Strong convergence of an extragradient-type algorithm for the multiple-sets split equality problem.

This paper introduces a new extragradient-type method to solve the multiple-sets split equality problem (MSSEP). Under some suitable conditions, the strong convergence of an algorithm can be verified in the infinite-dimensional Hilbert spaces. Moreover, several numerical results are given to show the effectiveness of our algorithm.

Introduction

The split feasibility problem (SFP) was first presented by Censor et al. [1]; it is an inverse problem that arises in medical image reconstruction, phase retrieval, radiation therapy treatment, signal processing etc. The SFP can be mathematically characterized by finding a point x that satisfies the property if such a point exists, where C and Q are nonempty closed convex subsets of Hilbert spaces and , respectively, and is a bounded and linear operator.

There are various algorithms proposed to solve the SFP, see [2-4] and the references therein. In particular, Byrne [5, 6] introduced the CQ-algorithm motivated by the idea of an iterative scheme of fixed point theory. Moreover, Censor et al. [7] introduced an extension upon the form of SFP in 2005 with an intersection of a family of closed and convex sets instead of the convex set C, which is the original of the multiple-sets split feasibility problem (MSSFP).

Subsequently, an important extension, which goes by the name of split equality problem (SEP), was made by Moudafi [8]. It can be mathematically characterized by finding points and that satisfy the property if such points exist, where C and Q are nonempty closed convex subsets of Hilbert spaces and , respectively, is also a Hilbert space, and are two bounded and linear operators. When , the SEP reduces to SFP. For more information about the methods for solving SEP, see [9, 10].

This paper considers the multiple-sets split equality problem (MSSEP) which generalizes the MSSFP and SEP and can be mathematically characterized by finding points x and y that satisfy the property where are positive integers, and are nonempty, closed and convex subsets of Hilbert spaces and , respectively, is also a Hilbert space, , are two bounded and linear operators. Obviously, if , the MSSEP is just right MSSFP; if , the MSSEP changes into the SEP. Moreover, when and , the MSSEP reduces to the SFP.

One of the most important methods for computing the solution of a variational inequality and showing the quick convergence is an extragradient algorithm, which was first introduced by Korpelevich [11]. Moreover, this method was applied for finding a common element of the set of solutions for a variational inequality and the set of fixed points of a nonexpansive mapping, see Nadezhkina et al. [12]. Subsequently, Ceng et al. in [13] presented an extragradient method, and Yao et al. in [14] proposed a subgradient extragradient method to solve the SFP. However, all these methods to solve the problem have only weak convergence in a Hilbert space. On the other hand, a variant extragradient-type method and a subgradient extragradient method introduced by Censor et al. [15, 16] possess strong convergence for solving the variational inequality.

Motivated and inspired by the above works, we introduce an extragradient-type method to solve the MSSEP in this paper. Under some suitable conditions, the strong convergence of an algorithm can be verified in the infinite-dimensional Hilbert spaces. Finally, several numerical results are given to show the feasibility of our algorithm.

Preliminaries

Let H be a real Hilbert space whose inner product and norm are denoted by and , respectively. Let I denote the identity operator on H.

Next, we recall several definitions and basic results that will be available later.

Definition 2.1

A mapping goes by the name of

nonexpansive if

firmly nonexpansive if

contractive on x if there exists such that

monotone if

β-inverse strongly monotone if there exists such that

The following properties of an orthogonal projection operator were introduced by Bauschke et al. in [17], and they will be powerful tools in our analysis.

Proposition 2.2

[17]

Let be a mapping from H onto a closed, convex and nonempty subset C of H if then is called an orthogonal projection from H onto C. Furthermore, for any and ,

;

;

.

The following lemmas provide the main mathematical results in the sequel.

Lemma 2.3

[18]

Let C be a nonempty closed convex subset of a real Hilbert space H, let be α-inverse strongly monotone, and let be a constant. Then, for any ,

Moreover, when , is nonexpansive.

Lemma 2.4

[19]

Let and be bounded sequences in a Hilbert space H, and let be a sequence in which satisfies the condition . Suppose that for all and . Then .

The lemma below will be a powerful tool in our analysis.

Lemma 2.5

[20]

Let be a sequence of nonnegative real numbers satisfying the condition where , are sequences of real numbers such that

and or, equivalently,

or

is convergent. Then .

Main results

In this section, we propose a formal statement of our present algorithm. Review the multiple-sets split equality problem (MSSEP), without loss of generality, suppose in (1.3) and define . Hence, MSSEP (1.3) is equivalent to the following problem:

Moreover, set , , , the adjoint operator of G is denoted by , then the original problem (3.1) reduces to

Theorem 3.1

Let be the solution set of MSSEP (3.2). For an arbitrary initial point , the iterative sequence can be given as follows: where is a sequence in such that , and , , are sequences in H satisfying the following conditions: Then converges strongly to a solution of MSSEP (3.2).

Proof

In view of the property of the projection, we infer for any . Further, from the condition in (3.4), we get that , it follows that is nonexpansive. Hence,

Since as and from the condition in (3.4), , it follows that as , that is, . We deduce that which is equivalent to Substituting (3.7) in (3.5), we obtain By induction,

Consequently, is bounded, and so is .

Let . From Proposition 2.2, one can know that the projection operator is monotone and nonexpansive, and is nonexpansive.

Therefore, that is, where .

Indeed, For convenience, let . By Lemma 2.5 in Shi et al. [1], it follows that is nonexpansive and averaged. Hence, Moreover, Substituting (3.11) in (3.10), we infer that

By virtue of , it follows that . Moreover, and are bounded, and so is . Therefore, (3.12) reduces to Applying (3.13) and Lemma 2.4, we get Combining (3.14) with (3.8), we obtain

Using the convexity of the norm and (3.5), we deduce that which implies that Since , and , we infer that

Applying Proposition 2.2 and the property of the projection , one can easily show that where satisfies

From (3.5) and (3.16), we get which means that Since , and , we infer that

Finally, we show that . Using the property of the projection , we derive which equals It follows from (3.5) and (3.17) that

Since , we observe that , then that is to say, By virtue of , and is bounded, we obtain , which implies that Moreover, it follows that all the conditions of Lemma 2.5 are satisfied. Combining (3.18), (3.19) and Lemma 2.5, we can show that . This completes the proof. □

Numerical experiments

In this section, we provide several numerical results and compare them with Tian’s [21] algorithm (3.15)’ and Byrne’s [22] algorithm (1.2) to show the effectiveness of our proposed algorithm. Moreover, the sequence given by our algorithm in this paper has strong convergence for the multiple-sets split equality problem. The whole program was written in Wolfram Mathematica (version 9.0). All the numerical results were carried out on a personal Lenovo computer with Intel(R)Pentium(R) N3540 CPU 2.16 GHz and RAM 4.00 GB.

In the numerical results, , , where , are all given randomly, are positive integers. The initial point , and , , , , in Theorem 3.1, in Tian’s (3.15)’ and in Byrne’s (1.2). The termination condition is . In Tables 1-4, the iterative steps and CPU are denoted by n and t, respectively.

Table 1

	n	t
Sequence (3.3)	60	0.078
Tian’s (3.15)’	117	0.093
Byrne’s (1.2)	1,845	1.125

Table 4

	n	t
Sequence (3.3)	123	0.25
Tian’s (3.15)	948	0.906
Byrne’s (1.2)	13,496	2.437

Table 2

	n	t
Sequence (3.3)	120	0.156
Tian’s (3.15)	294	0.29
Byrne’s (1.2)	8,533	2.734

Table 3

	n	t
Sequence (3.3)	63	0.093
Tian’s (3.15)	426	0.469
Byrne’s (1.2)	2,287	1.313