| The vast majority of methods which are now widely used to solve problems of optimization are monotonous,when the objective function is in the presence of slender curved Canyon in the feasible domain,monotonic algorithms will be greatly lost its computational efficiency.So non-monotonic algorithms have aroused the attention of many researches,many effective non-monotonic algorithms are proposed,in particular combining non-monotone line search techniques and trust region method with the outstanding performance results of calculation.In this paper,we combine non-monotonic line search techniques and non-monotonic trust region methods,each step of trust region methods employs line searches,make each step of iteration fully decrease,then accelerate the iterative speed.In the traditional trust region methods for unconstrained optimization,the solution of sub-problems is crucial to the trust region methods,is also the main part of the cost of the computation for the methods.If the trial stepδk is not acceptable,sets xk+1=xk,reduces the trust region radius,and resolves the sub-problems.Therefore,the sub-problems may be solved several times before an acceptable step is found,and these repetitious processes are likely to increase the total cost of computation for large scale problems.On the contrary,the line searches identifies a new pilot xk+1only requires a very small amount of computation.Therefore,combining the trust region methods and line search techniques,can reduce the times of resolving sub-problems, and reduce the cost of computation.The introduction of line searches requires the solution of sub-problems must be a dropped direction,that is gkTδk(?)O,In this paper,we study the existed methods of solving sub-problems,prove Zhang and Xun 1999's dogleg method and Steihaug 1983's conjugate gradient method to meet the condition of adequate decrease,then the solution can be used in line search.On this basis,we propose a new algorithm that employs both non-monotonic line techniques and non-monotonic trust region methods,and we show that the new algorithm preserves the properties of global convergence and local super-linear convergence speed.Numerical results are also presented. |
