Extremality, Stationarity and Generalized Separation of Collections of Sets

  • Hoa T. Bui
  • Alexander Y. KrugerEmail author


The core arguments used in various proofs of the extremal principle and its extensions as well as in primal and dual characterizations of approximate stationarity and transversality of collections of sets are exposed, analysed and refined, leading to a unifying theory, encompassing all existing approaches to obtaining ‘extremal’ statements. For that, we examine and clarify quantitative relationships between the parameters involved in the respective definitions and statements. Some new characterizations of extremality properties are obtained.


Extremal principle Approximate stationarity Transversality Regularity Separation 

Mathematics Subject Classification

49J52 49J53 49K40 90C30 



The research was 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. Alexander Y. Kruger benefited from the support of the FMJH Program PGMO and from the support of EDF. We wish to thank PhD student Nguyen Duy Cuong from Federation University Australia for careful reading of the manuscript and helping us eliminate numerous typos, and the anonymous referees for their constructive comments and suggestions.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Authors and Affiliations

  1. 1.Centre for Informatics and Applied Optimization, School of Science, Engineering and Information TechnologyFederation University AustraliaBallaratAustralia

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