Some Researches On Newton-type Methods For Solving Nonlinear Complementarity Problems

In this thesis, we disscuss Newton-type methods for solving nonlinear comple-mentarity problems.In the Introduction, we start from the current situation and background about complementarity problems, the importance of research in supplementary question is explained, the main idea and development of Newton-type methods are introduced. Finally, the organization of this thesis is offered briefly.In Chapter1, by using the Fischer-Burmeister function, we propose a new nonmonotone Levenberg-Marquardt algorithm for nonlinear complementarity prob-lems. To obtain the global convergence, we use the trust region technique and nonmonotone line search technique. Under suitable assumptions without strict com-plementarity condition, we get the local superlinear/quadratic convergence of the algorithm. Some numerical results show that this method is effective for nonlinear complementarity problems.In Chapter2, by constructing a new smoothing NCP-function, a modified one-step smoothing Newton method is proposed for solving nonlinear complementarity problems with Po-function. The proposed algorithm solves only one linear system of equations and performs only one line search at per iteration. Without requir-ing strict complementarity assumption at the solution, the algorithm is proved to be convergent globally. Furthermore, making use of the smooth and semismooth technique, we also prove that the algorithm has local superlinear/quadratic conver-gence under mild assumptions. Some numerical examples are given to illustrate the performance and efficiency of the presented algorithm.In Chapter3, first, we reformulate nonlinear complementarity problems as a smoothing system of equations, by using a new nonmonotone line search, a new non-monotone smoothing Broyden-like method is proposed for solving nonlinear com- plementarity problems. The algorithm solve only one system of linear equations, perform only one line search and update only one matrix per iteration. The glob-al and local superlinear convergence of the algorithm are proved under suitable conditions. Some numerical examples are given to illustrate the performance and efficiency of the presented algorithm.In the last chapter, we make a conclusion of the work and point out how to further our research work, such as ideas, suggestions and problems need to be solved.