Smoothing Newton Methods For Solving The Nonlinear Complementarity Problem

Nonlinear Complementarity problem is an important class of mathematical optimization problem, which has wide applications in many fields such as engineering, economics and traffic equilibrium problem. Therefore, it is significant to study the algorithms for solving nonlinear complementarity problems. Up to now there have developed many methods to solve it and global and local convergence results have also bean obtained. In recent years more attention has been devoted to reformulating the nonlinear complementarity problem as a system of non-smooth equations by using some NCP function.In this paper, based on the smooth function given by Chen Xiao-hong in 2006, the concept of the smoothing Newton methods that exist and the semi-smooth theory, we receive a smoothing Newton method for solving P₀-fanction nonlinear complementarity problem. The algorithm has been proved to be well-defined. At the same time, we also prove the algorithm is global convergence under a control function that has existed. Then a modified smoothing Newton for solving nonlinear complemmtarity problem is proposed. The smoothing factor is seen as a parameter in our paper. The form and iteration scheme of the smoothing parameter are simple. It is proved that the method has global convergence in proper conditions.