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On Computing the Mordukhovich Subdifferential Using Directed Sets in Two Dimensions

In: Variational Analysis and Generalized Differentiation in Optimization and Control

Author

Listed:
  • Robert Baier

    (University of Bayreuth)

  • Elza Farkhi

    (Tel Aviv University)

  • Vera Roshchina

    (Universidade de Évora)

Abstract

The Mordukhovich subdifferential, being highly important in variational and nonsmooth analysis and optimization, often happens to be hard to calculate. We propose a method for computing the Mordukhovich subdifferential of differences of sublinear (DS) functions applying the directed subdifferential of differences of convex (DC) functions. We restrict ourselves to the two-dimensional case mainly for simplicity of the proofs and for the visualizations. The equivalence of the Mordukhovich symmetric subdifferential (the union of the corresponding subdifferential and superdifferential) to the Rubinov subdifferential (the visualization of the directed subdifferential) is established for DS functions in two dimensions. The Mordukhovich subdifferential and superdifferential are identified as parts of the Rubinov subdifferential. In addition, the Rubinov subdifferential may be constructed as the Mordukhovich one by Painlevé–Kuratowski outer limits of Fréchet subdifferentials. The results are applied to the case of DC functions. Examples illustrating the obtained results are presented. 2010 Mathematics Subject Classification. Primary 49J52; Secondary 26B25, 49J50, 90C26

Suggested Citation

  • Robert Baier & Elza Farkhi & Vera Roshchina, 2010. "On Computing the Mordukhovich Subdifferential Using Directed Sets in Two Dimensions," Springer Optimization and Its Applications, in: Regina S. Burachik & Jen-Chih Yao (ed.), Variational Analysis and Generalized Differentiation in Optimization and Control, pages 59-93, Springer.
    • Handle: RePEc:spr:spochp:978-1-4419-0437-9_3
    DOI: 10.1007/978-1-4419-0437-9_3
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    Cited by:

    1. Robert Baier & Elza Farkhi & Vera Roshchina, 2014. "Directed Subdifferentiable Functions and the Directed Subdifferential Without Delta-Convex Structure," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 391-414, February.
    2. Robert Baier & Elza Farkhi & Vera Roshchina, 2016. "From Quasidifferentiable to Directed Subdifferentiable Functions: Exact Calculus Rules," Journal of Optimization Theory and Applications, Springer, vol. 171(2), pages 384-401, November.

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