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About Extensions of the Extremal Principle

Abstract

In this paper, after recalling and discussing the conventional extremality, local extremality, stationarity and approximate stationarity properties of collections of sets, and the corresponding (extended) extremal principle, we focus on extensions of these properties and the corresponding dual conditions with the goal to refine the main arguments used in this type of results, clarify the relationships between different extensions, and expand the applicability of the generalized separation results. We introduce and study new more universal concepts of relative extremality and stationarity and formulate the relative extended extremal principle. Among other things, certain stability of the relative approximate stationarity is proved. Some links are established between the relative extremality and stationarity properties of collections of sets and (the absence of) certain regularity, lower semicontinuity, and Lipschitz-like properties of set-valued mappings.

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Acknowledgements

The authors thank the referees for careful reading of the manuscript and their constructive comments and suggestions.

Funding

The research was partially supported by the Australian Research Council, project DP160100854. Hoa T. Bui is supported by an Australian Government Research Training Program (RTP) Stipend and RTP Fee-Offset Scholarship through Federation University Australia.

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Correspondence to Alexander Y. Kruger.

Additional information

Dedicated to Professor Michel Théra on the occasion of his 70th birthday.

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Bui, H.T., Kruger, A.Y. About Extensions of the Extremal Principle. Vietnam J. Math. 46, 215–242 (2018). https://doi.org/10.1007/s10013-018-0278-y

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  • DOI: https://doi.org/10.1007/s10013-018-0278-y

Keywords

  • Extremality
  • Stationarity
  • Transversality
  • Regularity
  • Separability
  • Extremal principle
  • Ekeland variational principle

Mathematics Subject Classification (2010)

  • Primary: 49J52
  • 49J53
  • Secondary: 49K40
  • 90C30
  • 90C46