Two New Nonmonotone Line Search Methods

It is well known that the unconstrained optimization problems are a class of the most fundamental ones in optimization; the monotone line search methods are a class of the most important methods for these problems. In 1986, Grippo,Lampariello and Lucidi(GLL for short)first proposed a nonmonotone line search technique for Newton method and the GLL methodology has many applications.In this paper, we mainly propose two new nonmonotone line search techniques. First, we vary decrease at each iteration and obtain a new nonmonotone line search method based on the work of Grippo,Lampariello and Lucidi. Under some suitable assumptions, by considering that the gradient function of the objective function is Lipschitz continuous or not, we analyze the global convergence of the first algorithm. Second , we consider the Hesse matrix of the objective function and some techniques of the quasi-Newton method, and give a new nonmonotone line search method. Furthermore, we also prove the global convergence of the second algorithm. Finally, we further generalize the two algorithms.