Multiple-sets Split Feasibility Problem Research Articles

Bi-extrapolated subgradient projection algorithm for solving multiple-sets split feasibility problem

This paper deals with a bi-extrapolated subgradient projection algorithm by introducing two extrapolated factors in the iterative step to solve the multiple-sets split feasibility problem. The strategy is intend to improve the convergence. And its convergence is proved under some suitable conditions. Numerical results illustrate that the bi-extrapolated subgradient projection algorithm converges more quickly than the existing algorithms.

A Successive Projection Algorithm for Solving the Multiple-Sets Split Feasibility Problem

The multiple-sets split feasibility problem (MSFP) arises in many areas and it can be unified as a model for many inverse problems where the constraints are required on the solutions in the domain of a linear operator as well as in the operator's range. Some existing algorithms, in order to get the suitable step size, need to compute the largest eigenvalue of the related matrix, estimate the Lipschitz constant, or use some step-size search scheme, which usually requires many inner iterations. In this article, we introduce a successive projection algorithm for solving the multiple-sets split feasibility problem. In each iteration of this algorithm, the step size is directly computed, which is not needed to compute the largest eigenvalue of the matrix or estimate the Lipschitz constant. It also does not need any step-size search scheme. Its theoretical convergence results are also given.

Strong convergence theorems for generalized nonexpansive mappings on star-shaped set with applications

In this paper, we study attractive points for a class of generalized nonexpansive mappings on star-shaped sets and establish strong convergence theorems of the Halpern iterative sequences generated by these mappings in a real Hilbert space. We modify Halpern’s iterations for finding an attractive point of a mapping T satisfying condition (E) on a star-shaped set C in a real Hilbert space H and provide an affirmative answer to an open problem posed by Akashi and Takahashi in a recent work of (Appl. Math. Comput. 219(4):2035-2040, 2012) for nonexpansive and nonspreading mappings. Furthermore, we study the approximation of common attractive points of generalized nonexpansive mappings and derive a strong convergence theorem by a new iteration scheme for these mappings. As applications of our results, we study multiple sets split monotone inclusion problems for inverse strongly monotone mappings, multiple sets split optimization problems, and multiple sets split feasibility problems. Our results contain many original results on multiple sets split feasibility problem in the literature. Our results also improve and generalize many well-known results in the current literature. MSC:47H10, 37C25.

A relaxed self-adaptive CQ algorithm for the multiple-sets split feasibility problem

The multiple-sets split feasibility problem (MSFP) is to find a point belongs to the intersection of a family of closed convex sets in one space, such that its image under a linear transformation belongs to the intersection of another family of closed convex sets in the image space. Many iterative methods can be employed to solve the MSFP. Jinling Zhao et al. proposed a modification for the CQ algorithm and a relaxation scheme for this modification to solve the MSFP. The strong convergence of these algorithms are guaranteed in finite-dimensional Hilbert spaces. Recently López et al. proposed a relaxed CQ algorithm for solving split feasibility problem, this algorithm can be implemented easily since it computes projections onto half-spaces and has no need to know a priori the norm of the bounded linear operator. However, this algorithm has only weak convergence in the setting of infinite-dimensional Hilbert spaces. In this paper, we introduce a new relaxed self-adaptive CQ algorithm for solving the MSFP where closed convex sets are level sets of some convex functions such that the strong convergence is guaranteed in the framework of infinite-dimensional Hilbert spaces. Our result extends and improves the corresponding results.

Multiple-set split feasibility problems for κ-asymptotically strictly pseudo-nonspreading mappings in Hilbert spaces

Some weak and strong convergence theorems for solving multiple-set split feasibility problems for κ-asymptotically strictly pseudo-nonspreading mappings in infinite-dimensional Hilbert spaces are proved. The results presented in the paper extend and improve the corresponding results of Xu (Inverse Probl. 22(6):2021-2034, 2006), Osilike and Isiogugu (Nonlinear Anal. 74:1814-1822, 2011), Chang et al. (Abstr. Appl. Anal. 2012:491760, 2012), and others.

A new simultaneous subgradient projection algorithm for solving a multiple-sets split feasibility problem

In this paper, we present a simultaneous subgradient algorithm for solving the multiple-sets split feasibility problem. The algorithm employs two extrapolated factors in each iteration, which not only improves feasibility by eliminating the need to compute the Lipschitz constant, but also enhances flexibility due to applying variable step size. The convergence of the algorithm is proved under suitable conditions. Numerical results illustrate that the new algorithm has better convergence than the existing one.

An Iterative Algorithm for the Split Equality and Multiple-Sets Split Equality 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.

Modified Alternating Direction Methods for the Modified Multiple-Sets Split Feasibility Problems

In this paper, we propose two new multiple-sets split feasibility problem models and new solution methods. The first model is more separable than the original one, which enables us to apply a modified alternating direction method with parallel steps to solve it. Then, to overcome the difficulty of computing projections onto the constraint sets, a special version of this modified method with the strategy of projection onto half-space is given. The second model consists in finding a least Euclidean norm solution of the multiple-sets split feasibility problem, for which we provide another modified alternating direction method. Numerical results presented at the last show the efficiency of our methods.

A New Algorithm for Solving The Multiple-Sets Split Feasibility Problem in Banach Spaces

In this article, we propose a new algorithm for solving the multiple-sets split feasibility problem in Banach spaces. Under some mild conditions we establish the norm convergence of the proposed algorithm. One advantage of the proposed algorithm is that its norm convergence does not need the weak-to-weak continuity of the duality mapping.

Solutions for a variational inclusion problem with applications to multiple sets split feasibility problems

In this paper, we first study the set of common solutions for two variational inclusion problems in a real Hilbert space and establish a strong convergence theorem of this problem. As applications, we study unique minimum norm solutions of the following problems: multiple sets split feasibility problems, system of convex constrained linear inverse problems, convex constrained linear inverse problems, split feasibility problems, convex feasibility problems. We establish iteration processes of these problems and show the strong convergence theorems of these iteration processes.MSC:47J20, 47J25, 47H05, 47H09.

Hierarchical problems with applications to mathematical programming with multiple sets split feasibility constraints

In this paper, we establish a strong convergence theorem for hierarchical problems, an equivalent relation between a multiple sets split feasibility problem and a fixed point problem. As applications of our results, we study the solution of mathematical programming with fixed point and multiple sets split feasibility constraints, mathematical programming with fixed point and multiple sets split equilibrium constraints, mathematical programming with fixed point and split feasibility constraints, mathematical programming with fixed point and split equilibrium constraints, minimum solution of fixed point and multiple sets split feasibility problems, minimum norm solution of fixed point and multiple sets split equilibrium problems, quadratic function programming with fixed point and multiple set split feasibility constraints, mathematical programming with fixed point and multiple set split feasibility inclusions constraints, mathematical programming with fixed point and split minimax constraints.

The constrained multiple-sets split feasibility problem and its projection algorithms

Abstract The projection algorithms for solving the constrained multiple-sets split feasibility problem are presented. The strong convergence results of the algorithms are given under some mild conditions. Especially, the minimum norm solution of the constrained multiple-sets split feasibility problem can be found.

Damped projection method for split common fixed point problems

The paper deals with the split common fixed-point problem (SCFP) introduced by Censor and Segal. Motivated by Eicke’s damped projection method, we propose a cyclic iterative scheme and prove its strong convergence to a solution of SCFP under some mild assumptions. An application of the proposed method to multiple-set split feasibility problems is also included.

On Two Projection Algorithms for the Multiple-Sets Split Feasibility Problem

We present a projection algorithm which modifies the method proposed by Censor and Elfving (1994) and also introduce a self-adaptive algorithm for the multiple-sets split feasibility problem (MSFP). The global rates of convergence are firstly investigated and the sequences generated by two algorithms are proved to converge to a solution of the MSFP. The efficiency of the proposed algorithms is illustrated by some numerical tests.

Multiple-Set Split Feasibility Problems forκ-Strictly Pseudononspreading Mapping in Hilbert Spaces

The purpose of this paper is to prove some weak and strong convergence theorems for solving the multiple-set split feasibility problems forκ-strictly pseudononspreading mapping in infinite-dimensional Hilbert spaces by using the proposed iterative method. The main results presented in this paper extend and improve the corresponding results of Xu et al. (2006), of Osilike et al. (2011), and of many other authors.

Multiple-set split feasibility problems for a finite family of demicontractive mappings in Hilbert spaces

In this paper, we introduce an iterative algorithm for solving the multiple-set split feasibility problems for a finite family demicontractive mappings in Hilbert spaces. The results presented in this paper improve and extend some recent corresponding results in (4), (7), (9), (10), (13), (14).

An improved method for solving multiple-sets split feasibility problem

The multiple-sets split feasibility problem (MSSFP) has a variety of applications in the real world such as medical care, image reconstruction and signal processing. Censor et al. proposed solving the MSSFP by a proximity function, and then developed a class of simultaneous methods for solving split feasibility. In our paper, we improve a simultaneous method for solving the MSSFP and prove its convergence.

Modified projection methods for the split feasibility problem and the multiple-sets split feasibility problem

The split feasibility problem is to find x∈C with Ax∈Q, if such points exist, where A is a given M×N real matrix, C and Q are nonempty closed convex sets in RN and RM, respectively. Byrne (2002) [2] proposed a well-known CQ algorithm for solving this problem. In this paper, we propose a modification for the CQ algorithm, which computes the stepsize adaptively, and performs an additional projection step onto some simple closed convex set X⊆RN in each iteration. We further give a relaxation scheme for this modification to make it more easily implemented. Convergence results of both algorithms are analyzed, and preliminary numerical results are reported. We also extend these modified algorithms to solve the multiple-sets split feasibility problem, which is to find a point closest to the intersection of a family of closed convex sets in one space such that its image under a linear transformation will be closest to the intersection of another family of closed convex sets in the image space.

A simple projection method for solving the multiple-sets split feasibility problem

In this article, we first introduce a simple but practical projection method for solving the multiple-sets split feasibility problem, which is to find a point in the intersection of a family of closed convex sets in one space, such that its image under a linear transformation is in the intersection of another family of closed convex sets in the image space. In each iteration of this method, the step-size is directly computed, and is shown to be the best for the current direction. Then we consider the corresponding relaxed projection scheme for the proposed method. The theoretical convergence results are established. Preliminary numerical experiments show that this simple method and its relaxed scheme are easy to implement and practical.

Several acceleration schemes for solving the multiple-sets split feasibility problem

In this paper we present several acceleration schemes for solving the multiple-sets split feasibility problem (MSFP), which is to find a point which belongs to the intersection of a family of closed convex sets in one space, such that its image under a linear transformation belongs to the intersection of another family of closed convex sets in the image space. We first modify the existing method and give a self-adaptive algorithm to solve the MSFP, which computes the step-size by Armijo-like searches and performs an additional projection step onto some simple closed convex set X⊆RN at each iteration; then we present a special case of this algorithm. Convergence results are analyzed, and further discussions on accelerating relaxed algorithms are lead. Preliminary numerical experiments shows that these accelerating schemes are practical and promising for solving the MSFPs.