Abstract

We study various regularity properties, including subregularity and semiregularity, of set-valued mappings acting in extended quasi-metric spaces. It turns out that this abstract framework allows to unify criteria for the usual (sub/semi) regularity as well as their directional and Hölder counterparts. Ioffe-type critera are obtained by applying a general version of the Ekeland's variational principle. We provide a self-contained material gathering and extending the existing theory on the topic. We (briefly) illustrate the importance of these criteria for applications. For example, we consider local convergence of a directional version of a Newton-type method for solving a generalized equation which may be applied when the usual (non-directional) regularity does not hold.

Author information

Contact details are reproduced from the original publication and may be historical.

Radek Cibulka

NTIS and Dept. of Mathematics, Faculty of Applied Sciences, University of West Bohemia, 306 14 Pilsen, Czech Republic

cibi@kma.zcu.cz

Tomáš Roubal

NTIS and Dept. of Mathematics, Faculty of Applied Sciences, University of West Bohemia, 306 14 Pilsen, Czech Republic

roubalt@ntis.zcu.cz

Keywords

  • Open mapping theorem
  • Hölder and directional metric semiregularity
  • Hölder and directional metric subregularity
  • Hölder and directional metric regularity

Mathematics Subject Classification

47J22, 49J53, 49K40, 90C29

References

  1. J.-P. Aubin, H. Frankowska: Set-Valued Analysis, Birkhäuser, Boston (1990).Google Scholar
  2. M. Bianchi, G. Kassay, and R. Pini: Existence of equilibria via Ekeland’s principle, J. Math. Anal. Appl. 305(2) (2005) 502–512.https://doi.org/10.1016/j.jmaa.2004.11.042Google Scholar
  3. R. Cibulka, M. Durea, M. Panţiruc, R. Strugariu: On the stability of the directional regularity, Set-Valued Var. Analysis (2019), doi 10.1007/s11228-019-00507-2.https://doi.org/10.1007/s11228-019-00507-2Google Scholar
  4. R. Cibulka, M. Fabian, A. Kruger: On semiregularity of mappings, J. Math. Anal. Appl. 473(2) (2019) 811–836.https://doi.org/10.1016/j.jmaa.2018.12.071Google Scholar
  5. S. Cobzaş: Completeness in quasi-metric spaces and Ekeland Variational Principle, Topology Appl. 158(8) (2011) 1073–1084.https://doi.org/10.1016/j.topol.2011.03.003Google Scholar
  6. S. Cobzaş: Functional analysis in asymmetric normed spaces, in: Asymmetric Topology and its Applications, Department of Mathematics, Seconda Università di Napoli, Caserta, Quad. Mat. 26 (2011) 1–102.Google Scholar
  7. A. L. Dontchev, R. T. Rockafellar: Implicit Functions and Solution Mappings. A View from Variational Analysis, 2n ed., Springer, New York (2014).https://doi.org/10.1007/978-1-4939-1037-3Google Scholar
  8. M. Durea, M. Panţiruc, R. Strugariu: A new type of directional regularity for mappings and applications to optimization, SIAM J. Optim. 27(2) (2017) 1204–1229.https://doi.org/10.1137/16M1067342Google Scholar
  9. M. Durea, R. Strugariu: Chain rules for linear openness in general Banach spaces, SIAM J. Optim. 22(3) (2012) 899–913.https://doi.org/10.1137/11082470XGoogle Scholar
  10. M. Durea, R. Strugariu: Openness stability and implicit multifunction theorems: applications to variational systems, Nonlinear Analysis 75(3) (2012) 1246–1259.https://doi.org/10.1016/j.na.2011.02.019Google Scholar
  11. H. Frankowska: Some inverse mapping theorems, Ann. Inst. Henri Poincaré Anal. Non Linéaire 7 (1990) 183–234.https://doi.org/10.1016/s0294-1449(16)30300-6Google Scholar
  12. H. Frankowska, M. Quincampoix: Hölder metric regularity of set-valued maps, Math. Program., Ser. A 132(1-2) (2012) 333–354.https://doi.org/10.1007/s10107-010-0401-7Google Scholar
  13. H. Gfrerer: On directional metric regularity, subregularity and optimality conditions for nonsmooth mathematical programs, Set-Valued Var. Analysis 21(2) (2013) 151–176.https://doi.org/10.1007/s11228-012-0220-5Google Scholar
  14. V. N. Huynh, A. Y. Kruger, M. Théra: Slopes of multifunctions and extensions of metric regularity, Vietnam J. Math. 40(2-3) (2012) 355–369.Google Scholar
  15. V. N. Huynh, M. Théra: Directional metric regularity of multifunctions, Math. Oper. Res. 40(4) (2015) 969–991.https://doi.org/10.1287/moor.2014.0705Google Scholar
  16. V. N. Huynh, N. H. Tron, M. Théra: Metric regularity of the sum of multifunctions and applications, J. Optim. Theory Appl. 160(2) (2014) 355–390.https://doi.org/10.1007/s10957-013-0385-6Google Scholar
  17. A. D. Ioffe: Variational Analysis of Regular Mappings. Theory and Applications, Springer, Cham (2017).https://doi.org/10.1007/978-3-319-64277-2Google Scholar
  18. D. Klatte, B. Kummer: Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications, Kluwer Academic Publishers, Dordrecht (2002).Google Scholar

Suggested citation

R. Cibulka, T. Roubal. “Ioffe-Type Criteria in Extended Quasi-Metric Spaces.” Journal of Convex Analysis 27 (2020), No. 1, 205–226. https://doi.org/10.68381/jca27012

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Published by Heldermann Verlag, 2020. Rights now held by Banach Press LLC.