The Directed and Rubinov Subdifferentials of Quasidifferentiable Functions. Part I: Definition and Examples
R. Baier, E. Farkhi, V. Roshchina:
The Directed and Rubinov Subdifferentials of Quasidifferentiable Functions. Part I: Definition and Examples
Nonlinear Analysis: Theory, Methods & Applications
75
(3),
1074  1088,
2012
DOI:
10.1016/j.na.2011.04.074
MR Nummer:
2861321
Zentralblattnummer:
1236.49031
Keywords: subdifferentials; quasidifferentiable functions; differences of sets; directed sets; directed subdifferential; amenable and lowerC^k functions
Mathematics Subject Classification Code: 49J52 (26B25 90C26)
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Abstract:
We extend the definition of the directed subdifferential, originally introduced in [R. Baier, E. Farkhi: The directed subdifferential of DC functions, in: A. Leizarowitz, B. S. Mordukhovich, I. Shafrir, A. J. Zaslavski (Eds.), 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, in: AMS Contemp. Mathem. 513, AMS and BarIlan University, 2010, pp. 2743], for differences of convex functions (DC) to the wider class of quasidifferentiable functions. Such generalization efficiently captures differential properties of a wide class of functions including amenable and lower/upperC^{k} functions. While preserving the most important properties of the quasidifferential, such as exact calculus rules, the directed subdifferential lacks the major drawbacks of quasidifferential: nonuniqueness and “inflation in size” of the two convex sets representing the quasidifferential after applying calculus rules. The Rubinov subdifferential is defined as the visualization of the directed subdifferential.
Contents:
1.  Introduction  
2.  Preliminaries  
3. 
Quasidifferentiable functions


4. 
Directed sets and the directed subdifferential


5.  Directed subdifferential for lowerC^{k} and amenable functions  
6.  Conclusions 