Abstract and Applied Analysis

Volume 2012 (2012), Article ID 432501, 9 pages

doi:10.1155/2012/432501

Research Article

An Explicit Method for the Split Feasibility Problem with Self-Adaptive Step Sizes

Youli Yu

School of Mathematics and Information Engineering, Taizhou University, Linhai 317000, China

Received 12 September 2012; Accepted 24 October 2012

Academic Editor: Wenming Zou

Copyright © 2012 Youli Yu. 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.

Abstract.

An explicit iterative method with self-adaptive step-sizes for solving the split feasibility problem is presented. Strong convergence theorem is provided.

1. Introduction

Since its publication in 1994, the split feasibility problem has been studied by many authors. For some related works, please consult [1–18]. Among them, a more popular algorithm that solves the split feasibility problems is Byrne’s method [2]:

where and are two closed convex subsets of two real Hilbert spaces and , respectively, and is a bounded linear operator. The algorithm only involves the computations of the projections and onto the sets and , respectively, and is therefore implementable in the case where and have closed-form expressions.

Note that algorithm can be obtained from optimization. If we set

then the convex objective is differentiable and has a Lipschitz gradient given by

Thus, the algorithm can be obtained by minimizing the following convex minimization problem

We can use a gradient projection algorithm below to solve the split feasibility problem:

where , the step size at iteration , is chosen in the interval , where is the Lipschitz constant of .

However, we observe that the determination of the step size depends on the operator (matrix) norm (or the largest eigenvalue of ). This means that in order to implement the algorithm, one has first to compute (or, at least, estimate) the matrix norm of , which is in general not an easy work in practice. To overcome the above difficulty, the so-called self-adaptive method which permits step size being selected self-adaptively was developed. See, for example, [10, 14, 15, 19–23].

Inspired by the above results and the self-adaptive method, in this paper, we present an explicit iterative method with self-adaptive step sizes for solving the split feasibility problem. Convergence analysis result is given.

2. Preliminaries

Let and be two real Hilbert spaces and and two closed convex subsets of and , respectively. Let be a bounded linear operator. The split feasibility problem is to find a point such that

Next, we use to denote the solution set of the split feasibility problem, that is, .

We know that a point is a stationary point of problem (1.4) if it satisfies

Given . solves the split feasibility problem if and only if solves the fixed point equation

Next we adopt the following notation:

(i) means that converges strongly to ;(ii) means that converges weakly to ;(iii) is the weak -limit set of the sequence .

Recall that a function is called convex if

for all and , . It is known that a differentiable function is convex if and only if there holds the relation:

for all . Recall that an element is said to be a subgradient of at if

for all . If the function has at least one subgradient at is said to be sub-differentiable at . The set of subgradients of at the point is called the subdifferential of at , and is denoted by . A function is called sub-differentiable if it is subdifferentiable at all . If is convex and differentiable, then its gradient and subgradient coincide. A function is said to be weakly lower semi continuous (w-lsc) at if implies

is said to be w-lsc on if it is w-lsc at every point .

A mapping is called nonexpansive if

for all .

Recall that the (nearest point or metric) projection from onto , denoted , assigns, to each , the unique point with the property

It is well known that the metric projection of onto has the following basic properties:

(a) for all ;(b) for every ;(c) for all , .

Lemma 2.1 (see [24]). Assume that is a sequence of nonnegative real numbers such that

where is a sequence in and is a sequence such that

(1); (2) or .

Then .

Lemma 2.2 (see [25]). Let be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence of such that for all . For every , define an integer sequence as

Then as and for all

3. Main Results

In this section, we will introduce our algorithm and prove our main results.

Let and be nonempty closed convex subsets of real Hilbert spaces and , respectively. Let be a bounded linear operator. In the sequel, we assume that the split feasibility problem is consistent, that is .

Algorithm 3.1. For and given , let the sequence defined by

where and .

Remark 3.2. In the sequel, we may assume that for all . Note that this fact can be guaranteed if the sequence is infinite; that is, Algorithm 3.1 does not terminate in a finite number of iterations.

Theorem 3.3. Assume that the following conditions are satisfied:

(i) and ;(ii).

Then defined by (3.1) converges strongly to .

Proof. Let . It follows that for all . From (2.5), we deduce that

Thus, by (3.1) and (3.2), we have

It follows that

By induction, we deduce

Hence, is bounded.

At the same time, we note that

Therefore,

It follows that

Next, we will prove that . Set for all . Since and , we may assume without loss of generality that for some . Thus, we can rewrite (3.8) as

where .

Now, we consider two possible cases.

Case  1. Assume that is eventually decreasing; that is, there exists such that is decreasing for . In this case, must be convergent and from (3.9) it follows that

where is a constant such that . Letting in (3.10), we get

Since is bounded, there exists a subsequence of converging weakly to . Since, , we also have of converging weakly to . From the weak lower semicontinuity of , we have

Hence, ; that is, . This indicates that

Furthermore, by using the property of the projection , we deduce

From (3.8), we obtain

This together with Lemma 2.1 imply that .

Case  2. Assume that is not eventually decreasing. That is, there exists an integer such that . Thus, we can define an integer sequence for all as follows:

Clearly, is a nondecreasing sequence such that as and

for all . In this case, we derive from (3.10) that

It follows that

This implies that every weak cluster point of is in the solution set ; that is, . So, . On the other hand, we note that

From which we can deduce that

Since , we have from (3.9) that

Combining (3.21) and (3.22) yields

and hence

From (3.15), we have

Thus,

From Lemma 2.2, we have

Therefore, . That is, . This completes the proof.

Acknowledgment

The author was supported in part by the Youth Foundation of Taizhou University (2011QN11).

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