Modified extragradient method with Bregman distance for variational inequalities

Herein, we present a new parallel extragradient method for solving systems of variational inequalities and common fixed point problems for demicontractive mappings in real Hilbert spaces. The algorithm determines the next iterate by computing a computationally inexpensive projection onto a sub-level set which is constructed using a convex combination of finite functions and an Armijo line-search procedure. A strong convergence result is proved without the need for the assumption of Lipschitz continuity on the cost operators of the variational inequalities. Finally, some numerical experiments are performed to illustrate the performance of the proposed method.

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Some Proximal Methods for Solving Mixed Variational Inequalities

Abstract and Applied Analysis ◽

10.1155/2012/610852 ◽

2012 ◽

Vol 2012 ◽

pp. 1-9

Author(s):

Muhammad Aslam Noor ◽

Zhenyu Huang

Keyword(s):

Variational Inequalities ◽

Extragradient Method ◽

Fixed Point Problem ◽

Point Problem ◽

Proximal Methods ◽

Proximal Point ◽

Equivalent Formulation ◽

New Methods ◽

Special Cases ◽

Mixed Variational Inequalities

It is well known that the mixed variational inequalities are equivalent to the fixed point problem. We use this alternative equivalent formulation to suggest some new proximal point methods for solving the mixed variational inequalities. These new methods include the explicit, the implicit, and the extragradient method as special cases. The convergence analysis of these new methods is considered under some suitable conditions. Our method of constructing these iterative methods is very simple. Results proved in this paper may stimulate further research in this direction.

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Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

Abstract and Applied Analysis ◽

10.1155/2013/718624 ◽

2013 ◽

Vol 2013 ◽

pp. 1-17 ◽

Cited By ~ 1

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.

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