scholarly journals Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Zhao-Rong Kong ◽  
Lu-Chuan Ceng ◽  
Qamrul Hasan Ansari ◽  
Chin-Tzong Pang
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Fixed Point Set ◽  
Inequality Problem ◽  
Hybrid Extragradient Method ◽  
Point Set ◽  
Hierarchical Variational Inequalities

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI.

scholarly journals Algorithms for Solving the Variational Inequality Problem over the Triple Hierarchical Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Thanyarat Jitpeera ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Variational Inequality Problem ◽  
Strong Convergence Theorem ◽  
Fixed Point Set ◽  
Inequality Problem ◽  
Point Set ◽  
Monotone Variational Inequality

This paper discusses the monotone variational inequality over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping. The strong convergence theorem for the proposed algorithm to the solution is guaranteed under some suitable assumptions.

scholarly journals Iterative methods for a fixed point of hemicontractive-type mapping and a solution of a variational inequality problem

2016 ◽  
Vol 25 (2) ◽  
pp. 183-196
Author(s):  
TESFALEM HADUSH MECHE ◽  
◽  
MENGISTU GOA SANGAGO ◽  
HABTU ZEGEYE ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Iterative Process ◽  
Variational Inequality Problem ◽  
Monotone Mapping ◽  
Common Point ◽  
Fixed Point Set ◽  
Inequality Problem ◽  
Type Mapping ◽  
Point Set

In this paper, we introduce and study an iterative process for finding a common point of the fixed point set of a Lipschitz hemicontractive-type multi-valued mapping and the solution set of a variational inequality problem for a monotone mapping. Our results improve and extend most of the results that have been proved for this class of nonlinear mappings.

scholarly journals Strong Convergence of an Iterative Algorithm for Hierarchical Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Poom Kumam ◽  
Thanyarat Jitpeera
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Iterative Algorithm ◽  
Variational Inequality Problem ◽  
Fixed Point Set ◽  
Mild Conditions ◽  
Hierarchical Problems ◽  
Point Set

We introduce the triple hierarchical problem over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping. The strong convergence of the algorithm is proved under some mild conditions. Our results extend those of Yao et al., Iiduka, Ceng et al., and other authors.

scholarly journals Global optimization and applications to a variational inequality problem

2021 ◽  
Vol 19 (1) ◽  
pp. 1349-1358
Author(s):  
Azhar Hussain ◽  
Muhammad Adeel ◽  
Hassen Aydi ◽  
Dumitru Baleanu
Keyword(s):  
Fixed Point ◽  
Global Optimization ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Variational Inequality Problem ◽  
Best Proximity Point ◽  
Inequality Problem

Abstract In the present paper, we study the existence and convergence of the best proximity point for cyclic Θ \Theta -contractions. As consequences, we extract several fixed point results for such cyclic mappings. As an application, we present some solvability theorems in the topic of variational inequalities.

scholarly journals A Mean Extragradient Method for Solving Variational Inequalities

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 462
Author(s):  
Apichit Buakird ◽  
Nimit Nimana ◽  
Narin Petrot
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Weak Convergence ◽  
Numerical Experiments ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Mean Value ◽  
Inequality Problem ◽  
The Mean ◽  
Mean Value Method

We propose a modified extragradient method for solving the variational inequality problem in a Hilbert space. The method is a combination of the well-known subgradient extragradient with the Mann’s mean value method in which the updated iterate is picked in the convex hull of all previous iterates. We show weak convergence of the mean value iterate to a solution of the variational inequality problem, provided that a condition on the corresponding averaging matrix is fulfilled. Some numerical experiments are given to show the effectiveness of the obtained theoretical result.

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