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

*(English)*Zbl 1236.49031Summary: 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.

##### MSC:

49J52 | Nonsmooth analysis |

26B25 | Convexity of real functions of several variables, generalizations |

90C26 | Nonconvex programming, global optimization |

##### Keywords:

subdifferentials; quasidifferentiable functions; differences of sets; directed sets; directed subdifferential; amenable and lower-\(C^k\) functions
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\textit{R. Baier} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 3, 1074--1088 (2012; Zbl 1236.49031)

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