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 1824, 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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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 mixedtype 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 DiniHadamard subdifferential as its convex part, the DiniHadamard superdifferential as its concave part, and its convex hull equals the MichelPenot subdifferential. Hence, the Rubinov subdifferential contains less critical points in general than the MichelPenot subdifferential, while the sharp necessary and su±cient optimality conditions in terms of the DiniHadamard 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.
Contents:
1. 
Introduction


2.  Preliminaries  Some Known Subdifferentials  
3.  Directed Sets  
4.  The Directed Subdifferential  
5.  Optimality Conditions, Descent and Ascent Directions  
6.  Conclusions 