Applications Of Barzilai-Borwein Method And Filter Technology In Optimization

The two most important algorithm frameworks to optimization problems are line search method and trust region method, in which search direction, stepsize, and trust region subproblem respectively are keys of the two methods. In this paper, firstly, we give a brief introduction to the two methods, and then mainly pay attention on the improvement of the original algorithm frameworks.Following [31], we use gradient vector to define filter set, while make some improvements on the algorithm in [38], we apply the filter technique to unconstrained optimization problems. We present a new globalization nonmonotone Filter strategy of Barzilai-Borwein algorithm. We prove the global convergence . Extensive numerical results show the effectiveness of the proposed method for large scale unconstrained optimization problems.We study the application of nonmonotone technique and BB method to smooth nonlinear equations, while combing with inexact quasi-Newton methods. We present a new strategy of nonmonotone technique of inexact quasi-Newton method. We prove its global convergence and report detailed numerical results which show the effectiveness of our algorithm, especially, for large-scale optimization problems.