Studies On Smoothing Newton-type 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 smoothing Newton method, we proposed a class of new smoothing Newton methods for solving nonlinear complementarity problem based on a class of smoothing NCP functions.In this paper, NCP is converted into a series of smoothing nonlinear equations and Newton method is used to solve the equations. Under the assumption that F is a P₀ function, we propose a new smoothing Newton method based on a new smoothing NCP function. On the other hand, also we introduce a new Broyden-like method in this paper. The algorithm considered here is based on the smooth approximation Fischer-Burmeister function function and make use of the line search rule of Donghui Li in.In this paper, also we proposed a regularization smoothing Newton method for singular nonlinear complementarity problems by introducing a perturbed positive parameterμ, which is also smoothing parameter. Without the assumption that F is P₀ function, we propose a new smoothing Newton method underlying the Newton equation is unsolvable by modifying the regularization smoothing Newton method. The introduction of Kanzow and Pieper's gradient step makes our algorithm to be globally convergent. Under R-regularity, the method has a quadratic/superlinearly convergence rate. Under suitable conditions, all our methods achieve fast local convergence rate globally and superlinearly.The paper contain seven parts. In the fist chapter, the application background and the main algorithms of the complementarity problems is introduced. The second, third, forth and fifth sections are the most important parts of this paper, in which four algorithms are detailed for solving nonlinear complementarity problem, also the global and local convergence is established for method. In the sixth chapter, we propose some numerical experiment, and the results show the effectiveness of the proposed algorithms. In the last chapter, we conclude the paper.