The Study On Two Classes Of Algorithms For Complementarity Problems

The complementarity problem is an important branch of Operations Research,which has been widely applied to many practical problems. At present, manynumerical solution methods for solving the complementarity problem have beenproposed in the literature, where the reformulation method based on somereformulation function has been shown to have great superiority. In this paper, twoclasses of reformulation methods for solving complementarity problem arediscussed. This main results and contributions as follows:1. First, a family of generalized smoothing functions is introduced, and itsproperties are discussed. Then, a class of nonlinear complementarity problems withnon-Lipschtizian continuous function is considered and the complementarityproblem is reformulated as some smoothing equations with the smoothing functions,and a Newton algorithm involving non-monotone line search is proposed to solvethe equations in order to obtain a solution of original problem. With greater weaklycondition, this method is globally convergent and locally quadratically convergent.Finally, the method is used for solving some free boundary problem, and thenumerical results show that the proposed method in this paper is promising.2. We consider a neural network method for solving the nonlinearcomplementarity problem (NCP). This method tries to find an equilibrium point ofa first-order differential equation which derived from an equivalent unconstrainedminimization reformulation of the NCP. We establish the existence of the solutionand the convergence of the trajectory of the neural network, and discuss Lyapunovstability, asymptotic stability as well as exponential stability of the solution.