The directed and Rubinov subdifferentials of quasidifferentiable functions, Part I : Definition and examples

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Baier, Robert ; Farkhi, Elza ; Roshchina, Vera:
The directed and Rubinov subdifferentials of quasidifferentiable functions, Part I : Definition and examples.
Bayreuth , 2011

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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.