A Generalized Newton's Algorithm For Nonlinear Complementarity Problems

The nonlinear complementarity problem is a very important part in mathematical programming. It has been widely used in the field of economics and engineer. The research on it is always a hot topic in nonlinear and computational science. And good achievements have got at the study of the algorithms in solving complementarity problems.This paper mainly studieds nonsmooth complementarity problem. After studying all kinds of algorithms and semismooth theories, we studies the smooth Newton algorithm furtherly. Through the study of many complementarity functions, we could find that some of them are silimar in shape. Firstly, we advance a kind of new complementarity function. This function include famous Fisher function and the minimal function. According to this function, we can convert solving the complementarity problem to solving the nonlinear equations. However this function is not differential at the acnodes, and the equations we have got are nonsmooth. Because of the questions above, we struct a smooth approach of the former function. After converting the nonsmooth equations to the smooth ones, we use the generalized smooth Newton method to solve them. We prove that the algorithm had global convergence and local convergence. Numerical experiments show that the algorithm has good effects in solving the complementarity problems. However, when the derivative matrix is singular, this algorithm lost efficacy. So on the base of the first algorithm, we advance another one which hasn't any need for the derivative matrix. Choosing some proper parameters, we prove its global convergence. Numerical experiments also make the good properties of this algorithm clear.