## 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

Review: |

## 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 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 |