Smooth approximation of Lipschitz maps and their subgradients
File(s)
Published version
Author(s)
Edalat, Abbas
Type
Journal Article
Abstract
We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as the lower limit (i.e., limit inferior) of the classical derivative of the map where it exists. The new representations lead to significantly shorter proofs for the basic properties of the subgradient and the generalised Jacobian including the chain rule. We establish that a sequence of locally Lipschitz maps between finite dimensional Euclidean spaces converges to a given locally Lipschitz map in the L-topology—that is, the weakest refinement of the sup norm topology on the space of locally Lipschitz maps that makes the generalised Jacobian a continuous functional—if and only if the limit superior of the sequence of directional derivatives of the maps in a given vector direction coincides with the generalised directional derivative of the given map in that direction, with the convergence to the limit superior being uniform for all unit vectors. We then prove our main result that the subspace of Lipschitz C∞ maps between finite dimensional Euclidean spaces is dense in the space of Lipschitz maps equipped with the L-topology, and, for a given Lipschitz map, we explicitly construct a sequence of Lipschitz C∞ maps converging to it in the L-topology, allowing global smooth approximation of a Lipschitz map and its differential properties. As an application, we obtain a short proof of the extension of Green’s theorem to interval-valued vector fields. For infinite dimensions, we show that the subgradient of a Lipschitz map on a Banach space is upper continuous, and, for a given real-valued Lipschitz map on a separable Banach space, we construct a sequence of Gateaux differentiable functions that converges to the map in the sup norm topology such that the limit superior of the directional derivatives in any direction coincides with the generalised directional derivative of the Lipschitz map in that direction.
Date Issued
2022-02
Date Acceptance
2021-08-09
Citation
Journal of the ACM, 2022, 69 (1), pp.1-32
ISSN
0004-5411
Publisher
Association for Computing Machinery (ACM)
Start Page
1
End Page
32
Journal / Book Title
Journal of the ACM
Volume
69
Issue
1
Copyright Statement
© 2021 Copyright held by the owner/author(s). This work is licensed under a Creative Commons Attribution International 4.0 License.
License URL
Identifier
https://dl.acm.org/doi/10.1145/3481805
Subjects
Computation Theory & Mathematics
08 Information and Computing Sciences
Publication Status
Published
Date Publish Online
2021-12-22