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.
Similar content being viewed by others
References
Apetrii, M., Durea, M., Strugariu, R.: On subregularity properties of set-valued mappings. Set-Valued Var. Anal. 21, 93–126 (2013)
Borwein, J. M., Jofré, A.: A nonconvex separation property in Banach spaces. Math. Methods Oper. Res. 48, 169–179 (1998)
Borwein, J. M., Zhu, Q. J.: Techniques of Variational Analysis. Springer, New York (2005)
Cibulka, R., Fabian, M., Kruger, A. Y.: On semiregularity of mappings. arXiv:1711.04420 (2017)
Dontchev, A. L., Rockafellar, R. T.: Implicit Functions and Solution Mappings: A View from Variational Analysis, 2nd edn. Springer Series in Operations Research and Financial Engineering. Springer, New York (2014)
Dubovitskii, A.Ya., Milyutin, A.A.: Extremum problems in the presence of restrictions. USSR Comp. Maths. Math. Phys. 5, 1–80 (1965)
Fabian, M.: Subdifferentials, local ε-supports and Asplund spaces. J. Lond. Math. Soc. s2-34, 568–576 (1986)
Fabian, M.: Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss. Acta Univ. Carol. 30, 51–56 (1989)
Ioffe, A. D.: Fuzzy principles and characterization of trustworthiness. Set-Valued Anal. 6, 265–276 (1998)
Ioffe, A. D.: Metric regularity and subdifferential calculus. Russ. Math. Surveys 55, 501–558 (2000)
Ioffe, A. D.: Transversality in variational analysis. J. Optim. Theory Appl. 174, 343–366 (2017)
Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications Nonconvex. Optimization and its Applications, vol. 60. Kluwer Academic Publishers, Dordrecht (2002)
Kruger, A.Y.: Generalized differentials of nonsmooth functions. VINITI no. 1332-81. Minsk, 67 pp. (In Russian). Available at: https://asterius.ballarat.edu.au/akruger/research/publications.html (1981)
Kruger, A.Y.: ε-semidifferentials and ε-normal elements. VINITI no. 1331-81. Minsk, 76 pp. (In Russian). Available at https://asterius.ballarat.edu.au/akruger/research/publications.html (1981)
Kruger, A. Y.: Generalized differentials of nonsmooth functions and necessary conditions for an extremum. Sibirsk. Mat. Zh. 26, 78–90 (1985). In Russian; English transl.: Sib. Math. J. 26, 370–379 (1985)
Kruger, A.Y.: About extremality of systems of sets. Dokl. Nats. Akad. Nauk Belarusi 42, 24–28 (1998). In Russian. Available at https://asterius.ballarat.edu.au/akruger/research/publications.html
Kruger, A.Y.: Strict (ε,δ)-semidifferentials and extremality of sets and functions. Dokl. Nats. Akad. Nauk Belarusi 44, 19–22 (2000). In Russian. Available at https://asterius.ballarat.edu.au/akruger/research/publications.html
Kruger, A. Y.: Strict (ε,δ)-subdifferentials and extremality conditions. Optimization 51, 539–554 (2002)
Kruger, A. Y.: On Fréchet subdifferentials. J. Math. Sci. 116, 3325–3358 (2003)
Kruger, A. Y.: Weak stationarity: Eliminating the gap between necessary and sufficient conditions. Optimization 53, 147–164 (2004)
Kruger, A. Y.: Stationarity and regularity of set systems. Pac. J. Optim. 1, 101–126 (2005)
Kruger, A. Y.: About regularity of collections of sets. Set-Valued Anal. 14, 187–206 (2006)
Kruger, A. Y.: About stationarity and regularity in variational analysis. Taiwan. J. Math. 13, 1737–1785 (2009)
Kruger, A. Y., López, M. A.: Stationarity and regularity of infinite collections of sets. J. Optim. Theory Appl. 154, 339–369 (2012)
Kruger, A. Y., Luke, D. R., Thao, N. H.: About subtransversality of collections of sets. Set-Valued Var. Anal. 25, 701–729 (2017)
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. Available at https://asterius.ballarat.edu.au/akruger/research/publications.html
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, 60 pp. Minsk (1980). In Russian. Available at https://asterius.ballarat.edu.au/akruger/research/publications.html
Kruger, A. Y., Thao, N. H.: About uniform regularity of collections of sets. Serdica Math. J. 39, 287–312 (2013)
Kruger, A. Y., Thao, N. H.: Quantitative characterizations of regularity properties of collections of sets. J. Optim. Theory Appl. 164, 41–67 (2015)
Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation. I: Basic Theory. Grundlehren der Mathematischen Wissenschaften, vol. 330. Springer, Berlin Heidelberg (2006)
Mordukhovich, B. S., Shao, Y.: Extremal characterizations of Asplund spaces. Proc. Am. Math. Soc. 124, 197–205 (1996)
Phelps, R. R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics, 2nd edn., vol. 1364. Springer-Verlag, Berlin Heidelberg (1993)
Rockafellar, R. T., Wets, R. J. -B.: Variational Analysis. Springer, Berlin Heidelberg (1998)
Pierra, G.: Decomposition through formalization in a product space. Math. Program. 28, 96–115 (1984)
Zheng, X. Y., Ng, K. F.: The Lagrange multiplier rule for multifunctions in Banach spaces. SIAM J. Optim. 17, 1154–1175 (2006)
Zheng, X. Y., Ng, K. F.: A unified separation theorem for closed sets in a Banach space and optimality conditions for vector optimization. SIAM J. Optim. 21, 886–911 (2011)
Zheng, X. Y., Yang, Z., Zou, J.: Exact separation theorem for closed sets in Asplund spaces. Optimization 66, 1065–1077 (2017)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Michel Théra on the occasion of his 70th birthday.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10013-018-0278-y
Keywords
- Extremality
- Stationarity
- Transversality
- Regularity
- Separability
- Extremal principle
- Ekeland variational principle