2026/07/23 by Ming Lei, Ting Kei Pong, Man-Chung Yue +2
#math.OC
We introduce a new generalized Hessian, called the Goldstein second-order δ-subdifferential, and an associated notion of (ε1,ε2,δ)-second-order stationary point for continuously differentiable functions with locally Lipschitz gradients. We propose a zeroth-order algorithm based on cubic regularization and Gaussian smoothing with homotopy to find such approximate second-order stationary points for Lipschitz differentiable functions, and derive the iteration complexity under a mild coercivity-type assumption on the objective function.