Novel Accelerated Cyclic Iterative Approximation for Hierarchical Variational Inequalities Constrained by Multiple-Set Split Common Fixed Point Problems

In this paper, we investigate a class of hierarchical variational inequalities (HVIPs, i.e., strongly monotone variational inequality problems defined on the solution set of multiple-set split common fixed-point problems) with quasi-pseudocontractive mappings in real Hilbert spaces, with special cases being able to be found in many important engineering practical applications, such as image recognizing, signal processing, and machine learning. In order to solve HVIPs of potential application value, inspired by the primal-dual algorithm, we propose a novel accelerated cyclic iterative algorithm that combines the inertial method with a correction term and a self-adaptive step-size technique. Our approach eliminates the need for prior knowledge of the bounded linear operator norm. Under appropriate assumptions, we establish strong convergence of the algorithm. Finally, we apply our novel iterative approximation to solve multiple-set split feasibility problems and verify the effectiveness of the proposed iterative algorithm through numerical results.

Our contribution in this paper is to propose an iterative algorithm which does not require prior knowledge of operator norm and prove strong convergence theorem for approximating a solution of split common fixed point problem of demicontractive mappings in a real Hilbert space. So many authors have used algorithms involving the operator norm for solving split common fixed point problem, but as widely known the computation of these algorithms may be difficult and for this reason, authors have recently started constructing iterative algorithms with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm. We introduce a new algorithm for solving the split common fixed point problem for demicontractive mappings with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm and then prove strong convergence of the sequence in real Hilbert spaces. Finally, we give some applications of our result and numerical example at the end of the paper.

In this paper, we introduce a new iterative algorithm from primal-dual methods for solving the split equality common fixed-point problem of quasi-nonexpansive mappings in real Hilbert space. Our algorithm includes the simultaneous iterative algorithm as special case which has been proposed by Moudafi and Al-Shemas for solving the split equality common fixed-point problem. We use a way of selecting the stepsizes such that the implementation of our algorithm does not need any prior information about bounded linear operator norms. It avoids the difficult task of estimating the operator norms. Under suitable conditions, we get the weak convergence of the proposed algorithm. The performance of the proposed algorithm is also illustrated by preliminary numerical experiments. The results presented in the paper improve and extend some corresponding results.

We extend the notion of asymptotically nonexpansive mapping to the more general class, namely, e-enriched asymptotically nonexpansive mappings. It is shown, with an example, that the class of e-enriched asymptoticallynonexpansive mappings is more general than the class of asymptotically nonexpansive mappings. Certain weak and strong convergence theorems are then proved for the iterative approximation of split common fixed point problem involving the class of(e,ϑ)-enriched strictly quasi-pseudocontractive mappings and the class of e-enriched asymptotically nonexpansive mappings in the domain of two Banach spaces. Furthermore, a significant result for the hierarchical variational inequality problem is obtained as a consequence of our main result.

The purpose of this paper is to propose an algorithm for solving the split common fixed point problem for strict quasi-ϕ-pseudocontractive mappings in Banach spaces. It is proved that the sequence generated by the proposed iterative algorithm converges strongly to a solution of the split common fixed point problem. Then, the main result is used to study the split common null point problem and the split quasi-inclusion problem. Finally, a numerical example is provided to illustrate our main result. The results presented in this paper extend and improve some recent corresponding results.

Let , and be real Hilbert spaces, let and be two bounded linear operators. Moudafi introduced simultaneous iterative algorithms with weak convergence for the following split common fixed-point problem:Section.Display where and are two firmly quasi-nonexpansive operators with nonempty fixed-point sets and . Note that, by taking and , we recover the split common fixed-point problem originally introduced by Cesnor and Segal. However, to employ Moudafi’s algorithms, one needs to know a prior norm (or at least an estimate of the norm) of the bounded linear operators. To estimate the norm of an operator is very difficult, if it is not an impossible task. In this paper, we will continue to consider the split common fixed-point problem (1) governed by the general class of quasi-nonexpansive operators. We introduce a simultaneous iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information about the operator norms. The weak convergence result of algorithm is obtained and some applied nonlinear analysis examples are stated.

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

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.

In this paper, we study the split common fixed point problem, which is to find a fixed point of a quasi-pseudocontractive mapping in one space whose image under a linear transformation is a fixed point of anther quasi-pseudocontractive mapping in the image space. We design and analyze a new iterative algorithm for solving this split common fixed point problem. A weak convergence theorem is given.MSC:49J53, 49M37, 65K10, 90C25.

The split common fixed point problem is an inverse problem that consists in finding an element in a fixed point set such that its image under a bounded linear operator belongs to another fixed-point set. In this paper, we present new iterative algorithms for solving the split common fixed point problem of demimetric mappings in Hilbert spaces. Moreover, our algorithm does not need any prior information of the operator norm. Weak and strong convergence theorems are given under some mild assumptions. The results in this paper are the extension and improvement of the recent results in the literature.

In this paper, we study the triple hierarchical variational inequality problem (THVIP, for short) with the hierarchical variational inequality constraint for finitely many nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. Based on Korpelevich's extragradient method, hybrid steepest-descent method, Mann's implicit iteration method and Halpern's iteration method, a general implicit hybrid iterative algorithm for solving this THVIP is proposed. Under mild conditions, the strong convergence of the iterative sequences generated by the algorithm is established. Our results improve and extend the corresponding results in the earlier and recent literature.

A Simple Computational Algorithm With Inertial Extrapolation for Generalized Split Common Fixed Point Problems

In a real Hilbert space, we investigate the Tseng’s extragradient algorithms with hybrid adaptive step-sizes for treating a Lipschitzian pseudomonotone variational inequality problem and a strict pseudocontraction fixed-point problem, which are symmetry. By imposing some appropriate weak assumptions on parameters, we obtain a norm solution of the problems, which solves a certain hierarchical variational inequality.

This paper introduces a novel iterative scheme that integrates an inertial technique and Nesterov acceleration for solving the multi-output set split common fixed point problem with k-semi-contractive mappings. The primary objectives are to enhance the convergence rate and reduce computational complexity. Firstly, under the condition of fixed step size, two iterative algorithms are designed, and their weak convergence and strong convergence are proved respectively. The strong convergence result is achieved by introducing relaxation parameters and external control sequences. Secondly, to avoid the difficulty of operator norm calculation, a variable step size iterative scheme is further proposed, which improves the practicality of the algorithm by dynamically adjusting the step size, and its convergence is strictly analyzed. The theoretical proof shows that under the conditions of satisfying the subclosed principle and appropriate parameters, the sequence generated by the proposed algorithm can converge to the solution of the problem. Numerical experiments verify the acceleration effect of the inertial term and the advantages of variable step size.

In this paper, we first introduce a new algorithm with a viscosity iteration method for solving the split common fixed point problem (SCFP) for a finite family of nonexpansive semigroups. We also present a new algorithm for solving the SCFP for an infinite family of quasi-nonexpansive mappings. We establish strong convergence of these algorithms in an infinite-dimensional Hilbert spaces. As application, we obtain strong convergence theorems for split variational inequality problems and split common null point problems. Our results improve and extend the related results in the literature.