Set-Valued Analysis

, Volume 2, Issue 1–2, pp 381–393 | Cite as

Convergence of generalized gradients

  • Tullio Zolezzi
Article

Abstract

For the graphs of Clarke's generalized gradients we prove that
$$lim sup_{n \to + \infty } gph \partial f_n \subset gph \partial f in (E, strong) \times (E^* , weak).$$
provided that the sequencefn of locally Lipschitz functions on a Banach spaceE with separable dual is strongly epi-convergent tof, equi-lower semidifferentiable and locally equibounded. This result extends [21] to the infinite-dimensional setting, and finds applications to the continuous behavior of the multiplier rule and of the generalized gradients of integral functionals under data perturbations.

Mathematics Subject Classification (1991)

49J52 

Key words

Generalized gradient epi-convergence stability of multipliers integral functionals 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ambrosetti, A. and Sbordone, C.: Γ-convergenza e G convergenza per problemi non lineari di tipo ellittico,Boll. Un. Mat. Ital. 13A (1976), 352–362.Google Scholar
  2. 2.
    Attouch, H.:Variational Convergence for Functions and Operators, Pitman, New York, 1984.Google Scholar
  3. 3.
    Attouch, H. and Beer, G.: On the convergence of subdifferentials of convex functions,Sém. Anal. Convexe 21, exposé 8 (1991).Google Scholar
  4. 4.
    Attouch, H., Ndoutoume, J.L. and Thera, M.: Epigraphical convergence of functions and convergence of their derivatives in Banach spaces, preprint, 1991.Google Scholar
  5. 5.
    Aubin, J.P. and Frankowska, H.:Set-Valued Analysis, Birkhäuser, Boston, Basel, Berlin, 1990.Google Scholar
  6. 6.
    Boccardo, L. and Marcellini, P.: Sulla convergenza delle soluzioni di disequazioni variazionali,Ann. Mat. Pura Appl. 110 (1976), 137–159.Google Scholar
  7. 7.
    Borwein, J.M. and Preiss, D.: A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions,Trans. Amer. Math. Soc. 303 (1987), 517–527.Google Scholar
  8. 8.
    Clarke, F.H.: A new approach to Lagrange multipliers,Math. Oper. Res. 1 (1976), 165–174.Google Scholar
  9. 9.
    Clarke, F.H.:Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983.Google Scholar
  10. 10.
    Degiovanni, M., Marino, A. and Tosques, M.: Evolution equations with lack of convexity,Nonlinear Anal. 9 (1985), 1401–1443.Google Scholar
  11. 11.
    Deville, R.: Stability of subdifferentials of non convex functions in Banach spaces, to appear.Google Scholar
  12. 12.
    Dunford, N. and Schwartz, J.:Linear Operators, Part I, Interscience, New York, 1958.Google Scholar
  13. 13.
    Ioffe, A.D.: Absolutely continuous subgradients of nonconvex integral functionals,Nonlinear Anal. 11 (1987), 245–257.Google Scholar
  14. 14.
    Ioffe, A.D.: Proximal analysis and approximate subdifferentials,J. London Math. Soc. 41 (1990), 175–192.Google Scholar
  15. 15.
    Lebourg, G.: Sous-dérivabilité de fonctions semi-continues et convergence de dérivées: quelques résultats en densité,C.R. Acad. Sci. Paris Sér. A 288 (1979), 753–755.Google Scholar
  16. 16.
    Matzeu, M.: Su un tipo di continuita' dell'operatore subdifferenziale,Boll. Un. Mat. Ital. 14B (1977), 480–490.Google Scholar
  17. 17.
    Penot, J.-P.: On the convergence of subdifferentials of convex functions, Nonlinear Anal.,21 (1993), 87–101.Google Scholar
  18. 18.
    Poliquin, R.A.: An extension of Attouch's theorem and its application to second-order epidifferentiation of convexly composite functions,Trans. Amer. Math. Soc. 332 (1991), 861–874.Google Scholar
  19. 19.
    Zolezzi, T.: Well posed optimization problems for integral functionals,J. Optim. Theory Appl. 31 (1980), 417–430.Google Scholar
  20. 20.
    Zolezzi, T.: On stability analysis in mathematical programming,Math. Programming Study 21 (1984), 227–242.Google Scholar
  21. 21.
    Zolezzi, T.: Continuity of generalized gradients and multipliers under perturbations,Math. Oper. Res. 10 (1985), 664–673.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Tullio Zolezzi
    • 1
  1. 1.Dipartimento di MatematicaGenovaItaly

Personalised recommendations