Split Variational Inclusion Research Articles

New iterative algorithms for solving split variational inclusions

New iterative algorithms for solving split variational inclusions

Modified splitting algorithms for approximating solutions of split variational inclusions in Hilbert spaces

Modified splitting algorithms for approximating solutions of split variational inclusions in Hilbert spaces

An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces

Abstract In this article, we study the split variational inclusion and fixed point problems using Bregman weak relatively nonexpansive mappings in the p p -uniformly convex smooth Banach spaces. We introduce an inertial shrinking projection self-adaptive iterative scheme for the problem and prove a strong convergence theorem for the sequences generated by our iterative scheme under some mild conditions in real p p -uniformly convex smooth Banach spaces. The algorithm is designed to select its step size self-adaptively and does not require the prior estimate of the norm of the bounded linear operator. Finally, we provide some numerical examples to illustrate the performance of our proposed scheme and compare it with other methods in the literature.

On some fast iterative methods for split variational inclusion problem and fixed point problem of demicontractive mappings

On some fast iterative methods for split variational inclusion problem and fixed point problem of demicontractive mappings

The AA-Viscosity Algorithm for Fixed-Point, Generalized Equilibrium and Variational Inclusion Problems

The aim of this paper is to propose an inertial-type AA-viscosity algorithm for approximating the common solutions of the split variational inclusion problem, the generalized equilibrium problem and the common fixed-point problem of nonexpansive mappings. The strong convergence of an iterative sequence obtained through the proposed method is proved under some mild assumptions. Consequently, approximations of the solution of the split feasibility problem, the relaxed split feasibility problem, the split common null point problem and the split minimization problem are given. The applicability of our proposed algorithm has been illustrated with the help of a numerical example. Our iterative method was then compared graphically with different comparable methods in the existing literature.

Tseng-type algorithms for solving variational inequalities over the solution sets of split variational inclusion problems with an application to a bilevel optimization problem

Tseng-type algorithms for solving variational inequalities over the solution sets of split variational inclusion problems with an application to a bilevel optimization problem

Strong convergence for split variational inclusion problems under hybrid algorithms with applications

In this article, we present two inertial modifications of regularized algorithms for the split variational inclusion problem (SVIP, for short) in real Hilbert spaces (RHSs). When the circumstances are right, strong convergence theorems are demonstrated. The major findings are used to solve the split minimization, split common fixed point problem (SCFPP), split minimization problem (SMP), and split feasibility problems (SFP) in applications. The proposed algorithms are contrasted with a number of other existing algorithms in the literature in order to test their numerical performances. Finally, the computer tests demonstrate that the suggested algorithms outperform alternative strategies in terms of speed and efficiency.

Dynamic stepsize iteration process for solving split common fixed point problems with applications

In this paper, we study the split common fixed point problem for two nonlinear mappings in p-uniformly convex and uniformly smooth Banach spaces. We propose an algorithm which uses dynamic stepsize, it allows to be easily implemented without prior information about operator norm. We further apply our result to solve the split variational inclusion problem, equilibrium problem and convexly constrained linear inverse problem. Moreover, we provide numerical examples to verify efficiency of our algorithm.

An Algorithm That Adjusts the Stepsize to Be Self-Adaptive with an Inertial Term Aimed for Solving Split Variational Inclusion and Common Fixed Point Problems

In this research paper, we present a new inertial method with a self-adaptive technique for solving the split variational inclusion and fixed point problems in real Hilbert spaces. The algorithm is designed to choose the optimal choice of the inertial term at every iteration, and the stepsize is defined self-adaptively without a prior estimate of the Lipschitz constant. A convergence theorem is demonstrated to be strong even under lenient conditions and to showcase the suggested method’s efficiency and precision. Some numerical tests are given. Moreover, the significance of the proposed method is demonstrated through its application to an image reconstruction issue.

An algorithm for approximating solutions of the split variational inclusion problem

We introduce and study an iterative algorithm to approximate a solution to the split variational inclusion problem in real Hilbert spaces. The strong convergence of the proposed algorithm is proved via some suitable conditions. As applications, we derive consequences for several special cases such as the split variational inequality problem, the split minimum point problem, and the split common fixed point problem. Finally, we present some numerical experiments to demonstrate the performance of the proposed algorithm.

New inertial modification of regularized algorithms for solving split variational inclusion problem

In this paper, we introduce two new self-adaptive iterative algorithms for finding the solution of a split variational inclusion problem in real Hilbert spaces. The strong convergence theorems are proved under suitable conditions. In applications, the main results are applied for solving the split feasibility problem, the split minimization problem, and the split common fixed point problem. In order to test the numerical performances, the proposed algorithms are compared with several existing algorithms in the literature. Finally, the computational experiments show that the proposed algorithms are fast and more efficient than other approaches.

Inertial Iterative Algorithms for Split Variational Inclusion and Fixed Point Problems

This paper aims to present two inertial iterative algorithms for estimating the solution of split variational inclusion (SpVIsP) and its extended version for estimating the common solution of (SpVIsP) and fixed point problem (FPP) of a nonexpansive mapping in the setting of real Hilbert spaces. We establish the weak convergence of the proposed algorithms and strong convergence of the extended version without using the pre-estimated norm of a bounded linear operator. We also exhibit the reliability and behavior of the proposed algorithms using appropriate assumptions in a numerical example.

Perturbation Resilience of Self-Adaptive Step-Size Algorithms for Solving Split Variational Inclusion Problems and their Applications

In this paper, two viscosity proximal type algorithms involving the superiorization method are presented for solving the split variational inclusion problems in real Hilbert spaces. Strong convergence theorems and bounded perturbation resilience analysis of the proposed algorithms are obtained under mild conditions. The split feasibility problems, the split minimization problems, and the variational inequality problems are concerned as the applications, and several numerical experiments are performed to show the efficiency and implementation of the proposed algorithms.

A Strongly Convergent Viscosity-Type Inertial Algorithm with Self Adaptive Stepsize for Solving Split Variational Inclusion Problems in Hilbert Spaces

A Strongly Convergent Viscosity-Type Inertial Algorithm with Self Adaptive Stepsize for Solving Split Variational Inclusion Problems in Hilbert Spaces

Inertial Viscosity Approximation Methods for General Split Variational Inclusion and Fixed Point Problems in Hilbert Spaces

The purpose of this paper is to find a common element of the fixed point set of a nonexpansive mapping and the set of solutions of the general split variational inclusion problem in symmetric Hilbert spaces by using the inertial viscosity iterative method. Some strong convergence theorems of the proposed algorithm are demonstrated. As applications, we use our results to study the split feasibility problem and the split minimization problem. Finally, the numerical experiments are presented to illustrate the feasibility and effectiveness of our theoretical findings, and our results extend and improve many recent ones.

Variational inequalities over the solution sets of split variational inclusion problems

We study variational inequalities over the solution sets of split variational inclusion problems with multiple output sets. In order to find a solution to such problems, we introduce a new parallel iterative algorithm by combining the inertial forward–backward splitting method with the steepest descent method. Our algorithm does not depend on the norm of the transfer mappings. Some of its applications to solving split common null point, split minimum point, and split common fixed point problems are also given.

Inertial algorithms with adaptive stepsizes for split variational inclusion problems and their applications to signal recovery problem

With the help of the Meir–Keeler contraction method and the Mann‐type method, two adaptive inertial iterative schemes are introduced for finding solutions of the split variational inclusion problem in Hilbert spaces. The strong convergence of the suggested algorithms is guaranteed by a new stepsize criterion that does not require calculation of the bounded linear operator norm. Some numerical experiments and applications in signal recovery problems are given to demonstrate the efficiency of the proposed algorithms.

A modified Ishikawa iteration scheme for b‐enriched nonexpansive mapping to solve split variational inclusion problem and fixed point problem in Hilbert spaces

In this article, an Ishikawa iteration scheme is modified for ‐enriched nonexpansive mapping to solve a fixed point problem and a split variational inclusion problem in real Hilbert spaces. Under some suitable conditions, we obtain convergence theorem. Moreover, to demonstrate the effectiveness and performance of proposed scheme, we apply the scheme to solve a split feasibility problem and compare it with some existing iterative schemes.

Bounded perturbation resilience of a regularized forward-reflected-backward splitting method for solving variational inclusion problems with applications

ABSTRACT The forward-reflected-backward splitting method recently introduced for solving variational inclusion problems involves just one forward evaluation and one backward evaluation of the monotone operator and the maximal monotone operator, respectively, per iteration. This structure gives it some advantage over the earlier proposed methods. However, it only provides weak convergence, in general. Our aim in this paper is to improve the forward-reflected-backward splitting method in order to obtain strong convergence. To this end, we first study a regularized variational inclusion problem of finding the zero of the sum of two monotone operators. We then propose a regularized forward-reflected-backward splitting method for approximating a solution to the problem and prove the strong convergence of our iterative scheme under some suitable assumptions on the parameters. Moreover, we show that our algorithm has the bounded perturbation resilience property. Furthermore, we apply our results to convex minimization, split feasibility, split variational inclusion, and image deblurring problems, and illustrate the performance of our algorithm with several numerical examples.

Algorithms for Approximating Solutions of Split Variational Inclusion and Fixed-Point Problems

In this paper, the split fixed point and variational inclusion problem is considered. With the help of fixed point technique, Tseng-type splitting method and self-adaptive rule, an iterative algorithm is proposed for solving this split problem in which the involved operators S and T are demicontractive operators and g is plain monotone. Strong convergence theorem is proved under some mild conditions.