Hyperplane projection algorithm for split equality problems in Hilbert spaces

In this paper, we are concerned with the split equality problem (SEP) in Hilbert spaces. By converting it to a coupled fixed-point equation, we propose a new algorithm for solving the SEP. Whenever the convex sets involved are level sets of given convex functionals, we propose two new relaxed alternating algorithms for the SEP. The first relaxed algorithm is shown to be weakly convergent and the second strongly convergent. A new idea is introduced in order to prove strong convergence of the second relaxed algorithm, which gives an affirmative answer to Moudafi’s question. Finally, preliminary numerical results show the efficiency of the proposed algorithms.

A new convex feasibility problem, the split equality problem (SEP), has been proposed by Moudafi and Byrne. The SEP was solved through the ACQA and ARCQA algorithms. In this paper the SEPs are extended to infinite-dimensional SEPs in Hilbert spaces and we established the strong convergence of a proposed algorithm to a solution of general split equality problems (GSEPs).

In this paper, we propose a projection dynamical system for solving the split equality problem, or more generally the approximate split equality problem, in Hilbert spaces. The proposed dynamical system endows with the continuous behavior with time for Moudafi’s alternating CQ-algorithm and Byrne and Moudafi’s extended CQ-algorithm. Under mild conditions, we prove that the trajectory of the dynamical system converges weakly to a solution of the approximate split equality problem as time variable t goes to $+\infty $. We further derive the exponential-type convergence provided that a bounded linear regularity property holds for the approximate split equality problem. Several numerical examples are given to demonstrate the validity and transient behavior of the proposed method.

ABSTRACTIn this paper, we consider the varying stepsize gradient projection algorithm (GPA) for solving the split equality problem (SEP) in Hilbert spaces, and study its linear convergence. In particular, we introduce a notion of bounded linear regularity property for the SEP, and use it to establish the linear convergence property for the varying stepsize GPA. We provide some mild sufficient conditions to ensure the bounded linear regularity property, and then conclude the linear convergence rate of the varying stepsize GPA. To the best of our knowledge, this is the first work to study the linear convergence for the SEP.

Recently, Moudafi introduced the split equality problem (SEP): find such that Ax = By. In this paper, we show that finding a solution of the SEP is equivalent to finding a solution of a coupled fixed-point equation which is an extension of the coupled equation proposed by Yu and Wang. Based on this coupled fixed-point equation, we propose two new relaxed alternating algorithms for the SEP, and we prove that the first relaxed algorithm is weakly convergent and the second is strongly convergent.

We study a generalized model of a split problem which is a combination of a split feasibility problem with multiple output sets and a split equality problem in real Hilbert spaces. Employing an unconstrained optimization approach, we propose two new iterative algorithms for solving such split problems. Our algorithms do not depend on the norms of the transfer mappings.

The split feasibility problem is an inverse problem which arises in signal processing and medical image reconstruction. So there is practical value in studying it. While both the split equality problem and the split variational inclusion problem are generalized form of the split feasibility problem which are more meaningful than the split feasibility problem. In this paper, fusing the two problems, we research a split inclusion problem and propose relevant methods for solving it. What counts is that not only the proposed algorithms have strong convergence, but also the limit points of the algorithms are the minimal norm solution of the split inclusion problem.

The multiple-sets split equality problem (MSSEP) requires finding a pointx∈∩i=1NCi,y∈∩j=1MQjsuch thatAx=By, whereNandMare positive integers,{C1,C2,…,CN}and{Q1,Q2,…,QM}are closed convex subsets of Hilbert spacesH1,H2, respectively, andA:H1→H3,B:H2→H3are two bounded linear operators. WhenN=M=1, the MSSEP is called the split equality problem (SEP). If B=I, then the MSSEP and SEP reduce to the well-known multiple-sets split feasibility problem (MSSFP) and split feasibility problem (SFP), respectively. One of the purposes of this paper is to introduce an iterative algorithm to solve the SEP and MSSEP in the framework of infinite-dimensional Hilbert spaces under some more mild conditions for the iterative coefficient.

In this paper, we consider split equality generalized mixed equilibrium problem, which is more general than many problems such as split feasibility problem, split equality problem, split equilibrium problem, and so on. We propose a new modified algorithm to obtain strong and weak convergence theorems for split equality generalized mixed equilibrium problem for nonexpansive mappings in Hilbert spaces. Also, we give some applications to other problems. Our results extend some results in the literature.

Let H 1 , H 2 , H 3 be real Hilbert spaces, C ⊆ H 1 , Q ⊆ H 2 be two nonempty closed convex sets, and let A : H 1 → H 3 , B : H 2 → H 3 be two bounded linear operators. The split equality problem (SEP) is finding x ∈ C , y ∈ Q such that A x = B y . Recently, Moudafi has presented the ACQA algorithm and the RACQA algorithm to solve SEP. However, the two algorithms are weakly convergent. It is therefore the aim of this paper to construct new algorithms for SEP so that strong convergence is guaranteed. Firstly, we define the concept of the minimal norm solution of SEP. Using Tychonov regularization, we introduce two methods to get such a minimal norm solution. And then, we introduce two algorithms which are viewed as modifications of Moudafi’s ACQA, RACQA algorithms and KM-CQ algorithm, respectively, and converge strongly to a solution of SEP. More importantly, the modifications of Moudafi’s ACQA, RACQA algorithms converge strongly to the minimal norm solution of SEP. At last, we introduce some other algorithms which converge strongly to a solution of SEP.

The multiple-sets split equality problem (MSSEP) requires finding a point x ∈ ⋂ i = 1 N C i , y ∈ ⋂ j = 1 M Q j , such that A x = B y , where N and M are positive integers, { C 1 , C 2 , … , C N } and { Q 1 , Q 2 , … , Q M } are closed convex subsets of Hilbert spaces H 1 , H 2 , respectively, and A : H 1 → H 3 , B : H 2 → H 3 are two bounded linear operators. When N = M = 1 , the MSSEP is called the split equality problem (SEP). If let B = I , then the MSSEP and SEP reduce to the well-known multiple-sets split feasibility problem (MSSFP) and split feasibility problem (SFP), respectively. Recently, some authors proposed many algorithms to solve the SEP and MSSEP. However, to implement these algorithms, one has to find the projection on the closed convex sets, which is not possible except in simple cases. One of the purposes of this paper is to study the SEP and MSSEP for a family of quasi-nonexpansive multi-valued mappings in the framework of infinite-dimensional Hilbert spaces, and propose an algorithm to solve the SEP and MSSEP without the need to compute the projection on the closed convex sets.

In this paper, we studied the split equality problems (SEP) with a new proposed iterative algorithm and established the strong convergence of the proposed algorithm to solution of the split equality problems (SEP).

The split equality problem has board applications in many areas of applied mathematics. Many researchers studied this problem and proposed various algorithms to solve it. From the literature we know that most algorithms for the split equality problems came from the idea of the projected Landweber algorithm proposed by Byrne and Moudafi (Working paper UAG, 2013), and few algorithms came from the idea of the alternating CQ-algorithm given by Moudafi (Nonlinear Anal. 79:117-121, 2013). Hence, it is important and necessary to give new algorithms from the idea of the alternating CQ-algorithm. In this paper, we first present a hybrid projected Landweber algorithm to study the split equality problem. Next, we propose a hybrid alternating CQ-algorithm to study the split equality problem. As applications, we consider the split feasibility problem and linear inverse problem. Finally, we give numerical results for the split feasibility problem to demonstrate the efficiency of the proposed algorithms.

The multiple-sets split equality problem, a generalization and extension of the split feasibility problem, has a variety of specific applications in real world, such as medical care, image reconstruction, and signal processing. It can be a model for many inverse problems where constraints are imposed on the solutions in the domains of two linear operators as well as in the operators’ ranges simultaneously. Although, for the split equality problem, there exist many algorithms, there are but few algorithms for the multiple-sets split equality problem. Hence, in this paper, we present a relaxed two points projection method to solve the problem; under some suitable conditions, we show the weak convergence and give a remark for the strong convergence method in the Hilbert space. The interest of our algorithm is that we transfer the problem to an optimization problem, then, based on the model, we present a modified gradient projection algorithm by selecting two different initial points in different sets for the problem (we call the algorithm as two points algorithm). During the process of iteration, we employ subgradient projections, not use the orthogonal projection, which makes the method implementable. Numerical experiments manifest the algorithm is efficient.

In this paper, we consider the relaxed gradient projection algorithm to solve the split equality problem in Hilbert spaces, and we investigate its linear convergence. In particular, we use the concept of the bounded linear regularity property for the split equality problem to prove the linear convergence property for the above algorithm. Furthermore, we conclude the linear convergence rate of the relaxed gradient projection algorithm. Finally, some numerical experiments are given to test the validity of our results.