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Extremality, Stationarity and Generalized Separation of Collections of Sets

  • Hoa T. Bui
  • Alexander Y. KrugerEmail author
Article
  • 84 Downloads

Abstract

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.

Keywords

Extremal principle Approximate stationarity Transversality Regularity Separation 

Mathematics Subject Classification

49J52 49J53 49K40 90C30 

Notes

Acknowledgements

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.

References

  1. 1.
    Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton, NJ (1970)Google Scholar
  2. 2.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing Co. Inc., River Edge, NJ (2002).  https://doi.org/10.1142/9789812777096 CrossRefzbMATHGoogle Scholar
  3. 3.
    Penot, J.P.: Analysis–from Concepts to Applications. Springer, Cham (2016)zbMATHGoogle Scholar
  4. 4.
    Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367–426 (1996).  https://doi.org/10.1137/S0036144593251710 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kruger, A.Y., Luke, D.R., Thao, N.H.: Set regularities and feasibility problems. Math. Program. Ser. B 168(1–2), 279–311 (2018).  https://doi.org/10.1007/s10107-016-1039-x MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Drusvyatskiy, D., Ioffe, A.D., Lewis, A.S.: Transversality and alternating projections for nonconvex sets. Found. Comput. Math. 15(6), 1637–1651 (2015).  https://doi.org/10.1007/s10208-015-9279-3 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bauschke, H.H., Dao, M.N., Noll, D., Phan, H.M.: Proximal point algorithm, Douglas–Rachford algorithm and alternating projections: a case study. J. Convex Anal. 23(1), 237–261 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Kruger, A.Y., Thao, N.H.: Regularity of collections of sets and convergence of inexact alternating projections. J. Convex Anal. 23(3), 823–847 (2016)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Noll, D., Rondepierre, A.: On local convergence of the method of alternating projections. Found. Comput. Math. 16(2), 425–455 (2016).  https://doi.org/10.1007/s10208-015-9253-0 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dubovitskii, A.Y., Miljutin, A.A.: Extremum problems in the presence of restrictions. USSR Comput. Maths. Math. Phys. 5, 1–80 (1965)CrossRefGoogle Scholar
  11. 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. 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. 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. 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
  15. 15.
    Kruger, A.Y.: Generalized differentials of nonsmooth functions and necessary conditions for an extremum. Sibirsk. Mat. Zh. 26(3), 78–90 (1985). [in Russian; English transl.: Siberian Math. J. 26 (1985), 370–379]MathSciNetzbMATHGoogle Scholar
  16. 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
  17. 17.
    Fabian, M.: Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss. Acta Univ. Carolinae 30, 51–56 (1989)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Mordukhovich, B.S., Shao, Y.: Extremal characterizations of Asplund spaces. Proc. Am. Math. Soc. 124(1), 197–205 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ioffe, A.D.: Fuzzy principles and characterization of trustworthiness. Set-Valued Anal. 6, 265–276 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Borwein, J.M., Jofré, A.: A nonconvex separation property in Banach spaces. Math. Methods Oper. Res. 48(2), 169–179 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory. Fundamental Principles of Mathematical Sciences, vol. 330. Springer, Berlin (2006)Google Scholar
  22. 22.
    Kruger, A.Y., López, M.A.: Stationarity and regularity of infinite collections of sets. J. Optim. Theory Appl. 154(2), 339–369 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kruger, A.Y.: On Fréchet subdifferentials. J. Math. Sci. 116(3), 3325–3358 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kruger, A.Y.: Weak stationarity: eliminating the gap between necessary and sufficient conditions. Optimization 53(2), 147–164 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kruger, A.Y.: About stationarity and regularity in variational analysis. Taiwan. J. Math. 13(6A), 1737–1785 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kruger, A.Y., Luke, D.R., Thao, N.H.: About subtransversality of collections of sets. Set-Valued Var. Anal. 25(4), 701–729 (2017).  https://doi.org/10.1007/s11228-017-0436-5 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zheng, X.Y., Ng, K.F.: Perturbation analysis of error bounds for systems of conic linear inequalities in Banach spaces. SIAM J. Optim. 15(4), 1026–1041 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zheng, X.Y., Ng, K.F.: The Lagrange multiplier rule for multifunctions in Banach spaces. SIAM J. Optim. 17(4), 1154–1175 (2006).  https://doi.org/10.1137/060651860 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Li, G., Ng, K.F., Zheng, X.Y.: Unified approach to some geometric results in variational analysis. J. Funct. Anal. 248(2), 317–343 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Li, G., Tang, C., Yu, G., Wei, Z.: On a separation principle for nonconvex sets. Set-Valued Anal. 16, 851–860 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    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(3), 886–911 (2011).  https://doi.org/10.1137/100811155 MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Bui, H.T., Kruger, A.Y.: About extensions of the extremal principle. Vietnam J. Math. 46(2), 215–242 (2018).  https://doi.org/10.1007/s10013-018-0278-y MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  34. 34.
    Penot, J.P.: Calculus Without Derivatives. Graduate Texts in Mathematics, vol. 266. Springer, New York (2013).  https://doi.org/10.1007/978-1-4614-4538-8 CrossRefzbMATHGoogle Scholar
  35. 35.
    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)zbMATHGoogle Scholar
  36. 36.
    Ioffe, A.D.: Variational Analysis of Regular Mappings. Theory and Applications. Springer Monographs in Mathematics. Springer, New York (2017)CrossRefzbMATHGoogle Scholar
  37. 37.
    Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  39. 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
  40. 40.
    Rockafellar, R.T.: Directionally Lipschitzian functions and subdifferential calculus. Proc. Lond. Math. Soc. (3) 39(2), 331–355 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics, vol. 1364, 2nd edn. Springer, Berlin (1993)zbMATHGoogle Scholar
  42. 42.
    Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. Springer, New York (2005)zbMATHGoogle Scholar
  43. 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
  44. 44.
    Kruger, A.Y.: About regularity of collections of sets. Set-Valued Anal. 14(2), 187–206 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Kruger, A.Y.: Stationarity and regularity of set systems. Pac. J. Optim. 1(1), 101–126 (2005)MathSciNetzbMATHGoogle Scholar
  46. 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
  47. 47.
    Kruger, A.Y.: Strict \((\varepsilon,\delta )\)-subdifferentials and extremality conditions. Optimization 51(3), 539–554 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Kruger, A.Y., Thao, N.H.: About uniform regularity of collections of sets. Serdica Math. J. 39, 287–312 (2013)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Kruger, A.Y., Thao, N.H.: Quantitative characterizations of regularity properties of collections of sets. J. Optim. Theory Appl. 164(1), 41–67 (2015).  https://doi.org/10.1007/s10957-014-0556-0 MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Lewis, A.S., Luke, D.R., Malick, J.: Local linear convergence for alternating and averaged nonconvex projections. Found. Comput. Math. 9(4), 485–513 (2009).  https://doi.org/10.1007/s10208-008-9036-y MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Bauschke, H.H., Luke, D.R., Phan, H.M., Wang, X.: Restricted normal cones and the method of alternating projections: theory. Set-Valued Var. Anal. 21(3), 431–473 (2013).  https://doi.org/10.1007/s11228-013-0239-2 MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Hesse, R., Luke, D.R.: Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems. SIAM J. Optim. 23(4), 2397–2419 (2013).  https://doi.org/10.1137/120902653 MathSciNetCrossRefzbMATHGoogle Scholar
  53. 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)

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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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