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The Directed Subdifferential of DC functions

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Baier, Robert ; Farkhi, Elza:
The Directed Subdifferential of DC functions.
In: Leizaarowitz, A. ; Mordukhovich, B. S. ; Shafrir, I. ; Zaslavski, A. J. (Hrsg.): Nonlinear analysis and optimization II. Optimization : a conference in celebration of Alex Ioffe's 70th and Simeon Reich's 60th birthdays, June 18-24, 2008, Haifa, Israel. - Providence, R.I. : American Mathematical Society , 2010 . - S. 27-43 . - (Contemporary Mathematics ; 514 )
ISBN 9780821881934
DOI: https://doi.org/10.1090/conm/514

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Abstract

The space of directed sets is a Banach space in which convex compact subsets of |R are embedded. Each directed set is visualized as a (nonconvex) subset of |R, which is comprised of a convex, a concave and a mixed-type part.
Following an idea of A. Rubinov, the directed subdifferential of a difference of convex (DC) functions is defined as the directed difference of the corresponding embedded convex subdifferentials. Its visualization is called the Rubinov subdifferential. The latter contains the Dini-Hadamard subdifferential as its convex part, the Dini-Hadamard superdifferential as its concave part, and its convex hull equals the Michel-Penot subdifferential. Hence, the Rubinov subdifferential contains less critical points in general than the Michel-Penot subdifferential, while the sharp necessary and sufficient optimality conditions in terms of the Dini-Hadamard subdifferential are recovered by the convex part of the directed subdifferential.
Furthermore, the directed subdifferential could distinguish between points that are candidates for a maximum and those for a minimum. It also allows to easily detect ascent and descent directions from its visualization. Seven out of eight axioms that A. Ioffe demanded for a subdifferential are satisfied as well as the sum rule with equality.

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Publikationsform: Aufsatz in einem Buch
Begutachteter Beitrag: Ja
Zusätzliche Informationen: Contents:
1. Introduction
1.1 Basic Notations
2. Preliminaries - Some Known Subdifferentials
3. Directed Sets
4. The Directed Subdifferential
5. Optimality Conditions, Descent and Ascent Directions
6. Conclusions
Keywords: nonsmooth analysis; subdifferential calculus; difference of convex (DC) functions; optimality conditions; ascent and descent directions
Fachklassifikationen: Mathematics Subject Classification Code: 49J52 (90C26 90C46 49J50)
Institutionen der Universität: Fakultäten > Fakultät für Mathematik, Physik und Informatik > Mathematisches Institut > Lehrstuhl Mathematik V (Angewandte Mathematik)
Fakultäten
Fakultäten > Fakultät für Mathematik, Physik und Informatik
Fakultäten > Fakultät für Mathematik, Physik und Informatik > Mathematisches Institut
Titel an der UBT entstanden: Ja
Themengebiete aus DDC: 500 Naturwissenschaften und Mathematik > 510 Mathematik
Eingestellt am: 25 Feb 2021 08:41
Letzte Änderung: 25 Mai 2021 13:23
URI: https://eref.uni-bayreuth.de/id/eprint/63448