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Characterizations of Some Transversality-Type Properties

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Abstract

Primal characterizations (necessary and sufficient conditions) and slope characterizations of subtransversality, intrinsic transversality and transversality are obtained. The metric nature of intrinsic transversality is established. The relation of intrinsic transversality and tangential transversality is clarified. The equivalence of our characterization of intrinsic transversality and the primal characterization of intrinsic transversality of Thao et al. in the setting of Hilbert spaces is proved, while in general Banach spaces our characterization is less restrictive.

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References

  1. Azé, D., Corvellec, J. -N.: Characterizations of error bounds for lower semicontinuous func- tions on metric spaces. ESAIM Control Optim. Calc. Var. 10, 409–425 (2004)

    Article  MathSciNet  Google Scholar 

  2. Azé, D., Corvellec, J. -N., Lucchetti, R. E.: Variational pairs and applications to stability in nonsmooth analysis. Nonlinear Anal. Theory Methods Appl. 49A(5), 643–670 (2002)

    Article  MathSciNet  Google Scholar 

  3. Bivas, M., Krastanov, M., Ribarska, N.: On tangential transversality. J. Math. Anal. Appl. 481(1), 123445 (2020)

    Article  MathSciNet  Google Scholar 

  4. Bui, H. T., Kruger, A. Y.: Extremality, stationarity and generalized separation of collections of sets. J. Optim. Theory Appl. 182, 211—264 (2019)

    MathSciNet  MATH  Google Scholar 

  5. Cibulka, R., Fabian, M.: On primal regularity estimates for set-valued mappings. J. Math. Anal. Appl. 438(1), 444–464 (2016)

    Article  MathSciNet  Google Scholar 

  6. Cuong, N. D., Kruger, A. Y.: Dual sufficient characterizations of transversality properties. Positivity 24, 1313–1359 (2020)

    Article  MathSciNet  Google Scholar 

  7. Cuong, N. D., Kruger, A. Y.: Primal necessary characterizations of transversality properties. Positivity 25, 531–558 (2021)

    Article  MathSciNet  Google Scholar 

  8. Cuong, N. D., Kruger, A. Y.: Transversality properties: primal sufficient conditions. Set-Valued Var. Anal. 29, 221–256 (2021)

    Article  MathSciNet  Google Scholar 

  9. De Giorgi, E., Marino, A., Tosques, M.: Problemi di evoluzione in spazi metricie curve di massima pendenza. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 68, 180–187 (1980)

    MATH  Google Scholar 

  10. Drusvyatskiy, D., Ioffe, A. D., Lewis, A. S.: Alternating projections and coupling slope http://www.optimization-online.org/DB_HTML/2014/01/4217.html (2014)

  11. Drusvyatskiy, D., Ioffe, A. D., Lewis, A. S.: Transversality and alternating projections for nonconvex sets. Found. Comput. Math. 15, 1637–1651 (2015)

    Article  MathSciNet  Google Scholar 

  12. Ioffe, A. D.: Metric regularity and subdifferential calculus. Russ. Math. Surv. 55(3), 501–558 (2000)

    Article  MathSciNet  Google Scholar 

  13. Ioffe, A.: Transversality in variational analysis. J. Optim. Theory Appl. 174(2), 343–366 (2017)

    Article  MathSciNet  Google Scholar 

  14. Ioffe, A.: Variational Analysis of Regular Mappings: Theory and Applications. Springer Monographs in Mathematics. Springer, Berlin (2017)

    Book  Google Scholar 

  15. Ivanov, M., Zlateva, N.: Long orbit or empty value principle, fixed point and surjectivity theorems. Compt. Rend. Acad. Bulg. Sci. 69(5), 553–562 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Jech, T.: Set Theory. Academic Press, New York (1978)

    MATH  Google Scholar 

  17. Kruger, A. Y.: About regularity of collections of sets. Set-Valued Anal. 14, 187–206 (2006)

    Article  MathSciNet  Google Scholar 

  18. Kruger, A. Y.: Error bounds and metric subregularity. Optimization 64(1), 49–79 (2015)

    Article  MathSciNet  Google Scholar 

  19. Kruger, A. Y.: About intrinsic transversality of pairs of sets. Set-Valued Var. Anal. 26, 111–142 (2018)

    Article  MathSciNet  Google Scholar 

  20. Kruger, A. Y., Luke, D. R., Thao, N. H.: Set regularities and feasibility problems. Math. Program. B 168, 279–311 (2018)

    Article  MathSciNet  Google Scholar 

  21. Thao, N. H., Bui, H. T., Cuong, N. D., Verhaegen, M.: Some new characterizations of intrinsic transversality in Hilbert spaces. Set-Valued Var. Anal. 28, 5–39 (2020)

    Article  MathSciNet  Google Scholar 

  22. Van Ngai, H., Théra, M.: Error bounds in metric spaces and application to the perturbation stability of metric regularity. SIAM J. Optim. 19(1), 1–20 (2008)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are grateful to Prof. A. Dontchev for his useful comments and suggestions and to the unknown referees for their careful reading of the manuscript and for the useful remarks that helped to improve it.

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Authors and Affiliations

  1. Faculty of Mathematics and Informatics, Sofia University, James Bourchier Boul. 5, 1126, Sofia, Bulgaria

    Stoyan Apostolov, Mira Bivas & Nadezhda Ribarska

  2. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, G. Bonchev str., bl. 8, 1113, Sofia, Bulgaria

    Mira Bivas & Nadezhda Ribarska

Authors
  1. Stoyan Apostolov
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  2. Mira Bivas
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  3. Nadezhda Ribarska
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Correspondence to Stoyan Apostolov.

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This work was supported by the Bulgarian National Scientific Fund under Grant KP-06-H22/4/04.12.2018 and by the Sofia University “St. Kliment Ohridski” fund “Research & Development” under contract 80-10-40 /22.03.2021.

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Apostolov, S., Bivas, M. & Ribarska, N. Characterizations of Some Transversality-Type Properties. Set-Valued Var. Anal 30, 1041–1060 (2022). https://doi.org/10.1007/s11228-022-00633-4

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  • DOI: https://doi.org/10.1007/s11228-022-00633-4

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