Based On The Alpha Quantile Nonmonotone Line Search And Its Application

In this paper, a new nonmonotone line search’s technique presented based on α quantile. At the same time, we study the application of the new nonmonotone line search’s technique in the quasi-Newton methods and the conjugate gradient methods. The new nonmonotone line search belongs to Armijo-type line searchs, its idea is from Grippo etal who put it forward in1986. However, the presented new nonmontone line search’s technique in this paper has a superior advantages in choosing the step size due to the flexility of α quantile.Firstly, we introduced some relevant concepts for the conjugate gradient method and the quasi-newton method, after giving a review on the recent advances in the conjugate gradient method, the quasi-newton method and the line search techniques, we also summarized the main work in this paper.Secondly, we study the application of the new nonmonotone line search’s technique based on a quantile in spectral conjugate method As it is known, spectral conjugate gradient method attracted a large amount of people’s attention due to its less storage requirements and simple computation. But, most of the spectral conjugate gradient algorithms were based on the Armijo type line search or the Wolfe-Powell type line search, and both of them belong to monotonical line search.Based on this, a kind of nonmonotone spectral conjugate gradient algorithm with three parameters was developed for solving unconstrained optimization problems, and under some mild assumptions, the global convergence theorem of this algorithm was proved. Preliminary numerical experiments showed that the developed algorithm is valuable.Thirdly, we study the application of the new nonmonotone line search’s technique based on α quantile in quasi-Newton methods. And under some mild assumptions, the global convergence theorem of this algorithm was proved. Numerical experiments showed that the developed algorithm is effective.