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On Computing the Mordukhovich Subdifferential Using Directed Sets in Two Dimensions

  • Robert Baier
  • Elza Farkhi
  • Vera Roshchina
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 47)

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

Keywords

Convex Function Directional Derivative Outer Limit Positive Homogeneity Sublinear Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

Acknowledgements

The authors thank the referees for their valuable remarks. They also express their gratitude to Boris Mordukhovich for encouraging them to explore the connections between the Rubinov subdifferential and his subdifferential as well as to Vladimir Demyanov, who initiated the research to find connections between the quasidifferential and the Mordukhovich subdifferential. This work was partially supported by The Hermann Minkowski Center for Geometry at Tel Aviv University.

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Chair of Applied MathematicsUniversity of BayreuthBayreuthGermany
  2. 2.School of Mathematical Sciences, Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael
  3. 3.Centro de Investigação em Matemática e AplicaçõesUniversidade de ÉvoraÉvoraPortugal

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