Numerical Solution Of Nonlinear Complementarity Problem And Its Application

This paper deals with the numerical methods for solving the nonlinear complementarity problem and constrained minimax problem. The algorithms are proposed. The convergence of the algorithms is proved. The numerical results also show the effectiveness of the algorithms.In the first chapter, the nonlinear complementarity problem and constrained minimax problems are introduced. The related definitions and theorems are also given.A new auxiliary function is given for the nonlinear complementarity problem in the second chapter. Some properties of this function is studied. By using this smoothing approximate function the nonlinear complementarity problem is reformulated to the nonsmooth system of equations. Then a non-monotone smoothing Newton method is developed. With suitable conditions, the global convergence and local quadratic convergence of the method are proved.In the third chapter, two methods for the constrained minimax problem are given. First, the constrainal minimax problem is converted into an equivalent nonlinear programming problem by using an auxiliary function. Second, the aggregate function is used to approximate the maximum function. Then by using the smooth approximation function the KKT conditions of the constrined minimax problems are reformulated to a smooth nonlinear system. Finally, a new quasi Newton method is built to solve the two systems of equations.Finally, the algorithms given in this paper are tested on numerical examples. Numerical results show the effectiveness of the algorithms.