The Ball-Relaxed Gradient-Projection Algorithm for Split Feasibility Problem

Abstract

In this paper, we concern with the split feasibility problem (SFP) whenever the convex sets involved are composed of level sets. By applying Gradient-projection algorithm which is used to solve constrained convex minimization problem of a real valued convex function, we construct two new algorithms for the split feasibility problem and prove that both of them are convergent weakly to a solution of the feasibility problem. In the end, as an application, we obtain a new algorithm for solving the split equality problem.

1. Introduction

The split feasibility problem (SFP) was first introduced by Censor and Elfving [1]. And it is formulated as finding a point satisfying the property: where and are nonempty, closed, and convex subsets of real Hilbert spaces and , respectively, and is a bounded linear operator from to .

Many inverse problems arising from real world can be summarized as SFP. So, the SFP has attracted many scholars to study it. Of course, various algorithms by far have been invented to solve the SFP (see, e.g., [2–9]). And one of the most famous methods for solving the SFP is algorithm which is introduced by Byrne. Take an initial guess and define a recursively equation as where and denotes the largest eigenvalue of the matrix . Later, Byrne introduced another recursively equation as where . Byrne has shown that the sequence produced by (2) or (3) is convergent weakly to a solution of the SFP (1). From (2) and (3), we know that we have first to compute or estimate the largest eigenvalue of the matrix and the operator norm before implementing algorithms. However, computing or estimating the matrix norm is in general not an easy work in practice. So the conditions that Byrne put on his proposed two algorithms seem restrictive. Recently, Lopez et al. [10] proposed a novel way to construct another stepsize which has no connection with eigenvalue and matrix norm. And Lopez et al. suggested the following stepsize: and showed the weak convergence of the algorithm (3).

In this paper, we consider the SFP whenever both and are level sets for given strongly convex function, that is, where is an -strongly convex lower semicontinuous function and is a -strongly convex lower semicontinuous function. However, in general, computing the orthogonal projections onto and is not an easy task. To overcome this difficulty, Yu et al. [11] have considered the case when and are two sequences of balls defined, respectively, by where and are strongly convex functions with constant and , respectively. Yu et al. [11] have shown that is a ball whose centre is and radius is . And is also a ball whose centre is and radius is . It is easy to see that and for any . Since and are balls, the associated projections can be easily calculated.

The rest of this paper is organized as follows. In Section 2, we review some basic definitions and lemmas that we will use in the remaining sections. In Section 3, we introduce two new ball-relaxed gradient-projection algorithms for solving the SFP and prove the weak convergence of our algorithms. In Section 4, according to the algorithms that we suggest in Section 3, we obtain a new iterative algorithm for solving the split equality problem and establish its weak convergence.

2. Preliminaries

Throughout this paper, we always assume that is a real Hilbert space with the inner product and norm . We denote by the identity operator on and by the set of all cluster points of . The notation “”stands for strong convergence, and “” stands for weak convergence.

Definition 1 ([12]). Let be a nonempty subset of , and let . Then, is (1)Nonexpansive if it is Lipschitz continuous with constant 1, i.e.,(2)Firmly nonexpansive if(3)-inverse strongly monotone (-ism, ) if

Definition 2. Let be a nonempty closed convex subset of . Then, an orthogonal projection is defined by

Lemma 3. Let be a nonempty closed convex subset of ; then, (1)For every and in (2) and both are nonexpansive

Definition 4 (see [12]). Let and be a proper function. (1) is convex if(2) is strongly convex with constant where if(3)A vector is a subgradient of a point if(4)The subdifferential of is the set-valued operator(5) is subdifferentiable at if ; the elements of are the subgradients at

Definition 5. Let be a proper function; then, we can obtain the following: (1) is lower semicontinuous at a point if implies(2) is weakly lower semicontinuous at a point if implies(3) is lower semicontinuous on if it is lower semicontinuous at any point (4) is weakly lower semicontinuous on if it is weakly lower semicontinuous at any point

Lemma 6 (see [12]). Assume that is a proper convex function; then, is a lower semicontinuous function if and only if it is a weakly lower semicontinuous function.

Definition 7. Let be a nonempty closed convex subset in , and is a sequence in , if for any and , we have and then, we say that the sequence is Fejér-monotone with respect to .

Lemma 8 (see [12]). Assume that a sequence in is Fejér-monotone with respect to which is a nonempty closed convex subset of ; then, is weakly convergent to a point of if and only if its any weak cluster point belongs to .

3. The Algorithm Proposed and Proved

In this section, we still concern with the case the involved subsets are composed of level sets, that is, the case whenever and are given by (5) and (6). In this case, we shall assume that problem (1) is consistent, namely, its solution set, denoted by , is nonempty. Besides, we need to assume that and are bounded on bounded sets.

We know that, in , when consider the constrained convex minimization problem of a real valued convex function , one of the most famous methods is the gradient projection algorithm (GPA) that generates a sequence according to the recursive formula where the parameter is a sequence of positive real numbers. And if is Lipschitz continuous with constant and the parameter sequence satisfies , Xu [13] showed that the sequence generated by the GPA converges weakly to a minimizer of .

We know that the solution of the SFP amounts to unconstrained minimization of and . By a simple calculation, we have is Lipschitz continuous with . In this case, method (21) is reduced to

From the above equation, we know that the implementation of the above iterative algorithm needs to calculate the projections onto and first. Since and are both level sets defined by (5) and (6), it is very difficult to calculate the projection onto the level set at each step. To facilitate the computation of the projection onto and , we will compute the projections onto and which were defined by (7) and (8) instead of and . Now, we give our ball-relaxed gradient-projection algorithm for solving the SFP (1).

Algorithm 1. Let be arbitrary, and generate according to the following iterative formula. where and are defined by (7) and (8), and the parameter sequence satisfies , where and are two positive real numbers and .

Theorem 9. Let be the sequence generated by Algorithm 1; then, converges weakly to a solution of the SFP (1).

Proof. On the one hand, we will show that the sequence is Fejér-monotone with respect to .
Now, denoted by ; then, (24) is equivalent to . For any , that is, , and , we have From Lemma 3, we obtain Besides, Substituting (26) and (27) into (25), we have Since , we can obtain . The above inequality (28) implies that is Fejér-monotone with respect to .
On the other hand, we will show that any weak cluster point of the sequence belongs to the solution set , i.e., .
The inequality (28) implies that the sequence is bounded and converges to some finite limit. Passing to the limit in (28), we have that is, Besides, since the sequence is Fejér-monotone with respect to , it is bounded, and so is the sequence . Then, there exists a subsequence of convergent weakly to . What is more, due to and are bounded on bounded sets, there are two constants and , such that and for all , with .
From (7) and the fact that , we obtain Since is convex and lower semicontinuous, then it is also weakly lower semicontinuous by Lemma 6. This together with (31) implies that It turns out that .
Similarly, from (8) and the fact that , we obtain Since is convex and lower semicontinuous, then it is also weakly lower semicontinuous by Lemma 6. This together with (33) implies that It turns out that . Then, which implies that . From Lemma 8, we can get converges weakly to as . This completes the proof.

From Algorithm 1, we know that we have first to estimate the operator norm so that we can select appropriate parameters to implement Algorithm 1. However, computing or estimating the matrix norm is not an easy work in practice. To overcome this difficulty, we construct a variable stepsize that does not require the matrix norm.

Algorithm 2. Let be arbitrary, and generate according to the following iterative formula. where and are defined by (7) and (8), and the parameter sequence is given by where is a sequence of positive real numbers and is any positive real number.

Theorem 10. Let be the sequence generated by Algorithm 2; if , then converges weakly to a solution of the SFP (1).

Proof. For any , that is, , and , we have From Lemma 8, we obtain Substituting (36) and (38) into (37), we have The above inequality (39) implies that is Fejér-monotone with respect to .
Nextly, we show that any weak cluster point of the sequence belongs to the solution set , i.e., . The inequality (39) implies that the sequence is bounded and converges to some finite limit. Passing to the limit in (39), we have while so, we obtain The rest proof is similar to Theorem 9.

Remark 11. Recently, there are some new results about SFP, such as Shehu and Gibali introduced a relaxed method with alternated inertial step for solving SFP in [14]; Shehu et al. introduced some new computational technique for solving proximal SFP by using a modified proximal split feasibility algorithm in [15–17]. Our computational techniques are different from theirs.

4. Application in the Split Equality Problem

The split equality (SEP) is an inverse problem that requests to finding where and are two linear bounded operators, and and are two nonempty closed convex subsets.

We know that, for any and in , if we defined where and are defined as the same to (43). Then, it has been proved that is a bounded linear operator [11], and the SEP can be regarded as a special SFP: find , such that where , . Motivated by (35), we can propose a new method for solving the SEP (43).

Algorithm 3. For any arbitrary initial guess , define recursively by the following. where and are, respectively, given by (7) and (8), and parameter sequence is chosen as where is a sequence of positive real numbers, and is any positive real number.

Theorem 11. Let be the sequence generated by Algorithm 3, and if , then converges weakly to a solution of the SEP (43).

Proof. Let , and is a bounded linear operator which is defined by ; then, Algorithm 3 can be rewritten as where . Hence, by applying Theorem 10, we can conclude that converges weakly to a solution of the SEP (43).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed to each part of this work equally, and they all read and approved the final manuscript.

Acknowledgments

This research is supported by the Teacher Education Curriculum Reform Project of Henan Province (2021-JSJYYB-019), the Postgraduate Education Reform and Quality Improvement Project of Henan Province (YJS2022ZX33), the higher education teaching reform and practice project (postgraduate education) of Henan Normal University (YJS2019JG01), and the Scientific Research Project for Postgraduates of Henan Normal University (YL202121).