A Smoothing Method For Nonlinear Complementarity Problems

Theories and algorithms of the complementarity problem have been widely used in the field of economics, game theory and mathematical programming. The research on it is always the hot spots in nonlinear and computational science. Many results have been achieved. Our test deals with the numerical methods of the complementarity problem. Based on the concept of the smoothing Newton methods that exist and the semismooth theory, we do further research. First of all, we propose a new smooth approximation function of the famous Fischer-Burmeister function. And it is the generalization of the existed approximation function. At the same time, we study some of its attributes. Then we use it to convert the complementarity problem into nonlinear equations. And we use the smoothing Newton method to solve the equations. We also introduce a control function, prove the global convergence, and prove the local superlinear convergence under some conditions. Then we propose another new algorithm by using an existed approximation functions, which has the same characters as the first algorithms does. Finally, the result of the numerical experiments indicates the efficiency of the first algorithm.The paper contains four parts. In the first chapter, the application background and the main algorithms of the complementarity problems is introduced. In Chapter 2, some basic definitions and theories of complementarity problems are introduced. The 3rd chapter is the most important part of this paper, in which a new class of smoothing Newton method is detailed, also the global and local superlinear convergence is established for the method. In the 4th chapter, we propose some numerical experiment, and the results show the effectiveness of the proposed algorithms. In the last chapter, we conclude the paper.