Abstract

The paper deals with an extension of the available theory of SCD (subspace containing derivatives) mappings to mappings between spaces of different dimensions. This extension enables us to derive workable sufficient conditions for the isolated calmness of implicitly defined multifunctions around given reference points. This stability property differs substantially from isolated calmness at a point and, possibly in conjunction with the Aubin property, offers a new useful stability concept. The application area includes a broad class of parameterized generalized equations, where the respective conditions ensure a rather strong type of Lipschitzian behavior of their solution maps.

Author information

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

Helmut Gfrerer

Institute of Computational Mathematics, Johannes Kepler University, Linz, Austria

helmut.gfrerer@jku.at

Jiří V. Outrata

(1) Inst. Information Theory and Automation, Czech Acad. of Sciences, Prague, Czech Republic
(2) Centre for Informatics and Appl. Optimization, Federation University, Ballarat, Australia

outrata@utia.cas.cz

Keywords

  • Strong metric subregularity and isolated calmness on a neighborhood
  • generalized derivatives
  • semismoothness*
  • implicit multifunctions

Mathematics Subject Classification

65K10, 65K15, 90C33

References

  1. J.-P. Aubin, H. Frankowska: Set-Valued Analysis, Birkhäuser, Boston (1990).Google Scholar
  2. A. L. Dontchev, R. T. Rockafellar: Characterizations of strong regularity for variational inequalities over polyhedral convex sets, SIAM J. Optim. 6 (1996) 1087–1105.https://doi.org/10.1137/S1052623495284029Google Scholar
  3. A. L. Dontchev, R. T. Rockafellar: Implicit Functions and Solution Mappings, Springer, Berlin (2014).https://doi.org/10.1007/978-1-4939-1037-3Google Scholar
  4. H. Gfrerer, J. V. Outrata: On Lipschitzian properties of implicit multifunctions, SIAM J. Optim. 26 (2016) 2160–2189.https://doi.org/10.1137/15M1052299Google Scholar
  5. H. Gfrerer, J. V. Outrata: On a semismooth* Newton method for solving generalized equations, SIAM J. Optim. 31 (2021) 489–517.https://doi.org/10.1137/19M1257408Google Scholar
  6. H. Gfrerer, J. V. Outrata: On (local) analysis of multifunctions via subspaces contained in graphs of generalized derivatives, J. Math. Anal. Appl. 508 (2022), art. no. 125895.https://doi.org/10.1016/j.jmaa.2021.125895Google Scholar
  7. H. Gfrerer, J. J. Ye: New constraint qualifications for mathematical programs with equilibrium constraints via variational analysis, SIAM J. Optim. 27 (2017) 842–865.https://doi.org/10.1137/16M1088752Google Scholar
  8. M. S. Gowda: Inverse and implicit function theorems for H-differentiable and semismooth functions, Optim. Methods Software 19 (2004) 443–461.https://doi.org/10.1080/10556780410001697668Google Scholar
  9. A. Jourani: Radiality and semismoothness, Control Cybern. 36 (2007) 669–680.Google Scholar
  10. B. S. Mordukhovich: Variational Analysis and Generalized Differentiation. I: Basic Theory, Springer, Berlin (2006).https://doi.org/10.1007/3-540-31247-1Google Scholar
  11. B. S. Mordukhovich: Variational Analysis and Applications, Springer, Cham (2018).https://doi.org/10.1007/978-3-319-92775-6Google Scholar
  12. L. Qi, J. Sun: A nonsmooth version of Newton’s method, Math. Program., 58 (1993) 353–367.https://doi.org/10.1007/BF01581275Google Scholar
  13. S. M. Robinson: Some continuity properties of polyhedral multifunctions, Math. Prog. Study 14 (1981)206–214.https://doi.org/10.1007/BFb0120929Google Scholar
  14. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  15. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar

Suggested citation

H. Gfrerer, J. V. Outrata. “On the Isolated Calmness Property of Implicitly Defined Multifunctions.” Journal of Convex Analysis 30 (2023), No. 3, 1001–1023. https://doi.org/10.68381/jca30046

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Published by Heldermann Verlag, 2023. Rights now held by Banach Press.