On New Proximal Point Methods for Solving the Variational Inequalities

Muhammad Aslam Noor,Khalida Inayat Noor,Eisa Al-Said

doi:10.1155/2012/412413

Muhammad Aslam Noor, Khalida Inayat Noor + Show 1 more

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https://doi.org/10.1155/2012/412413

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Abstract

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

Highlights

Variational inequalities, the origin of which can be traced back to Stampacchia 1, are being used to study a wide class of diverse unrelated problems arising in various branches of pure and applied sciences in a unified framework
It is well known that the variational inequalities are equivalent to the fixed point problem
This alternative equivalent formulation has played an important and fundamental role in the existence, numerical methods, and other aspects of the variational inequalities. This equivalent formulation has been used to suggest the projection iterative method, the implicit iterative method, and the extragradient method, which is due to Korpelevich 2, for solving the variational inequalities

Summary

Introduction

Variational inequalities, the origin of which can be traced back to Stampacchia 1 , are being used to study a wide class of diverse unrelated problems arising in various branches of pure and applied sciences in a unified framework. It is well known that the variational inequalities are equivalent to the fixed point problem This alternative equivalent formulation has played an important and fundamental role in the existence, numerical methods, and other aspects of the variational inequalities. We remark that the implicit iterative method and the explicit iterative method are two different and distinct methods We use this alternative equivalent formulation to suggest and analyze some new proximal point methods, which include the implicit and explicit methods as special cases. This is the main motivation of this paper. We hope that the ideas and techniques of this paper may stimulate further research in this area of pure and applied sciences

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Conclusion