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The directed and Rubinov subdifferentials of quasidifferentiable functions. II: Calculus. (English) Zbl 1242.49033
Summary: We continue the study of the directed subdifferential for quasidifferentiable functions started in R. Baier, E. Farkhi, and V. Roshchina [”The directed and Rubinov subdifferentials of quasidifferentiable functions, Part I: Definition and examples, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 3, 1074-1088 (2012; Zbl 1236.49031]]. Calculus rules for the directed subdifferentials of sum, product, quotient, maximum and minimum of quasidifferentiable functions are derived. The relation between the Rubinov subdifferential and the subdifferentials of Clarke, Dini, Michel–Penot, and Mordukhovich is discussed. Important properties implying the claims of Ioffe’s axioms as well as necessary and sufficient optimality conditions for the directed subdifferential are obtained.

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