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The Directed Subdifferential of DC functions

Title data

Baier, Robert ; Farkhi, Elza:
The Directed Subdifferential of DC functions.
In: Leizaarowitz, A. ; Mordukhovich, B. S. ; Shafrir, I. ; Zaslavski, A. J. (ed.): 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 . - pp. 27-43 . - (Contemporary Mathematics ; 514 )
ISBN 9780821881934
DOI: https://doi.org/10.1090/conm/514


Official URL: Volltext

Abstract in another language

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.

Further data

Item Type: Article in a book
Refereed: Yes
Additional notes: 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
Subject classification: Mathematics Subject Classification Code: 49J52 (90C26 90C46 49J50)
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics)
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 25 Feb 2021 08:41
Last Modified: 24 Mar 2021 06:34
URI: https://eref.uni-bayreuth.de/id/eprint/63448