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

Combari, C.; Alaoui, A.; Levy, A.; Poliquin, R.; Thibault, L.

doi:10.1090/S0002-9939-98-04324-X

Convex composite functions in Banach spaces and the primal lower-nice property
HTML articles powered by AMS MathViewer

by C. Combari, A. Elhilali Alaoui, A. Levy, R. Poliquin and L. Thibault PDF

Proc. Amer. Math. Soc. 126 (1998), 3701-3708 Request permission

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

Jean-Pierre Aubin and Ivar Ekeland, Applied nonlinear analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 749753
Siegfried Schaible and William T. Ziemba (eds.), Generalized concavity in optimization and economics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 652702
J. M. Borwein, Stability and regular points of inequality systems, J. Optim. Theory Appl. 48 (1986), no. 1, 9–52. MR 825383, DOI 10.1007/BF00938588
Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces, Appl. Math. Optim. 5 (1979), no. 1, 49–62. MR 526427, DOI 10.1007/BF01442543
A.B. Levy, Second-order variational analysis with applications to sensitivity in optimization, PhD. Thesis, University of Washington, 1994.
A. B. Levy, R. Poliquin, and L. Thibault, Partial extensions of Attouch’s theorem with applications to proto-derivatives of subgradient mappings, Trans. Amer. Math. Soc. 347 (1995), no. 4, 1269–1294. MR 1290725, DOI 10.1090/S0002-9947-1995-1290725-3
René A. Poliquin, Integration of subdifferentials of nonconvex functions, Nonlinear Anal. 17 (1991), no. 4, 385–398. MR 1123210, DOI 10.1016/0362-546X(91)90078-F
René 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), no. 2, 861–874. MR 1145732, DOI 10.1090/S0002-9947-1992-1145732-5
Lionel Thibault and Dariusz Zagrodny, Integration of subdifferentials of lower semicontinuous functions on Banach spaces, J. Math. Anal. Appl. 189 (1995), no. 1, 33–58. MR 1312029, DOI 10.1006/jmaa.1995.1003
Corneliu Ursescu, Multifunctions with convex closed graph, Czechoslovak Math. J. 25(100) (1975), no. 3, 438–441. MR 388032, DOI 10.21136/CMJ.1975.101337