scholarly journals Error Bound and Convergence Analysis of Matrix Splitting Algorithms for the Affine Variational Inequality Problem

1992 ◽  
Vol 2 (1) ◽  
pp. 43-54 ◽  
Author(s):  
Zhi-Quan Luo ◽  
Paul Tseng
Keyword(s):  
Variational Inequality ◽  
Convergence Analysis ◽  
Error Bound ◽  
Variational Inequality Problem ◽  
Matrix Splitting ◽  
Splitting Algorithms ◽  
Inequality Problem ◽  
Affine Variational Inequality

scholarly journals Some Results about the Isolated Calmness of a Mixed Variational Inequality Problem

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Yali Zhao ◽  
Hongying Li ◽  
Weiyi Qian ◽  
Xiaodong Fan
Keyword(s):  
Variational Inequality ◽  
Programming Problem ◽  
Convex Programming ◽  
Error Bound ◽  
Variational Inequality Problem ◽  
Convex Programming Problem ◽  
Mixed Variational Inequality ◽  
Problem Model ◽  
Inequality Problem ◽  
Isolated Calmness

It is well known that optimization problem model has many applications arising from matrix completion, image processing, statistical learning, economics, engineering sciences, and so on. And convex programming problem is closely related to variational inequality problem. The so-called alternative direction of multiplier method (ADMM) is an importance class of numerical methods for solving convex programming problem. When analyzing the rate of convergence of various ADMMs, an error bound condition is usually required. The error bound can be obtained when the isolated calmness of the inverse of the KKT mapping of the related problem holds at the given KKT point. This paper is to study the isolated calmness of the inverse KKT mapping onto the mixed variational inequality problem with nonlinear term defined by norm function and indicator function of a convex polyhedral set, respectively. We also consider the isolated calmness of the inverse KKT mapping onto classical variational inequality problem with equality and inequality constrains under strict Mangasarian-Fromovitz constraint qualification condition. The results obtained here are new and very interesting.

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.

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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