A Gaussian smoothing-based zeroth-order method for Goldstein second-order stationarity

Abstract

We introduce a new generalized Hessian, called the Goldstein second-order $\hat{I}´$-subdifferential, and an associated notion of $(\hat{I}{\mu }_1,\hat{I}{\mu }_2,\hat{I}´)$-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.