Adaptive scaling damped BFGS method without gradient Lipschitz continuity

Abstract

The Broyden–Fletcher–Goldfarb–Shanno (BFGS) method plays an important role among the quasi-Newton algorithms for nonconvex and unconstrained optimization problems. However, in the proof of global convergence, BFGS-type methods generally need to assume that the gradient of the objective function is Lipschitz continuous. This issue prompts us to try to find quasi-Newton method for gradient non-Lipschitz continuous and nonconvex optimization based on the classical BFGS formula. In this paper, we propose an adaptive scaling damped BFGS method for gradient non-Lipschitz continuous and nonconvex problems. With Armijo or Weak Wolfe–Powell (WWP) line search, global convergence can be obtained. Under suitable conditions the convergence rate is superlinear with WWP-type line search. Applications of the given algorithms include the tested optimization problems, which turn out the proposed method is powerful and promising.