The Applications of Variational Inequalities And Complementarity Problems

The (finite-dimensional) Variational Inequality (VI) problem is an important class of problems that arise in many applications. Some examples of relevant problems that can be written as a VI include Nash equilibrium problems, system of equations, nonlinear complementarity problems, fixed-point problems, saddle-point problems and so on. Among of them the nonlinear complementarity problem(NCP) is a system which includes nonlinear inequalities of functions and non negative variables along with a special equation that expresses the complementarity relationship between the variables and corresponding inequalities. This complementarity condition is the key to distinguish the NCP from of a general inequality system which lies at the heart of all constrained optimization problems in finit dimensions, provide a powerful framework for the modeling of equilibria of many kinds, and exhibits a natural link between smooth and nonsmooth mathematics. This paper addresses in two significant ways. It introduce some class of problems and presents an efficient Extragradient method to solve(VI) and a Semismooth Newton type method for solving nonsmooth equation. Also it extends these methods to the solution of CPs since a CP can be reformulated as a system of nonsmooth equations.