A General Inertial Proximal Point Algorithm for Mixed Variational Inequality Problem

2015 ◽  
Vol 25 (4) ◽  
pp. 2120-2142 ◽  
Author(s):  
Caihua Chen ◽  
Shiqian Ma ◽  
Junfeng Yang
Keyword(s):  
Variational Inequality ◽  
Proximal Point Algorithm ◽  
Variational Inequality Problem ◽  
Proximal Point ◽  
Mixed Variational Inequality ◽  
Inequality Problem

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 Stability and Convergence Analysis for Set-Valued Extended Generalized Nonlinear Mixed Variational Inequality Problems and Generalized Resolvent Dynamical Systems

2021 ◽  
Vol 2021 ◽  
pp. 1-21
Author(s):  
Iqbal Ahmad ◽  
Zahoor Ahmad Rather ◽  
Rais Ahmad ◽  
Ching-Feng Wen
Keyword(s):  
Dynamical System ◽  
Variational Inequality ◽  
Iterative Algorithm ◽  
Variational Inequality Problem ◽  
Stability And Convergence ◽  
Mixed Variational Inequality ◽  
Generalized Resolvent ◽  
Inequality Problem ◽  
System A ◽  
Type Algorithm

In this paper, we study a set-valued extended generalized nonlinear mixed variational inequality problem and its generalized resolvent dynamical system. A three-step iterative algorithm is constructed for solving set-valued extended generalized nonlinear variational inequality problem. Convergence and stability analysis are also discussed. We have shown the globally exponential convergence of generalized resolvent dynamical system to a unique solution of set-valued extended generalized nonlinear mixed variational inequality problem. In support of our main result, we provide a numerical example with convergence graphs and computation tables. For illustration, a comparison of our three-step iterative algorithm with Ishikawa-type algorithm and Mann-type algorithm is shown.

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 Strong Convergence to a Solution of a Variational Inequality Problem in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Yasunori Kimura ◽  
Kazuhide Nakajo
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Variational Inequality Problem ◽  
Nonempty Closed Convex Subset ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Proximal Point ◽  
Inequality Problem ◽  
Common Solution ◽  
Differentiable Norm

We consider the variational inequality problem for a family of operators of a nonempty closed convex subset of a 2-uniformly convex Banach space with a uniformly Gâteaux differentiable norm, into its dual space. We assume some properties for the operators and get strong convergence to a common solution to the variational inequality problem by the hybrid method proposed by Haugazeau. Using these results, we obtain several results for the variational inequality problem and the proximal point algorithm.

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