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 lower-C^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 18-24, 2008, Haifa, Israel, in: AMS Contemp. Mathem. 513, AMS and Bar-Ilan University, 2010, pp. 27-43], 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/upper-Ck functions. While preserving the most important properties of the quasidifferential, such as exact calculus rules, the directed subdifferential lacks the major drawbacks of quasidifferential: non-uniqueness 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
3.1 Definition and some basic properties
3.2 Examples
4. Directed sets and the directed subdifferential
4.1 Directed sets
4.2 The directed and Rubinov subdifferentials
5. Directed subdifferential for lower-Ck and amenable functions
6. Conclusions

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