Global convergence of a family of modified BFGS methods under a modified weak-Wolfe–Powell line search for nonconvex functions

Abstract

In this paper, we consider an unconstrained optimization problem and propose a new family of modified BFGS methods to solve it. As it is known, classic BFGS method is not always globally convergence for nonconvex functions. To overcome this difficulty, we introduce a new modified weak-Wolfe–Powell line search technique. Under this new technique, we prove global convergence of the new family of modified BFGS methods and the classic BFGS method, for nonconvex functions. Furthermore, all members of this family have at least $$o(\Vert s \Vert ^{5})$$ error order. Our obtained results from numerical experiments on 77 standard unconstrained problems, indicate that the algorithms developed in this paper are promising and more effective than some similar algorithms.