Level sets of nonsmooth functions, Part 2: Lipschitz and piecewise-differentiable manifolds

Abstract

This paper introduces piecewise-differentiable ($P{C}^r$) manifolds according to a unified general framework that also applies to nonsmooth Lipschitz manifolds and smooth manifolds. We present definitions of nonsmooth manifolds and embedded submanifolds for abstract sets as well as for subsets of $R^n$, and explore the relationships between them. The $P{C}^r$ and Lipschitz Rank Theorems from Part 1 of this series, in terms of the Clarke Jacobian and B-subdifferential generalized derivative sets, are used to characterize level sets of functions between nonsmooth manifolds as embedded submanifolds. We illustrate how the Level Set Theorems developed in this paper can be applied to functions on Euclidean space, including a piecewise-differentiable process model for distillation columns.