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Calculus for Directional Limiting Normal Cones and Subdifferentials

Set-Valued and Variational Analysis volume 27, pages 713–745 (2019)Cite this article

Abstract

The paper is devoted to the development of a comprehensive calculus for directional limiting normal cones, subdifferentials and coderivatives in finite dimensions. This calculus encompasses the whole range of the standard generalized differential calculus for (non-directional) limiting notions and relies on very weak (non-restrictive) qualification conditions having also a directional character. The derived rules facilitate the application of tools exploiting the directional limiting notions to difficult problems of variational analysis including, for instance, various stability and sensitivity issues. This is illustrated by some selected applications in the last part of the paper.

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Acknowledgments

The authors would like to express their gratitude to both reviewers as well as to the editor for their helpful suggestions. The research of the first two authors was supported by the Austrian Science Fund (FWF) under grant P29190-N32. The research of the third author was supported by the Grant Agency of the Czech Republic, projects 17-04301S and 17-08182S and the Australian Research Council, project DP160100854F.

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Authors and Affiliations

  1. Institute of Computational Mathematics, Johannes Kepler University Linz, A-4040, Linz, Austria

    Matúš Benko & Helmut Gfrerer

  2. Czech Academy of Sciences, Institute of Information Theory and Automation, 18208, Prague, Czech Republic

    Jiří V. Outrata

  3. Centre for Informatics and Applied Optimization, Federation University of Australia, Ballarat, Vic, 3350, Australia

    Jiří V. Outrata

Authors
  1. Matúš Benko
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  2. Helmut Gfrerer
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  3. Jiří V. Outrata
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Corresponding author

Correspondence to Matúš Benko.

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Benko, M., Gfrerer, H. & Outrata, J.V. Calculus for Directional Limiting Normal Cones and Subdifferentials. Set-Valued Var. Anal 27, 713–745 (2019). https://doi.org/10.1007/s11228-018-0492-5

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  • DOI: https://doi.org/10.1007/s11228-018-0492-5

Keywords

  • Generalized differential calculus
  • Directional limiting normal cone
  • Directional limiting subdifferential
  • Qualification conditions

Mathematics Subject Classification (2010)

  • 49J53
  • 49J52
  • 90C31