Strong growth and Goldstein subgradients in piecewise smooth optimization

Abstract

We explore a new growth condition for Lipschitz functions around local minimizers, recently proposed in the context of Goldstein-subgradient-based algorithms and their behavior in practice. On generic examples, these algorithms are often observed to converge approximately linearly. We focus on objectives that are piecewise twice continuously differentiable. In that case, the new growth condition holds when the objective is, in addition, strongly convex. In the nonconvex case, we prove that quadratic growth in conjunction with a regularity property for the active gradients suffices. Computational experiments illustrate the importance of the assumptions.