APPROXIMATION OF ZEROS OF SUM OF MONOTONE MAPPINGS WITH APPLICATIONS TO VARIATIONAL INEQUALITY AND IMAGE RESTORATION PROBLEMS

The hybrid steepest descent method is an algorithmic solution to the variational inequality problem over the fixed point set of nonlinear mapping and applicable to broad range of convexly constrained nonlinear inverse problems in real Hilbert space. In this paper, we show that the strong convergence theorem [Yamada, I. (2001). The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S., eds. Inherently Parallel Algorithm for Feasibility and Optimization and Their Applications. Elsevier, pp. 473–504] of the method for nonexpansive mapping can be extended to a strong convergence theorem of the method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings called quasi-shrinking mapping. We also present a convergence theorem of the method for paramonotone variational inequality problem over the bounded fixed point set of quasi-shrinking mapping. By these generalizations, we can approximate successively to the solution of the convex optimization problem over the fixed point set of wide range of subgradient projection operators in real Hilbert space.

We introduce and analyze a hybrid iterative algorithm by virtue of Korpelevich's extragradient method, viscosity approximation method, hybrid steepest-descent method, and averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inequality problems (VIPs), the solution set of general system of variational inequalities (GSVI), and the set of minimizers of convex minimization problem (CMP), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others.

Based on the classical subgradient extragradient method, we further study an iterative algorithm for solving variational inequality problems in real Hilbert spaces. Unlike the known results, we use different constant parameters to flexibly adjust the step size and thus accelerate the convergence rate of the algorithm. Furthermore, weak and strong convergence theorems are obtained under slightly different assumptions on the parameters and operators. To substantiate the effectiveness of our proposed algorithm, we perform several numerical experiments. These experiments elucidate the different effects of the three parameters and compare with existing methods. Besides, we demonstrate the practical applicability of our method by dealing with image restoration problems, thereby demonstrating its performance capabilities.

In this paper, let H be a real Hilbert space and let C be a nonempty, closed, and convex subset of H. We assume that $(A+B)^{-1}0\cap U\neq\emptyset$ , where $A:C\rightarrow H$ is an α-inverse-strongly monotone mapping, $B:H\rightarrow H$ is a maximal monotone operator, the domain of B is included in C. Let U denote the solution set of the constrained convex minimization problem. Based on the viscosity approximation method, we use a gradient-projection algorithm to propose composite iterative algorithms and find a common solution of the problems which we studied. Then we regularize it to find a unique solution by gradient-projection algorithm. The point $q\in (A+B)^{-1}0\cap U$ which we find solves the variational inequality $\langle(I-f)q, p-q\rangle\geq0$ , $\forall p\in(A+B)^{-1}0\cap U$ . Under suitable conditions, the constrained convex minimization problem can be transformed into the split feasibility problem. Zeros of the sum of two operators can be transformed into the variational inequality problem and the fixed point problem. Furthermore, new strong convergence theorems and applications are obtained in Hilbert spaces, which are useful in nonlinear analysis and optimization.

"In this paper, relaxed and relaxed inertial modified Tseng algorithms for approximating zeros of sum of two monotone operators whose zeros are fixed points or J-fixed points of some nonexpansive-type mappings are introduced and studied. Strong convergence theorems are proved in the setting of real Banach spaces that are uniformly smooth and 2-uniformly convex. Furthermore, applications of the theorems to the concept of J-fixed point, convex minimization, image restoration and signal recovery problems are also presented. In addition, some interesting numerical implementations of our proposed methods in solving image recovery and compressed sensing problems are presented. Finally, the performance of our proposed methods are compared with that of some existing methods in the literature."

Suppose that in a real Hilbert space H, the variational inequality problem with Lipschitzian and pseudomonotone mapping A and the common fixed-point problem of a finite family of nonexpansive mappings and a quasi-nonexpansive mapping with a demiclosedness property are represented by the notations VIP and CFPP, respectively. In this article, we suggest two Mann-type inertial subgradient extragradient iterations for finding a common solution of the VIP and CFPP. Our iterative schemes require only calculating one projection onto the feasible set for every iteration, and the strong convergence theorems are established without the assumption of sequentially weak continuity for A. Finally, in order to support the applicability and implementability of our algorithms, we make use of our main results to solve the VIP and CFPP in two illustrating examples.

In this article, the problem of solving a strongly monotone variational inequality problem over the solution set of a monotone inclusion problem in the setting of real Hilbert spaces is considered. To solve this problem, two methods, which are improvements and modifications of the Tseng splitting method, and projection and contraction methods, are presented. These methods are equipped with inertial terms to improve their speed of convergence. The strong convergence results of the suggested methods are proved under some standard assumptions on the control parameters. Also, strong convergence results are achieved without prior knowledge of the operator norm. Finally, the main results of this research are applied to solve bilevel variational inequality problems, convex minimization problems, and image recovery problems. Some numerical experiments to show the efficiency of our methods are conducted.

The purpose of this paper is to study and analyze two different kinds of extragradient-viscosity-type iterative methods for finding a common element of the set of solutions of the variational inequality problem for a monotone and Lipschitz continuous operator and the set of fixed points of a demicontractive mapping in real Hilbert spaces. Although the problem can be translated to a common fixed point problem, the algorithm’s structure is not derived from algorithms in this field but from the field of variational inequalities and hence can be computed quite easily. We extend several results in the literature from weak to strong convergence in real Hilbert spaces and moreover, the prior knowledge of the Lipschitz constant of cost operator is not needed. We prove two strong convergence theorems for the sequences generated by these new methods. Primary numerical examples illustrate the validity and potential applicability of the proposed schemes.

The main aim of this research is to introduce and investigate an inertial Tseng iterative method to approximate a common solution for the variational inequality problem for γ-inverse strongly monotone mapping and monotone inclusion problem in real Hilbert spaces. We establish a strong convergence theorem for our suggested iterative method to approximate a common solution for our proposed problems under some certain mild conditions. Furthermore, we deduce a consequence from the main convergence result. Finally, a numerical experiment is presented to demonstrate the effectiveness of the iterative method. The method and methodology described in this paper extend and unify previously published findings in this field.

Our purpose in this paper is to construct a new iterative scheme and prove strong convergence theorem using the iterative scheme for approximation of a common fixed point of a countable family of relatively nonexpansive mappings, which is also a solution to a finite system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space. We apply our results to approximate fixed point of a nonexpansive mapping, which is also solution to a finite system of equilibrium problems in a real Hilbert space, and approximate a common solution to finite system of variational inequality problems and convex minimization problems which is also a common solution to countable family of relatively nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space. Our results extend many known recent results in the literature.

Strong and weak convergence theorems for general mixed equilibrium problems and variational inequality problems and fixed point problems in Hilbert spaces

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.

For the purpose of this article, we introduce a modified form of a generalized system of variational inclusions, called the generalized system of modified variational inclusion problems (GSMVIP). This problem reduces to the classical variational inclusion and variational inequalities problems. Motivated by several recent results related to the subgradient extragradient method, we propose a new subgradient extragradient method for finding a common element of the set of solutions of GSMVIP and the set of a finite family of variational inequalities problems. Under suitable assumptions, strong convergence theorems have been proved in the framework of a Hilbert space. In addition, some numerical results indicate that the proposed method is effective.

In this paper, we present a new iterative scheme for finding a common point among the set of solution of equilibrium problems, the set of solution to a variational inequality problem and the fixed point set of k-strictly pseudo-contractive mappings in a real Hilbert space. We then prove that the proposed scheme converges strongly to a common element which is the solution of a variational inequality problem, system of equilibrium problems, and a fixed point of k-strictly pseudo-contractive mappings. These results improve and generalize recent works in this direction.

In this paper, we consider a variational inequality problem which is defined over the set of intersections of the set of fixed points of a ζ-strictly pseudocontractive mapping, the set of fixed points of a nonexpansive mapping and the set of solutions of a minimization problem. We propose an iterative algorithm with regularization to solve such a variational inequality problem and study the strong convergence of the sequence generated by the proposed algorithm. The results of this paper improve and extend several known results in the literature.