The News Study Of Smoothing Methods Complementarity Problems

Complementarity problems, known as a kind of optimization problems, are resulted from the promotion of linear programming and nonlinear programming, and its applications vary from transportation, engineering, economic to financial areas. Therefore, the study of algorithms to solve complementarity problems plays a role. In this paper, we explored the smoothing methods for nonlinear complementarity problems, which include the construction of a new smoothing function, the designation of Jacobian smoothing method and completely smooth Newton method. Also, the convergence of our algorithm has been analyzed. The main contents are as follows:In the first chapter, we presented a brief introduction to the background and recent development of complementarity problems, introduced some corresponding preliminaries, And main work done were introduced.In the second chapter, we constructed a new piecewise smooth approximation function for non-linear complementarity problems, and analyzed some properties of the smooth function. Based on the new smooth function, we establish a Jacobian smoothing method for solving nonlinear complementarity problems and proved that under appropriate conditions this algorithm is global convergent with superlinear convergence rate. Numerical results show that the algorithm is effective.The third chapter presents a new smoothing function, after analyzing the different properties between our new smoothing function and those existing smooth function, we showed the existence of a smooth path and continuity when it is applied to solve the nonlinear Po complementarity problems. Furthermore, we designed a non-monotonic and completely smoothing Newton method to solve a class of nonlinear P0complementarity problems. Global convergence and superlinear convergence under appropriate conditions are proved. With comparison to the existing research results, we showed he effectiveness of our algorithm.Finally, in Chapter Five, based on the questions for existing complementary problems, we put forward new lectures for further study.