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The directed and Rubinov subdifferentials of quasidifferentiable functions. I: Definition and examples. (English) Zbl 1236.49031
Summary: We extend the definition of the directed subdifferential, originally introduced in R. Baier, E. Farkhi [The directed subdifferential of DC functions, Leizarowitz, Arie (ed.) et al., 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. Providence, RI: American Mathematical Society (AMS); Ramat-Gan: Bar-Ilan University (ISBN 978-0-8218-4835-7/pbk). Contemporary Mathematics 514; Israel Mathematical Conference Proceedings, 27-43 (2010; Zbl 1222.49020)], for Differences of Convex (DC) functions to the wider class of quasidifferentiable functions. Such generalization efficiently captures differential properties of a wide class of functions including amenable and lower/upper-\(C^k\) functions. While preserving the most important properties of the quasidifferential, such as exact calculus rules, the directed subdifferential lacks the major drawbacks of quasidifferential: 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.

49J52 Nonsmooth analysis
26B25 Convexity of real functions of several variables, generalizations
90C26 Nonconvex programming, global optimization
Full Text: DOI
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