Mixed quasi invex equilibrium problems

Muhammad Aslam Noor

doi:10.1155/s0161171204310112

Abstract

We introduce a new class of equilibrium problems, known as mixed quasi invex equilibrium (or equilibrium-like) problems. This class of invex equilibrium problems includes equilibrium problems, variational inequalities, and variational‐like inequalities as special cases. Several iterative schemes for solving invex equilibrium problems are suggested and analyzed using the auxiliary principle technique. It is shown that the convergence of these iterative schemes requires either pseudomonotonicity or partially relaxed strong monotonicity, which are weaker conditions than the previous ones. As special cases, we also obtained the correct forms of the algorithms for solving variational‐like inequalities, which have been considered in the setting of convexity. In fact, our results represent significant and important refinements of the previously known results.

Highlights

It is well known that the equilibrium problems theory provides us with a unified, natural, innovative, and general framework to study various unrelated problems arising in finance, economics, network analysis, transportation, elasticity, and nonlinear optimization; see [3, 10, 11, 12, 19, 21, 27, 29, 34, 22, 36] and the references therein
It is well known that the preinvex functions and invex sets may not be convex functions and convex sets
Noor [23] has proved that the minimum of the differentiable preinvex functions on the invex sets in normed spaces can be characterized by a class of variational inequalities, known as variational-like inequalities

Summary

MUHAMMAD ASLAM NOOR

We introduce a new class of equilibrium problems, known as mixed quasi invex equilibrium (or equilibrium-like) problems. This class of invex equilibrium problems includes equilibrium problems, variational inequalities, and variational-like inequalities as special cases. Several iterative schemes for solving invex equilibrium problems are suggested and analyzed using the auxiliary principle technique. It is shown that the convergence of these iterative schemes requires either pseudomonotonicity or partially relaxed strong monotonicity, which are weaker conditions than the previous ones. We obtained the correct forms of the algorithms for solving variational-like inequalities, which have been considered in the setting of convexity. Our results represent significant and important refinements of the previously known results.

Introduction

Consider the function

MUHAMMAD ASLAM NOOR References