A Class Of Smoothing Newton Methods For Nonlinear Complementarity Problems

Complementarity problem was first proposed in 1963.Since then it has been the hotspot in the research of mathematical programming.Also many algorithms have been proposed.With the idea of Qi's smoothing Newton method,we propose a new class of smoothing Newton methods for the nonlinear complementarity problem based on a class of special functions.In this paper,NCP is converted into a series of smoothing nonlinear equations and Newton method is used to solve the equations.We use Newton direction and Gradient direction together in the algorithm which guarantee that our method is globally convergent.Under certain conditions,our method achieves fast local convergence rate. Also we propose a new NCP function and with the new function we convert NCP into a square system of equations.Levenberg-Marquardt type method is used to solve this system.The conditions which guarantee the solution of the square system to be the solution of NCP are studied. Some numerical results are also reported.The paper contains four parts.In the first section ,the application background and the main algorithms of the complementarity problems are introduced and also the main content of this paper is referred. The second section is the most important part of this paper,in which a new class of smoothing Newton methods are detailed, also the global and local superlinear convergence is established for the method.In the third section, we propose a new NCP function and consider the nonlinear complementarity problem as nonlinear least-square problems. We solve the problem by Levenberg–Marquardt type algorithms. In the last section, we conclude the paper.