The Directed and Rubinov Subdifferentials of Quasidifferentiable Functions. Part II: Calculus

R. Baier, E. Farkhi, V. Roshchina: The Directed and Rubinov Subdifferentials of Quasidifferentiable Functions. Part II: Calculus
Nonlinear Analysis: Theory, Methods & Applications 75 (3), 1058 - 1073, 2012

DOI: 10.1016/j.na.2011.04.073
MR Nummer: 2861320
Zentralblattnummer: 1242.49033
Keywords: subdifferentials; quasidifferentiable functions; differences of sets; directed sets; directed subdifferential; Rubinov subdifferential
Mathematics Subject Classification Code: 49J52 (26B25 90C26)
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Abstract:

We continue the study of the directed subdifferential for quasidifferentiable functions started in [R. Baier, E. Farkhi, V. Roshchina: The directed and Rubinov subdifferentials of quasidifferentiable functions, Part I: Definitions and examples, Nonlinear Anal., same volume]. 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.

Contents:

1. Introduction
2. Directed subdifferential of quasidifferentiable functions
2.1 Directed sets
2.2 Visualization of directed sets
2.3 Quasidifferentiable functions and the directed subdifferential
3. Calculus rules for the directed subdifferential for quasidifferentiable functions
4. Optimality conditions, descent and ascent directions
5. Connections to other subdifferentials
5.1 Relations between the directed subdifferential and other subdifferentials
5.2 Ioffe's axioms
6. Conclusions

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