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)



Convex composite functions in Banach spaces
and the primal lower-nice property

Authors: C. Combari, A. Elhilali Alaoui, A. Levy, R. Poliquin and L. Thibault
Journal: Proc. Amer. Math. Soc. 126 (1998), 3701-3708
MSC (1991): Primary 58C20; Secondary 49J52
MathSciNet review: 1451793
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Primal lower-nice functions defined on Hilbert spaces provide examples of functions that are ``integrable'' (i.e. of functions that are determined up to an additive constant by their subgradients). The class of primal lower-nice functions contains all convex and lower-$C^2$ functions. In finite dimensions the class of primal lower-nice functions also contains the composition of a convex function with a $C^2$ mapping under a constraint qualification. In Banach spaces certain convex composite functions were known to be primal lower-nice (e.g. a convex function had to be continuous relative to its domain). In this paper we weaken the assumptions and provide new examples of convex composite functions defined on a Banach space with the primal lower-nice property. One consequence of our results is the identification of new examples of integrable functions on Hilbert spaces.

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

  • 1. J.-P. Aubin and I. Ekeland: Applied nonlinear analysis, Wiley-Interscience (1984). MR 87a:58002
  • 2. J.M. Borwein, Convex relations in analysis and optimization, in Generalized concavity in Optimization and Economics (S. Schaible and W. T. Ziemba, eds.). Academic Press, New York (1981), 335-376. MR 83g:90075
  • 3. J.M. Borwein, Stability and regular points of inequality systems, Jour. Opt. Theo. and Appl 48 (1986), 9-52. MR 87m:58018
  • 4. F.H. Clarke: Optimization and nonsmooth analysis, Wiley, New-York (1983). MR 85m:49002
  • 5. S. Kurcyusz and J. Zowe, Regularity and stability for the mathematical programming problem in Banach spaces, Appl. Math. Optim. 5 (1979), 49-62. MR 82a:90153
  • 6. A.B. Levy, Second-order variational analysis with applications to sensitivity in optimization, PhD. Thesis, University of Washington, 1994.
  • 7. A.B. Levy, R.A Poliquin and L. Thibault, Partial extensions of Attouch's theorem with applications to proto-derivatives of subgradient mappings, Trans. Amer. Math. Soc. 347 (1995), 1269-1294. MR 95k:49035
  • 8. R.A. Poliquin, Integration of subdifferentials of nonconvex functions, Nonlinear Anal. Th. Meth. Appl. 17(1991), 385-398. MR 92j:49008
  • 9. R.A Poliquin, An extension of Attouch's theorem and its application to second-order epi-differentiation of convexly composite functions. Trans. Amer. Math. Soc. 332 (1992), 861-874. MR 93a:49013
  • 10. L. Thibault and D. Zagrodny, Integration of subdifferentials of lower semicontinuous functions on Banach spaces, J. Math. Anal. Appl. 189 (1995) 33-58. MR 95i:49032
  • 11. C. Ursescu, Multifunctions with convex closed graph, Czech. Math. J. 7 (25) (1975), 438-441. MR 52:8869

Similar Articles

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

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

Additional Information

C. Combari
Affiliation: Université de Montpellier II, Laboratoire Analyse Convexe, Place Eugene Bataillon, 34095 Montpellier Cedex 5, France

A. Elhilali Alaoui
Affiliation: Université de Montpellier II, Laboratoire Analyse Convexe, Place Eugene Bataillon, 34095 Montpellier Cedex 5, France
Address at time of publication: Falculté des Sciences et Techniques de Marrakech, Université Cadi Ayad, B.P. 618, Marrakech, Maroc

A. Levy
Affiliation: Department of Mathematics, Bowdoin College, Brunswick, Maine 04011

R. Poliquin
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

L. Thibault
Affiliation: Université de Montpellier II, Laboratoire Analyse Convexe, Place Eugene Bataillon, 34095 Montpellier Cedex 5, France

Keywords: Primal lower-nice functions, subdifferential, convex composite functions, integrable functions
Received by editor(s): February 16, 1996
Received by editor(s) in revised form: November 27, 1996
Additional Notes: The research of R. Poliquin was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP41983.
Communicated by: Dale Alspach
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society