Introduction to Variational Inequalities and Nonlinear Complementarity Problems and their algorithms

The finite-dimensional nonlinear complementarity problem (NCP) is a system of finitely many nonlinear inequalities in finitely many nonnegative variables along with a special equation that expresses the complementary relationship between the variables and corresponding inequalities. This complementarity condition is the key feature distinguishing the NCP from a general inequality system, lies at the heart of all constrained optimization problems in finite dimensions, provides a powerful framework for the modeling of equilibria of many kinds, and exhibits a natural link between smooth and nonsmooth mathematics. This paper addresses this need in two significant ways.it introduces problem classes and presents most efficient Newton type methods for solving nonsmooth equations and extends these methods to the solution of VIs and CPs, since, a VI/CP can be reformulated as a saystem of nonsmooth equations