Some remarks on regularized nonconvex variational inequalities

Javad Balooee,Jong Kyu Kim

doi:10.1186/1029-242x-2013-531

Abstract

In this paper, we investigate and analyze the nonconvex variational inequalities introduced by Noor in (Optim. Lett. 3:411-418, 2009) and (Comput. Math. Model. 21:97-108, 2010) and prove that the algorithms and results in the above mentioned papers are not valid. To overcome the problems in the above cited papers, we introduce and consider a new class of variational inequalities, named regularized nonconvex variational inequalities, instead of the class of nonconvex variational inequalities introduced in the above mentioned papers. We also consider a class of nonconvex Wiener-Hopf equations and establish the equivalence between the regularized nonconvex variational inequalities and the fixed point problems as well as the nonconvex Wiener-Hopf equations. By using the obtained equivalence formulations, we prove the existence of a unique solution for the regularized nonconvex variational inequalities and propose some projection iterative schemes for solving the regularized nonconvex variational inequalities. We also study the convergence analysis of the suggested iterative schemes under some certain conditions. MSC:47H05, 47J20, 49J40, 90C33.

Highlights

Variational inequality theory, introduced by Stampacchia [ ], has become a rich source of inspiration and motivation for the study of a large number of problems arising in economics, finance, transportation, network and structural analysis, elasticity and optimization
By using the obtained equivalence formulations, we prove the existence of a unique solution for regularized nonconvex variational inequalities (RNVI) and propose some projection iterative schemes for solving RNVI
We have proved that the problem ( . ) from [, ] is not equivalent to the fixed point problem ( . ) from [, ] and the Wiener-Hopf equation ( . ) from [ ]

Summary

Introduction

Variational inequality theory, introduced by Stampacchia [ ], has become a rich source of inspiration and motivation for the study of a large number of problems arising in economics, finance, transportation, network and structural analysis, elasticity and optimization. The main idea in this technique is to establish the equivalence between the variational inequalities and the fixed point problems using the concept of projection This alternative formulation enables us to suggest some iterative methods for computing an approximate solution. In , Clarke et al [ ] introduced and studied a new class of nonconvex sets, called proximally smooth sets; subsequently, Poliquin et al in [ ] investigated the aforementioned sets under the name of uniformly prox-regular sets. These have been successfully used in many nonconvex applications in areas such as optimization, economic models, dynamical systems, differential inclusions, etc.

For all r

NWHE has a solution z

Then we have