and (3.4), we arrive at

(3.5)

From (2.5), we see that

That is,

(2.6)

Combining (3.5) with (3.6), we arrive at

(3.7)

On the other hand, we see from the algorithm (3.1) that

It follows from the condition limn→∞ αn = limn→∞ βn = 0 that

which combines with (3.7) yields that

(3.8)

From z = limt→∞ Jtu and (3.8), we arrive at

(3.9)

Step 3. Show that xn z as n → ∞.

It follows from (3.2) that

This implies that

(3.10)

Applying Lemma 2.3 to (3.10), we obtain that xn z as n → ∞. This completes the proof.

As a corollary of Theorem 3.1, we have the following.

Corollary 3.2. Let H be a real Hilbert space, C a nonempty, closed and convex subset of H and T: C → 2H a maximal monotone operator with . Let PC be a metric projection from H onto C. For any xn H and λn > 0, find and en H conforming to the SVME (2.5), where {λn} ⊂ (0, ∞) with λn → ∞ as n → ∞ and

with supn≥0 ηn = η < 1. Let {αn} be a real sequence in (0,1) satisfying the following control conditions:

Let {xn} be a sequence generated by the following manner:

where u C is a fixed element. Then the sequence {xn} strongly converges to a zero point z of T, where z = limt→ ∞, Jtu, if and only if en → 0 as n → ∞.

Remark 3.3. Corollary 3.2 improves Theorem CKZ by relaxing the restriction imposed on the sequence {en}. In [34], Rockafellar obtained a weak convergence by assuming that , see [34] for more details.

Next, as applications of Theorem 3.1, we consider the problem of finding a minimizer of a convex function.

Let H be a Hilbert space, and f: H → (-∞, +∞] be a proper convex lower semi-continuous function. Then the subdifferential ∂f of f is defined as follows:

Theorem 3.4. Let H be a real Hilbert space and f: H → (-∞, +∞] a proper convex lower semi-continuous function. Let {λn} be a sequence in (0, +∞) with λn → ∞ as n → ∞ and {en} a sequence in H with en → ∞ as n → ∞. Assume that

with supn≥0 ηn = η < 1. Let be the solution of SVME (2.5) with T replacing by f. That is,

Let {αn} and {βn} be real sequences in [0, 1] satisfying αn + βn < 1 and the following control conditions:

Let {xn} be a sequence generated by the following manner:

where u H is a fixed element. If , the sequence {xn} converges strongly to a minimizer of f nearest to u.

Proof. Since f: H → (-∞, +∞] is a proper convex lower semi-continuous function, we have that the subdifferential ∂ f of f is maximal monotone by Theorem 1 of [34]. Notice that

is equivalent to the following

It follows that

By Theorem 3.1, we can obtain the desired conclusion immediately.

As a corollary of Theorem 3.4, we have the following.

Corollary 3.5. Let H be a real Hilbert space and f: H → (-∞, +∞] a proper convex lower semi-continuous function. Let {λn} be a sequence in (0, +∞) with λn → ∞ as n → ∞ and {en} a sequence in H with en → ∞ as n → ∞. Assume that

with supn≥0 ηn = η < 1. Let be the solution of SVME (2.5) with T replacing by f. That is,

Let {αn} be a real sequence in [0, 1] ssatisfying the following control conditions:

Let {xn} be a sequence generated by the following manner:

where u H is a fixed element. If , the sequence {xn} converges strongly to a minimizer of f nearest to u.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors contributed equally to this paper. All authors read and approved the final manuscript.

### Acknowledgements

The authors are grateful to the referees for their valuable comments and suggestions which improve the contents of the article.

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