Inequality Problems In Hilbert Spaces Research Articles

Solving Bilevel quasimonotone variational inequality problem in Hilbert spaces

In this paper, we propose and study a Bilevel quasimonotone Variational Inequality Problem (BVIP) in the frame work of Hilbert space. We introduce a new modified inertial iterative technique with self-adaptive step size for approximating a solution of the BVIP. In addition, we established a strong convergence result of the proposed iterative technique with an adaptive step-size conditions without prior knowledge of Lipschitz’s constant of the cost operators as well as the strongly monotonicity coefficient under some standard mild assumptions. Finally, we provide some numerical experiments to demonstrate the efficiency of our proposed methods in comparison with some recently announced results in the literature.

Tseng’s extragradient method with double projection for solving pseudomonotone variational inequality problems in Hilbert spaces

Tseng’s extragradient method with double projection for solving pseudomonotone variational inequality problems in Hilbert spaces

A generalized proximal point algorithm with new step size update for solving monotone variational inequalities in real Hilbert spaces

The aim of this paper is to study a generalized version of proximal method with a new step size update for solving variational inequality problem in Hilbert spaces. Weak convergence of the proposed scheme is attained under some appropriate assumptions imposed on the operators and parameters. At the same time, we set up an R-linear convergence rate of our method when the operator is strong monotone. Numerical experiments are given to demonstrate the efficacy of our proposed iterative algorithm. Several versions of recently proposed methods in the literature are special cases of our method.

Characterizations of the existence of solutions for variational inequality problems in Hilbert spaces

Characterizations of the existence of solutions for variational inequality problems in Hilbert spaces

Relaxed Single Projection Methods for Solving Bilevel Variational Inequality Problems in Hilbert Spaces

Relaxed Single Projection Methods for Solving Bilevel Variational Inequality Problems in Hilbert Spaces

A new inertial relaxed Tseng extrgradient method for solving quasi-monotone bilevel variational inequality problems in Hilbert spaces

A new inertial relaxed Tseng extrgradient method for solving quasi-monotone bilevel variational inequality problems in Hilbert spaces

Strong convergence theorems for split variational inequality problems in Hilbert spaces

<abstract><p>In this paper, we consider the variational inequality problem and the split common fixed point problem. Considering the common fixed points of an infinite family of nonexpansive mappings, instead of just the fixed point of one nonexpansive mapping, we generalize the results of Tian and Jiang. By removing a projection operator, we improve the efficiency of our algorithm. Finally, we propose a very simple modification to the extragradient method, which gives our algorithm strong convergence properties. We also provide some numerical examples to illustrate our main results.</p></abstract>

A neural network for solving the generalized inverse mixed variational inequality problem in Hilbert Spaces

<abstract><p>In this paper, we study and analyze the generalized inverse mixed variational inequality. The existence and uniqueness of the solution of such problem are proposed. The neural network associated with the generalized inverse mixed variational inequality is presented, and moreover, the Wiener-Hopf equation which the solution of the equation is equivalent to the solution of the generalized inverse mixed variational inequality, is considered. The stability and existence of solution of such neural network are proved. Finally, we introduce some algorithms which are constructed by the concept of the neural network and display a numerical example for understanding our results.</p></abstract>

On solving pseudomonotone equilibrium problems via two new extragradient-type methods under convex constraints

Abstract The primary objective of this study is to develop two new proximal-type algorithms for solving equilibrium problems in real Hilbert space. Both new algorithms are analogous to the well-known two-step extragradient algorithm for solving the variational inequality problem in Hilbert spaces. The proposed iterative algorithms use a new step size rule based on local bifunction information instead of the line search technique. Two weak convergence theorems for both algorithms are well-established by letting mild conditions. The main results are used to solve the fixed point and variational inequality problems. Finally, we present several computational experiments to demonstrate the efficiency and effectiveness of the proposed algorithms.

An Iterative Algorithm for Solving Fixed Point Problems and Quasimonotone Variational Inequalities

In this paper, we survey a common problem of the fixed point problem and the quasimonotone variational inequality problem in Hilbert spaces. We suggest an iterative algorithm for finding a common element of the solution of a quasimonotone variational inequality and the fixed point of a pseudocontractive operator. Convergence theorems are shown under some mild conditions. Several corollaries are also obtained.

A new iterative method for solving the multiple-set split variational inequality problem in Hilbert spaces

This paper proposes a new self-adaptive algorithm for solving the multiple-set split variational inequality problem in Hilbert spaces. Our algorithm uses dynamic step-sizes, chosen based on information of the previous step. In comparison with the work by Censor et al. [Numer Algorithms. 2012;59:301–323], the new algorithm gives strong convergence results and does not require information about the transformation operator's norm. Some applications of our main results regarding the solution of the multiple-set split feasibility problem and the split feasibility problem are presented and show that the iterative method converges strongly under weaker assumptions than the ones used recently by Xu [Inverse Probl. 2006;22:2021–2034] and by Buong [Numer Algorithms. 2017;76:783–798]. Numerical experiments on finite-dimensional and infinite-dimensional spaces and an application to discrete optimal control problems are reported to demonstrate the advantages and efficiency of the proposed algorithms over some existing results.

An iterative approach to the solution of split variational inequalities

In this paper, we investigate the split variational inequality problem in Hilbert spaces, in which an operator is ?-inverse strongly ?-monotone operator and another operator is pseudomonotone. We construct an iterative algorithm for solving the split variational inequality problem. We show the strong convergence of the suggested algorithm.

An Iterative Method for Solving Split Monotone Variational Inclusion Problems and Finite Family of Variational Inequality Problems in Hilbert Spaces

The purpose of this paper is to study the convergence analysis of an intermixed algorithm for finding the common element of the set of solutions of split monotone variational inclusion problem (SMIV) and the set of a finite family of variational inequality problems. Under the suitable assumption, a strong convergence theorem has been proved in the framework of a real Hilbert space. In addition, by using our result, we obtain some additional results involving split convex minimization problems (SCMPs) and split feasibility problems (SFPs). Also, we give some numerical examples for supporting our main theorem.

A new self adaptive Tseng’s extragradient method with double-projection for solving pseudomonotone variational inequality problems in Hilbert spaces

Abstract The purpose of this paper is to study a new Tseng’s extragradient method with two different stepsize rules for solving pseudomonotone variational inequalities in real Hilbert spaces. We prove a strong convergence theorem of the proposed algorithm under some suitable conditions imposed on the parameters. Moreover, we also give some numerical experiments to demonstrate the performance of our algorithm.

An Inertial Iterative Algorithm to Find Common Solution of a Split Generalized Equilibrium and a Variational Inequality Problem in Hilbert Spaces

In this paper, we introduce and study an iterative algorithm via inertial and viscosity techniques to find a common solution of a split generalized equilibrium and a variational inequality problem in Hilbert spaces. Further, we prove that the sequence generated by the proposed theorem converges strongly to the common solution of our problem. Furthermore, we list some consequences of our established algorithm. Finally, we construct a numerical example to demonstrate the applicability of the theorem. We emphasize that the result accounted in the manuscript unifies and extends various results in this field of study.

Accelerated Modified Tseng’s Extragradient Method for Solving Variational Inequality Problems in Hilbert Spaces

The aim of this paper is to propose a new iterative algorithm to approximate the solution for a variational inequality problem in real Hilbert spaces. A strong convergence result for the above problem is established under certain mild conditions. Our proposed method requires the computation of only one projection onto the feasible set in each iteration. Some numerical examples are presented to support that our proposed method performs better than some known comparable methods for solving variational inequality problems.

Strong Convergence of the Modified Inertial Extragradient Method with Line-Search Process for Solving Variational Inequality Problems in Hilbert Spaces

Strong Convergence of the Modified Inertial Extragradient Method with Line-Search Process for Solving Variational Inequality Problems in Hilbert Spaces

A Generalized Strong Convergence Algorithm in the Presence of Errors for Variational Inequality Problems in Hilbert Spaces

In this paper, we study the strong convergence of an algorithm to solve the variational inequality problem which extends a recent paper (Thong et al., Numerical Algorithms. 78, 1045-1060 (2018)). We reduce and refine some of their algorithm conditions and we prove the convergence of the algorithm in the presence of some computational errors. Then, using the MATLAB software, the result will be illustrated with some numerical examples. Also, we compare our algorithm with some other well-known algorithms.

Inertial modified projection algorithm with self-adaptive technique for solving pseudo-monotone variational inequality problems in Hilbert spaces

For solving pseudo-monotone variational inequality problems in Hilbert spaces, we introduce a new algorithm with self-adaptive technique. The algorithm is based on the projection method and inertial method. The strong convergence theorem is established under some appropriate conditions. Finally, we provide some numerical experiments to illustrate the effectiveness and advantages of the proposed algorithm.

Error Bounds for Inverse Mixed Quasi-Variational Inequality via Generalized Residual Gap Functions

In this paper, we investigate error bounds of an inverse mixed quasi variational inequality problem in Hilbert spaces. Under the assumptions of strong monotonicity of function couple, we obtain some results related to error bounds using generalized residual gap functions. Each presented error bound is an effective estimation of the distance between a feasible solution and the exact solution. Because the inverse mixed quasi-variational inequality covers several kinds of variational inequalities, such as quasi-variational inequality, inverse mixed variational inequality and inverse quasi-variational inequality, the results obtained in this paper can be viewed as an extension of the corresponding results in the related literature.