Split Feasibility Problem Research Articles

Variational inequality problem over the solution set of split monotone variational inclusion problem with application to bilevel programming problem

The purpose of this paper is to study variational inequality problem over the solution set of multiple-set split monotone variational inclusion problem. We propose an iterative algorithm with inertial method for finding an approximate solution of this problem in real Hilbert spaces. Strong convergence of the sequence of iterates generated from the proposed method is obtained under some mild assumptions. The iterative scheme does not require prior knowledge of operator norm. Also we present some applications of our main result to solve the bilevel programming problem, the bilevel monotone variational inequalities, the split minimization problem, the multiple-set split feasibility problem and the multiple set split variational inequality problem.

New Bregman projection algorithms for solving the split feasibility problem

New Bregman projection algorithms for solving the split feasibility problem

Extension of a Unique Solution in Generalized Neutrosophic Cone Metric Spaces

In order to solve issues that arise in various branches of mathematical analysis, such as split feasibility problems, variational inequality problems, nonlinear optimization issues, equilibrium problems, complementarity issues, selection and matching problems, and issues proving the existence of solutions to integral and differential equations, fixed point theory provides vital tools. In this study, we discuss topological structure and several fixed-point theorems in the context of generalized neutrosophic cone metric spaces. In these spaces, the symmetric properties play an important role. We examine the existence and a uniqueness of a solution by utilizing new types of contraction mappings under some circumstances. We provide an example in which we show the existence and a uniqueness of a solution by utilizing our main result. These results are more generalized in the existing literature.

Convergence theorem for split feasibility problem, equilibrium problem and zeroes of sum of monotone operators

The main purpose of this paper is to introduce a parallel iterative algorithm for approximating the solution of a split feasibility problem on the zero of monotone operators, generalized mixed equilibrium problem and fixed point problem. Using our algorithm, we state and prove a strong convergence theorem for approximating a common element in the set of solutions of a problem of finding zeroes of sum of two monotone operators,generalized mixed equilibrium problem and fixed point problem for a finite family of $\eta$-demimetric mappings in the frame work of a reflexive, strictly convex and smooth Banach spaces. We also give a numerical experiment applying our main result. Our result improves, extends and unifies other results in this direction in the literature.

A new relaxed method for the split feasibility problem in Hilbert spaces

In this paper, we introduce a new relaxed method for solving the split feasibility problem in Hilbert spaces. In our method, the projection to the halfspace is replaced by the one to the intersection of two halfspaces. We give convergence of the sequence generated by our method under some suitable assumptions. Finally, we give a numerical example for illustrating the efficiency and implementation of our algorithms in comparison with existing algorithms in the literature.

An inertial self-adaptive iterative algorithm for finding the common solutions to split feasibility and fixed point problems in specific Banach spaces

In this paper, we propose a new self adaptive iterative algorithm with an inertial technique for solving split feasibility problems and fixed point problems of relative nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. Under some appropriate assumptions imposed on the parameters and operators, we prove that the iterative sequence generated by our proposed algorithm converges strongly to a common solution of split feasibility problems and fixed point problems. Our algorithm does not require a prior knowledge of the norm of the bounded linear operator as we use a new self adaptive step size. Moreover, we give some numerical examples to show the effectiveness and feasibility of our algorithm.

Some convergence results for split common and fixed point problems in Hilbert space

In this survey article, we present an introduction of split feasibility problems, multisets split feasibility problems and split common fixed point problems. Parallel and cyclic algorithms for solving the split common fixed point problems for finite family of strictly pseudocontractive mappings in Hilbert spaces are presented, weak and strong convergence theorems are being proved. Application of the split common fixed point problems and some numerical examples are also presented.

Convergence rate of the relaxed CQ algorithm under Hölderian type error bound property

ABSTRACT The relaxed CQ algorithm is one of the most important algorithms for solving the split feasibility problem. We study the issue of strong convergence of the relaxed CQ algorithm in Hilbert spaces together with estimates on the convergence rate. Under a kind of Hölderian type bounded error bound property, strong convergence of the relaxed CQ algorithm is established. Furthermore, qualitative estimates on the convergence rate is presented. In particular, for the case when the involved exponent is equal to 1, the linear convergence of the relaxed CQ algorithm is established. Finally, numerical experiments are performed to show the convergence property of the relaxed CQ algorithm for the compressed sensing problem.

Regularization methods for solving the split feasibility problem with multiple output sets in Hilbert spaces

We study the split feasibility problem with multiple output sets in Hilbert spaces. In order to solve this problem, we introduce several new iterative processes by using the Tikhonov regularization method.

Image restorations using a modified relaxed inertial technique for generalized split feasibility problems

In this article, we propose a new efficient self‐adaptive method and prove that it converges strongly to a minimum‐norm solution of a generalized split feasibility problem in real Hilbert spaces. The proposed method is derived from a definite discrete dynamical system in time, which combines both the relaxation and inertial techniques for the purpose of increasing the rate of convergence of the iterative scheme. Furthermore, the method requires the monotonicity and Lipschitz continuity condition of the underlying single‐valued cost operator , and it employs some simple self‐adaptive stepsizes that are generated at each iteration by some easy computations. As a by‐product, we obtain methods for solving other classes of generalized split feasibility problems in real Hilbert spaces. Two major merits of our scheme in solving image restoration problems over related schemes are the higher signal‐to‐noise ratio value and lower CPU time for generating recovered images. Finally, we analyze our methods with different related strong convergent methods in the literature.

The method for solving the extension of general of the split feasibility problem and fixed point problem of the cutter

This paper first introduces a new iterative method for weak and strong convergence theorem to demonstrate the estimation potential for a fixed point of the cutter and the finite general split feasibility problem. Consequently, the set of fixed points of a quasi-nonexpansive mapping and the finite general split feasibility problem, the constrained minimization problem, and the general constrained minimization problems are proved using our main results. Finally, we give two numerical examples to advocate our main results.

On split feasibility problem for finite families of equilibrium and fixed point problems in Banach spaces

Abstract In this article, motivated by the works of Ali Akbar and Elahe Shahrosvand [Split equality common null point problem for Bregman quasi-nonexpansive mappings, Filomat 32 (2018), no. 11, 3917–3932], Eskandani et al. [A hybrid extragradient method for solving pseudomonotone equilibrium problem using Bregman distance, J. Fixed Point Theory Appl. 20 (2018), 132], B. Ali and M. H. Harbau [Convergence theorems for Bregman K-mappings and mixed equilibrium problems in reflexive Banach spaces, J. Funct. Spaces (2016) Article ID 5161682, 18 pages], and some other related results in the literature, we introduce a hybrid extragradient iterative algorithm that employs a Bregman distance approach for approximating a split feasibility problem for a finite family of equilibrium problems involving pseudomonotone bifunctions and fixed point problems for a finite family of Bregman quasi-asymptotically nonexpansive mappings using the concept of Bregman K-mapping in reflexive Banach spaces. Using our iterative algorithm, we state and prove a strong convergence result for approximating a common solution to the aforementioned problems. Furthermore, we give an application of our main result to variational inequalities and also report a numerical example to illustrate the convergence of our method. The result presented in this article extends and complements many related results in the literature.

The Split Feasibility Problem with Multiple Output Sets for Demicontractive Mappings

The Split Feasibility Problem with Multiple Output Sets for Demicontractive Mappings

Self-adaptive Forward–Backward Contraction-type Methods for Generalized Split Feasibility Problems

Self-adaptive Forward–Backward Contraction-type Methods for Generalized Split Feasibility Problems

A General Picard-Mann Iterative Method for Approximating Fixed Points of Nonexpansive Mappings with Applications

Fixed point theory provides an important structure for the study of symmetry in mathematics. In this article, a new iterative method (general Picard–Mann) to approximate fixed points of nonexpansive mappings is introduced and studied. We study the stability of this newly established method which we find to be summably almost stable for contractive mappings. A number of weak and strong convergence theorems of such iterative methods are established in the setting of Banach spaces under certain geometrical assumptions. Finally, we present a number of applications to address various important problems (zero of an accretive operator, mixed equilibrium problem, convex optimization problem, split feasibility problem, periodic solution of a nonlinear evolution equation) appearing in the field of nonlinear analysis.

Extragradient Methods for Solving Split Feasibility Problem and General Equilibrium Problem and Resolvent Operators in Banach Spaces

In this paper, we introduce a new extragradient algorithm by using generalized metric projection. We prove a strong convergence theorem for finding a common element of the solution set of split feasibility problem and the set of fixed points of relatively nonexpansive mapping and a finite family of resolvent operator and the set of solutions of an equilibrium problem.

Self-adaptive algorithms for solving split feasibility problem with multiple output sets

Self-adaptive algorithms for solving split feasibility problem with multiple output sets

An extended inertial Halpern-type ball-relaxed CQ algorithm for multiple-sets split feasibility problem

An extended inertial Halpern-type ball-relaxed CQ algorithm for multiple-sets split feasibility problem

Inertial iterative method with self-adaptive step size for finite family of split monotone variational inclusion and fixed point problems in Banach spaces

Abstract In this paper, we propose and study a new inertial iterative algorithm with self-adaptive step size for approximating a common solution of finite family of split monotone variational inclusion problems and fixed point problem of a nonexpansive mapping between a Banach space and a Hilbert space. This method combines the inertial technique with viscosity method and self-adaptive step size for solving the common solution problem. We prove a strong convergence result for the proposed method under some mild conditions. Moreover, we apply our result to study the split feasibility problem and split minimization problem. Finally, we provide some numerical experiments to demonstrate the efficiency of our method in comparison with some well-known methods in the literature. Our method does not require prior knowledge or estimate of the operator norm, which makes it easily implementable unlike so many other methods in the literature, which require prior knowledge of the operator norm for their implementation.

Two inertial-type algorithms for solving the split feasibility problem

In this paper, we study two inertial-type algorithms with a relaxed splitting method for solving the split feasibility problem in Hilbert spaces. Weak convergence is established without assuming conditions such as and , where is the inertial factor. Compared with existing results where inertial factors are usually less than one, the inertial factor in the first algorithm can be taken as nonnegative numbers greater than one. The efficiency and advantage of our algorithms are illustrated by numerical experiments.