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Using second-order information in gradient sampling methods for nonsmooth optimization
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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.
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On computing Goldstein approximate second-order stationary points of structured nonsmooth nonconvex programs
A randomized first-order algorithm computes Goldstein approximate second-order stationary points of L-smooth nonconvex functions with oracle complexity Õ(ΔL⁸n²/ε⁹ + ΔL⁶n³/ε⁷).
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