Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the graph convergence
of subdifferentials of convex functions


Authors: C. Combari and L. Thibault
Journal: Proc. Amer. Math. Soc. 126 (1998), 2231-2240
MSC (1991): Primary 49J52, 58C20, 90C25
DOI: https://doi.org/10.1090/S0002-9939-98-04724-8
MathSciNet review: 1485464
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper provides another proof of the Attouch Theorem relating the epigraphical limit of sequences of convex functions to the set limit of the graphs of the subdifferentials.


References [Enhancements On Off] (What's this?)

  • 1. H. Attouch, Convergence de fonctions convexes, des sous différentiels et semigroupes associés, C. R. Acad. Sci. Paris Sér. I Math. 284 (1977), 539-542. MR 57:13587
  • 2. -, Variational Convergence for Functions and Operators, Pitman, Boston-London-Melbourne (1984). MR 86f:49002
  • 3. H. Attouch and G. Beer, On the convergence of subdifferentials of convex functions, Archiv. Math. 60 (1993), 389-400. MR 94b:49018
  • 4. H. Attouch and A. Damlamian, Strong solutions for parabolic variational inequalities, Nonlinear Anal. Th. Meth. Appl. 2 (1978), 329-353. MR 80a:47093
  • 5. G. Beer, The slice topology: A viable alternative to Mosco convergence in nonreflexive spaces, Nonlinear Anal. Th. Meth. Appl. 19 (1992), 271-290. MR 93h:49025
  • 6. -, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers, Dordrecht-Boston-London (1993). MR 95k:49001
  • 7. G. Beer and J. Borwein, Mosco convergence and reflexivity, Proc. Amer. Math. Soc. 109 (1990), 427-436. MR 91c:46016
  • 8. A. Brondsted and R. T. Rockafellar, On the subdifferentiability of convex functions, Proc. Amer. Math. Soc. 16 (1965), 605-611. MR 31:2361
  • 9. C. Combari and L. Thibault, On the convergence of approximate subdifferentials of convex functions, to appear.
  • 10. B. Lemaire, Coupling optimization methods and variational convergence, in Trends in Mathematical Optimization, series of Num. Math. Vol. 84, K. H. Hoffmann, J.-B. Hiriart-Urruty, C. Lemarechal and J. Zowe, eds., 163-179, Basel-Boston (1988). MR 91b:90154
  • 11. A. B. Levy, R. A. Poliquin and L. Thibault, Partial extension of Attouch's theorem with applications to proto-derivatives of subgradient mappings, Trans. Amer. Math. Soc. 347 (1995), 1269-1294. MR 95k:49035
  • 12. M. Matzeu, Su un tipo di continuità dell' operatore subdifferenziale, Boll. U.M.I. 11 (1975), 107-124. MR 57:1220
  • 13. J.-J. Moreau, Fonctionelles convexes, mimeographed lecture notes, Collège de France, 1967.
  • 14. U. Mosco, On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971), 518-535. MR 44:817
  • 15. R. A. Poliquin, An extension of Attouch's theorem and its applications to epi-differentiation of convexly composite functions, Trans. Amer. Math. Soc. 332 (1992), 861-874. MR 93a:49013
  • 16. R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209-216. MR 41:7432
  • 17. -, Characterization of the subdifferentials of convex functions, Pacific J. Math. 17 (1966), 497-510. MR 33:1769
  • 18. -, Generalized second derivatives of convex functions and saddle functions, Trans. Amer. Math. Soc. 322 (1990), 851-877. MR 91b:90190
  • 19. L. Thibault and D. Zagrodny, Integration of subdifferentials of lower semicontinuous functions, J. Math. Anal. Appl. 189 (1995), 33-58. MR 95i:49032

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 49J52, 58C20, 90C25

Retrieve articles in all journals with MSC (1991): 49J52, 58C20, 90C25


Additional Information

C. Combari
Affiliation: Université Montpellier II, Laboratoire d’Analyse Convexe, Place Eugene Bataillon, 34095 Montpellier cedex 5, France

L. Thibault
Affiliation: Université Montpellier II, Laboratoire d’Analyse Convexe, Place Eugene Bataillon, 34095 Montpellier cedex 5, France
Email: thibault@math.univ-montp2.fr

DOI: https://doi.org/10.1090/S0002-9939-98-04724-8
Keywords: Epilimit inferior, epilimit superior, Mosco-convergence, slice convergence, Painlev\'e-Kuratowski convergence
Received by editor(s): January 5, 1996
Communicated by: Dale Alspach
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society