Multiple-sets Split Feasibility Problem Research Articles

Alternating proximal penalization algorithm for the modified multiple-sets split feasibility problems

In this paper, we present an extended alternating proximal penalization algorithm for the modified multiple-sets feasibility problem. For this method, we first show that the sequences generated by the algorithm are summable, which guarantees that the distance between two adjacent iterates converges to zero, and then we establish the global convergence of the algorithm provided that the penalty parameter tends to zero.

A general algorithm for multiple-sets split feasibility problem involving resolvents and Bregman mappings

In this paper, by using Bregman distance, we introduce a new iterative process involving products of resolvents of maximal monotone operators for approximating a common element of the set of common fixed points of a finite family of multi-valued Bregman relatively nonexpansive mappings and the solution set of the multiple-sets split feasibility problem and common zeros of maximal monotone operators. We derive a strong convergence theorem of the proposed iterative algorithm under appropriate situations. Finally, we mention several corollaries and two applications of our algorithm.

Convergence analysis of an iterative method for solving multiple-set split feasibility problems in certain Banach spaces

Our purpose in this paper is to introduce an iterative scheme for solving multiple-set split feasiblity problems in p-uniformly convex Banach spaces which are also uniformly smooth using Bregman distance techniques. We further obtain a strong convergence result for approximating solutions of multiple-set split feasiblity problems in the framework of p-uniformly convex Banach spaces which are also uniformly smooth.

Remark on the Successive Projection Algorithm for the Multiple-Sets Split Feasibility Problem

ABSTRACTRecently, assuming that the metric projection onto a closed convex set is easily calculated, Liu et al. (Numer. Func. Anal. Opt. 35:1459–1466, 2014) presented a successive projection algorithm for solving the multiple-sets split feasibility problem (MSFP). However, in some cases it is impossible or needs too much work to exactly compute the metric projection. The aim of this remark is to give a modification to the successive projection algorithm. That is, we propose a relaxed successive projection algorithm, in which the metric projections onto closed convex sets are replaced by the metric projections onto halfspaces. Clearly, the metric projection onto a halfspace may be directly calculated. So, the relaxed successive projection algorithm is easy to implement. Its theoretical convergence results are also given.

A parallel method for variational inequalities with the multiple-sets split feasibility problem constraints

In this paper, we propose a parallel method for solving a strongly variational inequality over the multiple-sets split feasibility problem. Strong convergence of the iterative process is proved. As a consequence, we get a strongly convergent algorithm for finding the minimum-norm solution of the multiple-sets split feasibility problem. A simple numerical example is given to illustrate the proposed parallel algorithm.

Iterative algorithms for the multiple-sets split feasibility problem in Hilbert spaces

In this paper, for the multiple-sets split feasibility problem, that is to find a point closest to a family of closed convex subsets in one space such that its image under a linear bounded mapping will be closest to another family of closed convex subsets in the image space, we study several iterative methods for finding a solution, which solves a certain variational inequality. We show that particular cases of our algorithms are some improvements for existing ones in literature. We also give two numerical examples for illustrating our algorithms.

Solving the general split common fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operator norms

Let H1, H2, H3 be real Hilbert spaces, let A : H1 ? H3, B : H2 ? H3 be two bounded linear operators. The general multiple-set split common fixed-point problem under consideration in this paper is to find x ??p,i=1F(Ui), y ??r,j=1 F(Tj) such that Ax = Bym, (1) where p, r ? 1 are integers, Ui : H1 ? H1 (1 ? i ? p) and Tj : H2 ? H2 (1 ? j ? r) are quasi-nonexpansive mappings with nonempty common fixed-point sets ?p,i=1 F(Ui) = ?p,i=1 {x ? H1 : Uix = x} and ?r,j=1F(Tj) = ?r,j=1 {x ? H2 : Tjx = x}. Note that, the above problem (1) allows asymmetric and partial relations between the variables x and y. If H2 = H3 and B = I, then the general multiple-set split common fixed-point problem (1) reduces to the multiple-set split common fixed-point problem proposed by Censor and Segal [J. Convex Anal. 16(2009), 587-600]. In this paper, we introduce simultaneous parallel and cyclic algorithms for the general split common fixed-point problems (1). We introduce a way of selecting the stepsizes such that the implementation of our algorithms does not need any prior information about the operator norms. We prove the weak convergence of the proposed algorithms and apply the proposed algorithms to the multiple-set split feasibility problems. Our results improve and extend the corresponding results announced by many others.

A cyclic iterative method for solving Multiple Sets Split Feasibility Problems in Banach Spaces

In this paper, we construct an iterative scheme and prove strong convergence theorem of the sequence generated to an approximate solution to a multiple sets split feasibility problem in a p-uniformly convex and uniformly smooth real Banach space. Some numerical experiments are given to study the efficiency and implementation of our iteration method. Our result complements the results of F. Wang (A new algorithm for solving the multiple-sets split feasibility problem in Banach spaces, Numerical Functional Anal. Optim. 35 (2014), 99–110), F. Scho¨pfer et al. (An iterative regularization method for the solution of the split feasibility problem in Banach spaces, Inverse Problems 24 (2008), 055008) and many important recent results in this direction.

Convergence of Inexact Mann Iterations Generated by Nearly Nonexpansive Sequences and Applications

ABSTRACTThe Mann iterates behave well for nonexpansive mappings for any initial guess in the domain. Our aim in this article is to extend this method to a broad class of inexact fixed point algorithms generated by nearly nonexpansive sequences in Banach spaces and to locate the weak limit of the iterates by its initial guesses. Due to the inexactness, our algorithms become efficiently applicable for a wider class of problems. As applications, we give convergence theorems for finding solutions of variational inclusion problems and constrained multiple-sets split feasibility problems. Our results are significant refinements and improvements of the corresponding results in the literature.

Strong Convergence Theorem for Multiple Sets Split Feasibility Problems in Banach Spaces

ABSTRACTThe purpose of this article is to study the iterative approximation of solution to multiple sets split feasibility problems in p-uniformly convex real Banach spaces that are also uniformly smooth. We propose an iterative algorithm for solving multiple sets split feasibility problems and prove a strong convergence theorem of the sequence generated by our algorithm under some appropriate conditions in p-uniformly convex real Banach spaces that are also uniformly smooth.

Strong convergence for generalized multiple-set split feasibility problem

In this paper we introduce a new algorithm based on viscosity approximation method for solving the generalized multiple-set split feasibility problem (GMSSFP)in an infinite dimensional Hilbert spaces . We establish the strong convergence for the algorithm to find a unique solution of the variational inequality which is the optimality condition for the minimization problem.

On global convergence rate of two acceleration projection algorithms for solving the multiple-sets split feasibility problem

In this paper, we introduce two fast projection algorithms for solving the multiple-sets split feasibility problem (MSFP). Our algorithms accelerate algorithms proposed in [8] and are proved to have a global convergence rate O(1=n2). Preliminary numerical experiments show that these algorithms are practical and promising.

Iterative methods for solving the multiple-sets split feasibility problem with splitting self-adaptive step size

AbstractWe introduce an iterative algorithm for solving the multiple-sets split feasibility problem with splitting self-adaptive step size. This step size is calculated directly from the iteration process without need to know the spectral norm of linear operators. We also generalize the chosen step size to a relaxed iterative algorithm. Theoretical convergence is proved in an infinite dimensional Hilbert space. Some numerical experiments are presented to verify the effectiveness of our proposed methods.

Iterative process for solving a multiple-set split feasibility problem

This paper deals with a variant relaxed CQ algorithm by using a new searching direction, which is not the gradient of a corresponding function. The strategy is to intend to improve the convergence. Its convergence is proved under some suitable conditions. Numerical results illustrate that our variant relaxed CQ algorithm converges more quickly than the existing algorithms.

A Cyclic and Simultaneous Iterative Method for Solving the Multiple-Sets Split Feasibility Problem

The iterative projection methods for solving the multiple-sets split feasibility problem have been paid much attention in recent years. In this paper, we introduce a cyclic and simultaneous iterative sequence with self-adaptive step size for solving this problem. The advantage of the self-adaptive step size is that it does not need to know the Lipschitz constant of the gradient operator in advance. Furthermore, we propose a relaxed cyclic and simultaneous iterative sequence with self-adaptive step size, respectively. The theoretical convergence results are established in an infinite-dimensional Hilbert spaces setting. Preliminary numerical experiments show that these iteration methods are practical and easy to implement.

A general convergence theorem for multiple-set split feasibility problem in Hilbert spaces

We establish strong convergence result of split feasibility problem for a family of quasi-nonexpansive multi-valued mappings and a total asymptotically strict pseudo-contractive mapping in infinite dimensional Hilbert spaces.

Several iterative algorithms for solving the split common fixed point problem of directed operators with applications

In this paper, we propose a new simultaneous iterative algorithm for solving the split common fixed point problem of directed operators. Inspired by the idea of cyclic iterative algorithm, we also introduce two iterative algorithms which combine the process of cyclic and simultaneous together. Under mild assumptions, we prove convergence of the proposed iterative sequences. As applications, we obtain several iteration schemes to solve the inverse problem of multiple-sets split feasibility problem. Numerical experiments are presented to confirm the efficiency of the proposed iterative algorithms.

Split equality problem and multiple-sets split equality problem for quasi-nonexpansive multi-valued mappings

Abstract 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.

General Method for Solving the Split Common Fixed Point Problem

The split common fixed point problem (also called the multiple-sets split feasibility problem) is to find a common fixed point of a finite family of operators in one real Hilbert space, whose image under a bounded linear transformation is a common fixed point of another family of operators in the image space. In the literature one can find many methods for solving this problem as well as for its special case, called the split feasibility problem. We propose a general method for solving both problems. The method is based on a block-iterative procedure, in which we apply quasi-nonexpansive operators satisfying the demi-closedness principle and having a common fixed point. We prove the weak convergence of sequences generated by this method and show that the convergence for methods known from the literature follows from our general result.

Variable KM-like algorithms for fixed point problems and split feasibility problems

The convergence analysis of a variable KM-like method for approximating common fixed points of a possibly infinitely countable family of nonexpansive mappings in a Hilbert space is proposed and proved to be strongly convergent to a common fixed point of a family of nonexpansive mappings. Our variable KM-like technique is applied to solve the split feasibility problem and the multiple-sets split feasibility problem. Especially, the minimum norm solutions of the split feasibility problem and the multiple-sets split feasibility problem are derived. Our results can be viewed as an improvement and refinement of the previously known results. MSC:47H10, 65J20, 65J22, 65J25.