scholarly journals On the Parallel Subgradient Extragradient Rule for Solving Systems of Variational Inequalities in Hadamard Manifolds

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1496
Author(s):  
Chun-Yan Wang ◽  
Lu-Chuan Ceng ◽  
Long He ◽  
Hui-Ying Hu ◽  
Tu-Yan Zhao ◽  
...  
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Parallel Algorithms ◽  
Vector Fields ◽  
Variational Inequality Problem ◽  
Hadamard Manifold ◽  
Hadamard Manifolds ◽  
Inequality Problem ◽  
Symmetric Structure ◽  
Systems Of Variational Inequalities

In a Hadamard manifold, let the VIP and SVI represent a variational inequality problem and a system of variational inequalities, respectively, where the SVI consists of two variational inequalities which are of symmetric structure mutually. This article designs two parallel algorithms to solve the SVI via the subgradient extragradient approach, where each algorithm consists of two parts which are of symmetric structure mutually. It is proven that, if the underlying vector fields are of monotonicity, then the sequences constructed by these algorithms converge to a solution of the SVI. We also discuss applications of these algorithms for approximating solutions to the VIP. Our theorems complement some recent and important ones in the literature.

scholarly journals Parallel Tseng’s Extragradient Methods for Solving Systems of Variational Inequalities on Hadamard Manifolds

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 43
Author(s):  
Lu-Chuan Ceng ◽  
Yekini Shehu ◽  
Yuanheng Wang
Keyword(s):  
Variational Inequalities ◽  
Parallel Algorithms ◽  
Line Search ◽  
Vector Fields ◽  
Hadamard Manifolds ◽  
Monotone Variational Inequalities ◽  
Extragradient Methods ◽  
Lipschitz Constants ◽  
Systems Of Variational Inequalities

The aim of this article is to study two efficient parallel algorithms for obtaining a solution to a system of monotone variational inequalities (SVI) on Hadamard manifolds. The parallel algorithms are inspired by Tseng’s extragradient techniques with new step sizes, which are established without the knowledge of the Lipschitz constants of the operators and line-search. Under the monotonicity assumptions regarding the underlying vector fields, one proves that the sequences generated by the methods converge to a solution of the monotone SVI whenever it exists.

scholarly journals New Tseng’s extragradient methods for pseudomonotone variational inequality problems in Hadamard manifolds

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Konrawut Khammahawong ◽  
Poom Kumam ◽  
Parin Chaipunya ◽  
Somyot Plubtieng
Keyword(s):  
Variational Inequality ◽  
Numerical Experiments ◽  
Vector Fields ◽  
Variational Inequality Problem ◽  
Variational Inequality Problems ◽  
Hadamard Manifolds ◽  
Extragradient Methods ◽  
Continuous Vector ◽  
Inequality Problem ◽  
Lipschitz Continuous

AbstractWe propose Tseng’s extragradient methods for finding a solution of variational inequality problems associated with pseudomonotone vector fields in Hadamard manifolds. Under standard assumptions such as pseudomonotone and Lipschitz continuous vector fields, we prove that any sequence generated by the proposed methods converges to a solution of variational inequality problem, whenever it exits. Moreover, we give some numerical experiments to illustrate our main results.

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 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 New Tseng-Degree Gradient Method in Variational Inequality Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Zhuang Shan ◽  
Lijun Zhu ◽  
Long He ◽  
Danfeng Wu ◽  
Haicheng Wei
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
High Precision ◽  
Gradient Method ◽  
Variational Inequality Problem ◽  
Real Hilbert Space ◽  
Monotone Operators ◽  
Inequality Problem ◽  
Convergence Performance

This paper focuses on the problem of variational inequalities with monotone operators in real Hilbert space. The Tseng algorithm constructed by Thong replaced a high-precision step. Thus, a new Tseng-like gradient method is constructed, and the convergence of the algorithm is proved, and the convergence performance is higher.

scholarly journals A New System of Multivalued Mixed Variational Inequality Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Xi Li ◽  
Xue-song Li
Keyword(s):  
Variational Inequality ◽  
Variational Inequality Problem ◽  
Existence Of Solutions ◽  
Projection Algorithm ◽  
Mixed Variational Inequality ◽  
Inequality Problem ◽  
Special Cases ◽  
Systems Of Variational Inequalities ◽  
Special Case ◽  
New System

We consider a new system of multivalued mixed variational inequality problem, which includes some known systems of variational inequalities as special cases. Under suitable conditions, the existence of solutions for the system of multivalued mixed variational inequality problem and the convergence of iterative sequences generated by the generalizedf-projection algorithm are proved. A perturbational algorithm for solving a special case of multivalued mixed variational inequality problem is formally constructed. The results concerned with the existence of solutions and the convergence of iterative sequences generated by the perturbational algorithm are also given. Some known results are improved and generalized.

scholarly journals On completely generalized multi-valued co-variational inequalities involving strongly accretive operators

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 657-663
Author(s):  
Rais Ahmad ◽  
Syed Irfan
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Banach Spaces ◽  
Iterative Algorithm ◽  
Variational Inequality Problem ◽  
Existence Of Solutions ◽  
Accretive Operators ◽  
Inequality Problem

In this paper we consider the completely generalized multi-valued co-variational inequality problem in Banach spaces and construct an iterative algorithm. We prove the existence of solutions for our problem involving strongly accretive operators and convergence of iterative sequences generated by the algorithm.

scholarly journals Existence theorems for vector variational inequalities

1996 ◽  
Vol 54 (3) ◽  
pp. 473-481 ◽  
Author(s):  
Aris Daniilidis ◽  
Nicolas Hadjisavvas
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Banach Spaces ◽  
Convex Subset ◽  
Variational Inequality Problem ◽  
Existence Of Solutions ◽  
Existence Theorems ◽  
Multivalued Operator ◽  
Inequality Problem ◽  
Vector Variational Inequality Problem

Given two real Banach spaces X and Y, a closed convex subset K in X, a cone with nonempty interior C in Y and a multivalued operator from K to 2L(x, y), we prove theorems concerning the existence of solutions for the corresponding vector variational inequality problem, that is the existence of some x0 ∈ K such that for every x ∈ K we have A(x − x0) ∉ − int C for some A ∈ Tx0. These results correct previously published ones.

ON THE CONSTRUCTION OF GAP FUNCTIONS FOR VARIATIONAL INEQUALITIES VIA CONJUGATE DUALITY

2007 ◽  
Vol 24 (03) ◽  
pp. 353-371 ◽  
Author(s):  
LKHAMSUREN ALTANGEREL ◽  
RADU IOAN BOŢ ◽  
GERT WANKA
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Variational Inequality Problem ◽  
Gap Functions ◽  
Conjugate Duality ◽  
Inequality Problem

In this paper, we deal with the construction of gap functions for variational inequalities by using an approach which bases on the conjugate duality. Under certain assumptions we also investigate a further class of gap functions for the variational inequality problem, the so-called dual gap functions.

scholarly journals Topological degree and application to a parabolic variational inequality problem

2001 ◽  
Vol 25 (4) ◽  
pp. 273-287 ◽  
Author(s):  
A. Addou ◽  
B. Mermri
Keyword(s):  
Variational Inequality ◽  
Topological Degree ◽  
Variational Inequality Problem ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Parabolic Variational Inequality ◽  
Inequality Problem ◽  
Monotone Map ◽  
New Formulation

We are interested in constructing a topological degree for operators of the formF=L+A+S, whereLis a linear densely defined maximal monotone map,Ais a bounded maximal monotone operators, andSis a bounded demicontinuous map of class(S+)with respect to the domain ofL. By means of this topological degree we prove an existence result that will be applied to give a new formulation of a parabolic variational inequality problem.

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