On the global convergence of the BFGS method or nonconvex unconstrained optimization problems

This paper is concerned with the open problem of whether the BFGS method with inexact line search converges globally when applied to nonconvex unconstrained optimization problems. We propose a cautious BFGS update and prove that the method with either a Wolfe-type or an Armijo-type line search converges globally if the function to be minimized has Lipschitz continuous gradients.