The Nonmonotone-type Methods For Unconstrained Optimization

The nonmonotone line search technique was first proposed by Grippo, Lampariello and Lucidi in 1986. Different from the monotonic one, the value of the objective function in non-monotone methods is permitted to increase among finite iterations. The trial point is accepted as long as its objective function is better than the maximum of some previous ones. Hence the small steps can be avoid when the trial points fell into the "very narrow valley". However, the traditional nonmonotone line search has several disadvantages. Firstly, a good function value generated in any iteration is essentially discarded due to the max which is in the nonmonotone line search conditions, moreover, in some cases, the numerical results are highly dependent on the selection of the degree of nonmonotonic. So, there are many progress can be made in this field.In this paper, we propose various iterative algorithms with nonmonotone line search tech-niques and prove that the algorithms are well-defined. We present a modified nonmonotone line search technique by a convex combination. Then the nonmonotone memory gradient method and nonmonotone spectral conjugate gradient method are given and their convergent properties are proved. At the end, some numerical results are reported to show our methods are efficient and better than some existed methods.