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.

Broyden-Fletcher-Goldfarb-Shanno (BFGS) 方法在非凸和无约束优化问题的拟牛顿算法中起着重要作用。但是在证明全局收敛时，BFGS类方法一般需要假设目标函数的梯度是Lipschitz连续的。这个问题促使我们在经典BFGS公式的基础上，尝试寻找梯度非Lipschitz连续非凸优化的拟牛顿法。在本文中，我们针对梯度非Lipschitz 连续和非凸问题提出了一种自适应缩放阻尼BFGS 方法。使用 Armijo 或 Weak Wolfe-Powell (WWP) 线搜索，可以获得全局收敛。在合适的条件下，WWP 型线搜索的收敛速度是超线性的。