On implicit functions in nonsmooth analysis

Dominik Dorsch (), Hubertus Th. Jongen (), Jan.-J. Rückmann () and Vladimir Shikhman ()
Additional contact information
Dominik Dorsch: RWTH Aachen University, Germany
Hubertus Th. Jongen: RWTH Aachen University , Germany
Jan.-J. Rückmann: School of Mathematics, The University of Birmingham, U.K.
Vladimir Shikhman: Université catholique de Louvain, CORE, Belgium

No 2013021, LIDAM Discussion Papers CORE from Université catholique de Louvain, Center for Operations Research and Econometrics (CORE)

Abstract: We study systems of equations, F (x) = 0, given by piecewise differentiable functions F : Rn → Rk, k ≤ n. The focus is on the representability of the solution set locally as an (n − k)-dimensional Lipschitz manifold. For that, nonsmooth versions of inverse function theorems are applied. It turns out that their applicability depends on the choice of a particular basis. To overcome this obstacle we introduce a strong full-rank assumption (SFRA) in terms of Clarke’s generalized Jacobians. The SFRA claims the existence of a basis in which Clarke’s inverse function theorem can be applied. Aiming at a characterization of SFRA, we consider also a full-rank assumption (FRA). The FRA insures the full rank of all matrices from the Clarke’s generalized Jacobian. The article is devoted to the conjectured equivalence of SFRA and FRA. For min-type functions, we give reformulations of SFRA and FRA using orthogonal projections, basis enlargements, cross products, dual variables, as well as via exponentially many convex cones. The equivalence of SFRA and FRA is shown to be true for min-type functions in the new case k = 3.

Keywords: Clarke's inverse function theorem; strong full-rank assumption; full-rank assumption; full-rank conjecture; Lipschitz manifold (search for similar items in EconPapers)
Date: 2013-05-22
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://sites.uclouvain.be/core/publications/coredp/coredp2013.html (application/pdf)

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:cor:louvco:2013021

Access Statistics for this paper

More papers in LIDAM Discussion Papers CORE from Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) Voie du Roman Pays 34, 1348 Louvain-la-Neuve (Belgium). Contact information at EDIRC.
Bibliographic data for series maintained by Alain GILLIS ().