The Directed Subdifferential of DC functions

R. Baier, E. Farkhi: The Directed Subdifferential of DC functions
in: 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, A. Leizarowitz, B. S. Mordukhovich, I. Shafrir, A. J. Zaslavski (eds.)
AMS Contemporary Mathematics 514, 2010, 27 - 43

DOI: 10.1090/conm/514
MR Nummer: 2668252
Zentralblattnummer: 1222.49020
Keywords: nonsmooth analysis; subdifferential calculus; difference of convex (DC) functions; optimality conditions; ascent and descent directions
Mathematics Subject Classification Code: 49J52 (90C26 90C46 49J50)
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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 su±cient 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.


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

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