Two Classes Of Numerical Algorithms For The Nonlinear Complementarity Problem

This paper studies the numerical algorithms for the nonlinear complementarity problem, including the LQP algorithm and the Levenberg-Marquardt algorithm. We present a new LQP algorithm and a new Levenberg-Marquardt algorithm. Convergence analyses will be elaborated in this paper and the difference between our algorithms and the method existed also be illustrated by some numerical experiments.For the LQP algorithm, we transform the nonlinear complementarity problem into a maximal monotone inclusion problem then solving a system of nonlinear equations. We propose a LQP prediction-correction projection method with two predictor and a corrector constituted by the convex combination of xk and a projection operator. Global convergence of the algorithm is proved. Have the aid of some numerical experiments, we illustrate the supervisory of this proposed algorithm contrast to the method in relevant literature and the numerical results show the superiority of our new algorithm.For the Levenberg-Marquardt algorithm, we reformulate the nonlinear complementarity problem as a system of smoothing nonlinear equations by using some nonlinear complementarity function. The introduction of the positive parameter μk makes the search direction dkL away from the iteration step of the generalized inverse of matrix dkMp.In order to prove the global convergence of the Levenberg-Marquardt algorithm, most of literatures needs the premise condition that F is a Po function. In this paper, we amend the search direction dkL, and propose a smoothing Levenberg-Marquardt algorithm by using the Armijo line search technique. For general mapping F not necessarily a P0 function, the algorithm has global convergence. Having the aid of some numerical experiments, we illustrate the supervisory of this proposed algorithm contrast to the algorithm in relevant literature.