A Note on Clarke’s Generalized Jacobian for the Inverse of Bi-Lipschitz Maps

Florian Behr () and Georg Dolzmann ()
Additional contact information
Florian Behr: University of Regensburg
Georg Dolzmann: University of Regensburg

Journal of Optimization Theory and Applications, 2024, vol. 200, issue 2, No 15, 852-857

Abstract: Abstract Clarke’s inverse function theorem for Lipschitz mappings states that a bi-Lipschitz mapping f is locally invertible about a point $$x_0$$ x 0 if the generalized Jacobian $$\partial f(x_0)$$ ∂ f ( x 0 ) does not contain singular matrices. It is shown that under these assumptions the generalized Jacobian of the inverse mapping at $$f(x_0)$$ f ( x 0 ) is the convex hull of the set of matrices that can be obtained as limits of sequences $$J_f(x_k)^{-1}$$ J f ( x k ) - 1 with f differentiable in $$x_k$$ x k and $$x_k$$ x k converging to $$x_0$$ x 0 . This identity holds as well if f is assumed to be locally bi-Lipschitz at $$x_0$$ x 0 .

Keywords: Inverse mapping; Lipschitz continuous mapping; Clarke Jacobian; 49J52; 26B10; 58C15 (search for similar items in EconPapers)
Date: 2024
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10957-023-02333-x Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:joptap:v:200:y:2024:i:2:d:10.1007_s10957-023-02333-x

Ordering information: This journal article can be ordered from
http://www.springer. ... cs/journal/10957/PS2

DOI: 10.1007/s10957-023-02333-x

Access Statistics for this article

Journal of Optimization Theory and Applications is currently edited by Franco Giannessi and David G. Hull

More articles in Journal of Optimization Theory and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().