Some Aspects of Extended General Variational Inequalities

Muhammad Aslam Noor

doi:10.1155/2012/303569

Abstract

Noor (“Extended general variational inequalities,” 2009, “Auxiliary principle technique for extended general variational inequalities,” 2008, “Sensitivity analysis of extended general variational inequalities,” 2009, “Projection iterative methods for extended general variational inequalities,” 2010) introduced and studied a new class of variational inequalities, which is called the extended general variational inequality involving three different operators. This class of variational inequalities includes several classes of variational inequalities and optimization problems. The main motivation of this paper is to review some aspects of these variational inequalities including the iterative methods and sensitivity analysis. We expect that this paper may stimulate future research in this field along with novel applications.

Highlights

Variational inequalities, which were introduced and studied in early sixties, contain a wealth of new ideas
We have introduced and considered a new class of variational inequalities, which is called the extended general variational inequalities
We have established the equivalent between the extended general variational inequalities and fixed point problem using the technique of the projection operator

Summary

Introduction

Variational inequalities, which were introduced and studied in early sixties, contain a wealth of new ideas. Noor 3 showed that the minimum of this type of differentiable nonconvex function on the nonconvex g-convex set can be characterized by the general variational inequalities This result shows that the general variational inequalities are closely associated with nonlinear optimization. Noor 13–16 has shown that the minimum of such type of differentiable nonconvex gh-convex functions can be characterized by a class of variational inequalities on the nonconvex gh-convex sets This fact motivated Noor 13–16 to introduce and study a new class of variational inequalities, called the extended general variational inequalities involving three different operators. This alternative equivalent form has been used to study the existence of a solution of the variational inequalities and related problems This technique and its variant forms have been used to develop several iterative methods for solving the extended general variational inequalities and optimization problems. We would like to emphasize that the results obtained and discussed in this paper may motivate and bring a large number of novel, innovative, and important applications, extensions, and generalizations in other fields

Basic Concepts

Projection Methods

Auxiliary Principle Technique

14 We now define the residue vector R u by

Conclusion