Abstract

We study regularity properties of the subdifferential of proper lower semicontinuous convex functions in Hilbert spaces. More precisely, we investigate the metric regularity and subregularity, the strong regularity and subregularity of such a subdifferential. We characterize each of these properties in terms of a growth condition involving the function.

Author information

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

Francisco Javier Aragón Artacho

Dept. of Statistics and Operations Research, University of Alicante, 03071 Alicante, Spain

francisco.aragon@ua.es

Michel H. Geoffroy

Laboratoire AOC, Dép. de Mathématiques, Université Antilles-Guyane, 97159 Pointe-à-Pitre, Guadeloupe – France

michel.geoffroy@univ-ag.fr

Keywords

  • Set-valued mapping
  • subdifferential
  • metric regularity
  • quadratic growth condition

Mathematics Subject Classification

49J52, 49J53

References

  1. J. P. Aubin, H. Frankowska: Set-Valued Analysis, Birkhäuser, Boston (1990).Google Scholar
  2. A. Auslender, R. Cominetti: First and second order sensitivity analysis of nonlinear programs under directional constraint qualification condition, Optimization 21 (1990) 351–363.https://doi.org/10.1080/02331939008843555Google Scholar
  3. D. Azé: A unified theory for metric regularity of multifunctions, J. Convex Analysis 13(2) (2006) 225–252.https://doi.org/10.68381/jca13023Google Scholar
  4. J. F. Bonnans, A. D. Ioffe: Second-order sufficiency and quadratic growth for nonisolated minima, Math. Oper. Res. 20(4) (1995) 801–817.https://doi.org/10.1287/moor.20.4.801Google Scholar
  5. J. F. Bonnans, A. D. Ioffe: Quadratic growth and stability in convex programming problems with multiple solutions, J. Convex Analysis 2(1-2) (1995) 41–57.https://doi.org/10.68381/jca02003Google Scholar
  6. J. F. Bonnans, A. Shapiro: Perturbation Analysis of Optimization Problems, Springer, New York (2000).https://doi.org/10.1007/978-1-4612-1394-9Google Scholar
  7. J. M. Borwein, Q. J. Zhu: Techniques of Variational Analysis, Springer, New York (2005).Google Scholar
  8. J. Borwein, Q. J. Zhu: A survey of subdifferential calculus with applications, Nonlinear Anal., Theory Methods Appl. 38(6A) (1999) 687–773.https://doi.org/10.1016/S0362-546X(98)00142-4Google Scholar
  9. J.-N. Corvellec, V. V. Motreanu: Nonlinear error bounds for lower semicontinuous functions on metric spaces, Math. Program., Ser. A, to appear; DOI 10.1007/s10107-007-0102-z.Google Scholar
  10. A. L. Dontchev: Characterizations of Lipschitz stability in optimization, in: Recent Developments in Well-Posed Variational Problems, R. Lucchetti et al. (ed.), Math. Appl. 331, Kluwer, Dordrecht (1995) 95–115.https://doi.org/10.1007/978-94-015-8472-2_4Google Scholar
  11. A. L. Dontchev, W. W. Hager: Implicit functions, Lipschitz maps, and stability in optimization, Math. Oper. Res. 19 (1994) 753–768.https://doi.org/10.1287/moor.19.3.753Google Scholar
  12. A. L. Dontchev, R. T. Rockafellar: Ample parameterization of variational inclusions, SIAM J. Optim. 12 (2001) 170–187.https://doi.org/10.1137/S1052623400371016Google Scholar
  13. A. L. Dontchev, R. T. Rockafellar: Characterization of strong regularity of variational inequalities over polyhedral convex sets, SIAM J. Optim. 6 (1996) 1087–1105.https://doi.org/10.1137/S1052623495284029Google Scholar
  14. A. L. Dontchev, R. T. Rockafellar: Regularity and conditioning of solution mappings in variational analysis, Set-Valued Anal. 12 (2004) 79–109.https://doi.org/10.1023/B:SVAN.0000023394.19482.30Google Scholar
  15. M. Durea: Applications of the Fréchet subdifferential, Serdica Math. J. 29(4) (2003) 301–314.Google Scholar
  16. A. D. Ioffe: Metric regularity and subdifferential calculus, Usp. Mat. Nauk 55(3) (2000) 103–162 (in Russian); Russ. Math. Surv. 55 (2000) 501–558 (in English).https://doi.org/10.4213/rm292https://doi.org/10.1070/RM2000v055n03ABEH000292Google Scholar
  17. A. Jourani, L. Thibault: Approximate subdifferential and metric regularity: the finite-dimensional case, Math. Program., Ser. A 47(2) (1990) 203–218.https://doi.org/10.1007/BF01580860Google Scholar
  18. A. Jourani, L. Thibault: Metric regularity and subdifferential calculus in Banach spaces, Set-Valued Anal. 3(1) (1995) 87–100.https://doi.org/10.1007/BF01033643Google Scholar
  19. P. Kenderov: Semi-continuity of set-valued monotone mappings, Fundam. Math. 88 (1975) 61–69.https://doi.org/10.4064/fm-88-1-61-69Google Scholar
  20. S. Lahrech, A. Benbrik: On the Mordukhovich subdifferential in binormed spaces and some applications, Int. J. Pure Appl. Math. 20(1) (2005) 31–39.Google Scholar
  21. A. B. Levy, R. A. Poliquin: Characterizing the single-valuedness of multifunctions, Set-Valued Anal. 5 (1997) 351–364.https://doi.org/10.1023/A:1008648821168Google Scholar
  22. B. S. Mordukhovich: Sensitivity analysis in nonsmooth optimization, in: Theoretical Aspects of Industrial Design, A. D. Field et al. (ed.), SIAM, Philadelphia (1992) 32–46.Google Scholar
  23. B. S. Mordukhovich: Variational Analysis and Generalized Differentiation. I: Basic Theory, Springer, Berlin (2006).https://doi.org/10.1007/3-540-31247-1Google Scholar
  24. S. M. Robinson: Some continuity properties of polyhedral multifunctions, Math. Program. Study 14 (1981) 206–214.https://doi.org/10.1007/BFb0120929Google Scholar
  25. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  26. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1997).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  27. A. Shapiro: Sensitivity analysis of nonlinear programs and differentiability properties of metric projections, SIAM J. Control Optimization 26 (1988) 628–645.https://doi.org/10.1137/0326037Google Scholar
  28. R. Zhang, J. Treiman: Upper-Lipschitz multifunctions and inverse subdifferentials, Nonlinear Anal., Theory Methods Appl. 24(2) (1995) 273–286.https://doi.org/10.1016/0362-546X(94)E0025-CGoogle Scholar

Suggested citation

F. J. Aragón Artacho, M. H. Geoffroy. “Characterization of Metric Regularity of Subdifferentials.” Journal of Convex Analysis 15 (2008), No. 2, 365–380. https://doi.org/10.68381/jca15026

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Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.