Extremality, Stationarity and Generalized Separation of Collections of Sets
- 47 Downloads
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.
KeywordsExtremal principle Approximate stationarity Transversality Regularity Separation
Mathematics Subject Classification49J52 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.
- 1.Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton, NJ (1970)Google Scholar
- 10.Dubovitskii, A.Y., Miljutin, A.A.: Extremum problems in the presence of restrictions. USSR Comput. Maths. Math. Phys. 5, 1–80 (1965)Google Scholar
- 11.Kruger, A.Y., Mordukhovich, B.S.: New necessary optimality conditions in problems of nondifferentiable programming. In: Numerical Methods of Nonlinear Programming, pp. 116–119. Kharkov (1979) (in Russian). https://asterius.ballarat.edu.au/akruger/research/publications.html
- 12.Kruger, A.Y., Mordukhovich, B.S.: Generalized normals and derivatives and necessary conditions for an extremum in problems of nondifferentiable programming. II. VINITI no. 494-80. Minsk (1980) (in Russian). https://asterius.ballarat.edu.au/akruger/research/publications.html
- 13.Kruger, A.Y., Mordukhovich, B.S.: Extremal points and the Euler equation in nonsmooth optimization problems. Dokl. Akad. Nauk BSSR 24(8), 684–687 (1980) (in Russian). https://asterius.ballarat.edu.au/akruger/research/publications.html
- 14.Kruger, A.Y.: Generalized differentials of nonsmooth functions. VINITI no. 1332-81. Minsk (1981) (in Russian). https://asterius.ballarat.edu.au/akruger/research/publications.html
- 16.Kruger, A.Y.: \(\varepsilon \)-semidifferentials and \(\varepsilon \)-normal elements. VINITI no. 1331-81. Minsk (1981) (in Russian). https://asterius.ballarat.edu.au/akruger/research/publications.html
- 21.Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory. Fundamental Principles of Mathematical Sciences, vol. 330. Springer, Berlin (2006)Google Scholar
- 39.Ioffe, A.D., Tikhomirov, V.M.: Theory of Extremal Problems, Studies in Mathematics and Its Applications, vol. 6. North-Holland Publishing Co., Amsterdam (1979)Google Scholar
- 43.Kruger, A.Y.: About extremality of systems of sets. Dokl. Nats. Akad. Nauk Belarusi, 42(1), 24–28 (1998) (in Russian). https://asterius.ballarat.edu.au/akruger/research/publications.html
- 46.Kruger, A.Y.: Strict \((\varepsilon ,\delta )\)-semidifferentials and extremality of sets and functions. Dokl. Nats. Akad. Nauk Belarusi 44(2), 19–22 (2000). (in Russian). https://asterius.ballarat.edu.au/akruger/research/publications.html
- 53.Bui, H.T., Lindstrom, S.B., Roshchina, V.: Variational analysis down under 2018 open problem session. J. Optim. Theory Appl. (2018). https://doi.org/10.1007/s10957-018-1399-x. (This issue)