A Kind Of Fine Modified Newton Method And Quasi-newton Method Research

Among the existing methods for solving unconstrained problems, modified Newton method and quasi-Newton method attracted a large amount of people’s attention due to its global convergence and quickly convergent. But, most of modified Newton and quasi-Newton algorithms were based on the monotone line search. Relatively, there is little research on the modified Newton algorithm based on the nonmonotone line search. In this paper, we studied new nonmonotone modified Newton and quasi-Newton methods.Firstly, we introduced some relevant concepts for modified Newton method and quasi-Newton method, after giving a review on the recent advances in the modified Newton method, the quasi-Newton method and the line search technique, we also summarized the main work in this paper.Secondly, when Newton method is used to solve a nonconvex minimum problem, the Hessian matrix of the objective function at each iterate point must not be positive definite. For this, a cautious modified Newton method is proposed in this paper, where the first and the second information of the objective function at each iterate points are employed to determine a search direction. It is a hybrid method based on the steepest descent method, the Newton method and the existing modified Newton method. Under some mild assumptions, the global convergence theory is established for the developed algorithm. Numerical experiments demonstrate the computational efficiency of the algorithm, particularly in comparison with the existing similar algorithms.Thirdly, with the superiority of nonmonotone line search in finding a solution of optimization problem, a class of nonmonotone cautious BFGS algorithms is developed. Different from the existing techniques of nonmonotone line search, the parameter, which is employed to control the magnitude of nonmonotonicity, is modified (not a fixed value) by the known information of the objective function and the gradient function so that the numerical performance of the developed algorithm is improved. Under some suitable assumptions, the global convergence is proved for this algorithm. Implementing the algorithm to solve some benchmark test problems, the results demonstrate that it is more effective than the similar algorithms.Finally, with the superiority of nonmonotone line search in finding a solution of optimization problem, a class of nonmonotone modified BFGS algorithms is developed to solve smooth nonlinear equations. Different from the existing techniques of nonmonotone line search, the parameter, which is employed to control the magnitude of nonmonotonicity, is modified (not a fixed value) by the known information of nonlinear system function so that the numerical performance of the developed algorithm is improved. Under some suitable assumptions, the global convergence is proved for this algorithm. Implementing the algorithm to solve some benchmark test problems, the results demonstrate that it is more effective than the similar algorithms.