Active manifolds, stratifications, and convergence to local minima in nonsmooth optimization

Abstract

We show that the subgradient method converges only to local minimizers when applied to generic Lipschitz continuous and subdifferentially regular functions that are definable in an o-minimal structure. At a high level, the argument we present is appealingly transparent: we interpret the nonsmooth dynamics as an approximate Riemannian gradient method on a certain distinguished submanifold that captures the nonsmooth activity of the function. In the process, we develop new regularity conditions in nonsmooth analysis that parallel the stratification conditions of Whitney, Kuo, and Verdier and extend stochastic processes techniques of Pemantle.

Version 1 of the arxiv report has been split into two parts. Version 2 of the arxiv report is Part 1 of the original submission. Part 2 will appear as a separate arxiv submission