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

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. Math., vol. 513, AMS, 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- 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.
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https://hal.inria.fr/hal-00724862
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Submitted on : Wednesday, August 22, 2012 - 7:09:32 PM
Last modification on : Friday, October 13, 2017 - 5:08:16 PM

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Robert Baier, Elza Farkhi, Vera Roshchina. The directed and Rubinov subdifferentials of quasidifferentiable functions, Part I: Definition and examples. Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 2012, 75 (3), pp.1074-1088. ⟨10.1016/j.na.2011.04.074⟩. ⟨hal-00724862⟩

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