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

We introduce a new generalized Hessian, called the Goldstein second-order $\delta$-subdifferential, and an associated notion of $(\epsilon_1,\epsilon_2,\delta)$-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.