Using second-order information in gradient sampling methods for nonsmooth optimization

Abstract: In this article, we introduce a novel concept for second-order information of a nonsmooth function inspired by the Goldstein eps-subdifferential. It comprises the coefficients of all existing second-order Taylor expansions in an eps-ball around a given point. Based on this concept, we define a model of the objective as the maximum of these Taylor expansions, and derive a sampling scheme for its approximation in practice. Minimization of this model induces a simple descent method, for which we show convergence for the case where the objective is convex or of max-type. While we do not prove any rate of convergence of this method, numerical experiments suggest superlinear behavior with respect to the number of oracle calls of the objective.