URN to cite this document: urn:nbn:de:bvb:703-epub-5605-7
Title data
Baier, Robert ; Farkhi, Elza:
The Directed Subdifferential of DC functions.
Bayreuth
,
2010
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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.
Further data
Item Type: | Preprint, postprint |
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Additional notes (visible to public): | erscheint 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 DOI: https://doi.org/10.1090/conm/514 |
Keywords: | nonsmooth analysis; subdifferential calculus; difference of convex (DC) functions; optimality conditions; ascent and descent directions |
DDC Subjects: | 500 Science 500 Science > 510 Mathematics |
Institutions of the University: | Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics) Faculties Faculties > Faculty of Mathematics, Physics und Computer Science Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics |
Language: | English |
Originates at UBT: | Yes |
URN: | urn:nbn:de:bvb:703-epub-5605-7 |
Date Deposited: | 25 May 2021 13:23 |
Last Modified: | 08 Jun 2021 10:15 |
URI: | https://epub.uni-bayreuth.de/id/eprint/5605 |