An Approximate Subgradient Method For Nonlinear Complementarity Problems

Nonlinear complementarity problems are one of the most important optimization problems,they are widely used in economic, transport, finance and other fields. In this paper,we use a new nonsmooth unconstrained optimization algorithm called an approximate subgradient algorithm to solve the nonlinear complementarity problems.We prove that the merit function for nonlinear complementarity problem is semismooth quasid-ifferentiable function when min-function is used as NCP-function,and also the convergence of the approximate subgradient method for nonlinear complementarity problems is proved.Numerical experiments on nonlinear complementarity problems solved by the proposed methed show that it is promising.The structure of the paper is as follows:In Chapter 1,we introduce the source of nonlinear complementarity problem,some knowledge of the basic definitions and theorems,as well as the methed for solving nonlinear complementarity problem.In Chapter 2,we introduce some basic definitions and theorems on nonsmooth optimization. We give a new algorithm which called approximate subgradient algorithm for solving nonsmooth unconstrained optimization problems,and also we introduce the basic framework of the algorithm and its convergence.In Chapter 3,it contains the main contribution of this thesis.Because nonlinear complementarity problem can be transformed into unconstrained optimization prob-lem,When min-function is used as NCP -function,the objective function of this unconstrained optimization problem is semismooth and quasidifferentiable.We use an approximate subgradient algorithm to solve the semismooth and quasidifferentiable unconstrained optimization problem which transformed by nonlinear complementarity prob-lem,and also we show the convergence result and the numerical experiments.In Chapter 4,we give the conclusion of this thesis as well as some considerations for further research.