Abstract

Some comparisons (equality, strong density or complete characterizations) are made between the Clarke and limiting subdifferentials for the class of integral functionals defined on reflexive Lebesgue spaces LpL_p, without any locally Lipschitzian assumptions on these functionals. Let IfI_f be the integral functional associated with a normal integrand f:Ω×E→R∪{±∞}f:\Omega\times E\to {\mathbb{R}}\cup\{\pm\infty\}, where EE is a finite dimensional space and Ω\Omega is endowed with an atomless σ\sigma-finite measure. At a given point xx such that almost everywhere the Clarke subdifferential of fωf_{\omega} at x(ω)x(\omega) coincide with the closed convex hull of the limiting subdifferential and such that the limiting subdifferential multifunction at xx, ω⇉∂Lfω(x(ω))\omega\rightrightarrows \partial^{L}f_{\omega}(x(\omega)), admits a q=p(p−1)−1q=p(p-1)^{-1} integrable selection, we show that the limiting subdifferential of the integral functional is strongly dense in the Clarke subdifferential. This property allows us to give an exact computation of the Clarke subdifferential by a legitimate disintegration formula. As a consequence, when for almost every ω\omega in Ω\Omega the function fω(.)f_{\omega}(.) is locally Lipschitz at x(ω)x(\omega) then the limiting subdifferential of IfI_f at xx is strongly dense in the Clarke subdifferential of IfI_f at xx. When the Lipschitz rate is qq-integrable, there is the coincidence of the Clarke subdifferential with the limiting subdifferential of IfI_f at xx and the disintegration formula is valid. Moreover such a disintegration formula is valid for the calculus of the Clarke normal (respectively tangent) cone at a given point to the set of pp-integrable selections of a measurable multifunction.

Author information

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

Emmanuel Giner

Laboratoire MIP, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex 04, France

giner@math.univ-toulouse.fr

Keywords

  • Nonsmooth Analysis
  • Fréchet
  • Clarke
  • limiting subdifferentials
  • non locally Lipschitzian integral functionals

Mathematics Subject Classification

26A16, 26A2, 26E15, 2820; 49J52, 54C3, 58C20

References

  1. J. P. Aubin, H. Frankowska: Set-Valued Analysis, Systems & Control: Foundations & Applications 2, Birkhäuser, Boston (1990).Google Scholar
  2. A. Bourass: Comparaison de Fonctionnelles Intégrales. Conditions de Croissance Caractéristiques et Application à la Semi Continuité dans les Espaces Intégraux de Type Orlicz, Thèse d’État, Université de Perpignan, Perpignan (1983).Google Scholar
  3. C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer, Berlin (1977).https://doi.org/10.1007/BFb0087685Google Scholar
  4. N. H Chieu: The Fréchet and limiting subdifferentials of integral functionals on the space L1 (Ω, E), J. Math. Anal. Appl. 360(2) (2009) 704–710.https://doi.org/10.1016/j.jmaa.2009.07.017Google Scholar
  5. F. H. Clarke: Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Avanced Texts, Wiley, New York (1983).Google Scholar
  6. F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, P. R. Wolenski: Nonsmooth Analysis and Control Theory, Springer, New York (1998).Google Scholar
  7. E. Giner: Etudes sur les Fonctionnelles Intégrales, Thèse de Doctorat, Université de Pau, Pau (1985).Google Scholar
  8. E. Giner: Calmness properties and contingent subgradients of integral functionals on Lebesgue spaces Lp, 1 ≤ p < ∞, Set-Valued Var. Anal. 17 (2010) 223–243.https://doi.org/10.1007/s11228-009-0116-1Google Scholar
  9. E. Giner: On the Clarke subdifferential of an integral functional on Lp, 1 ≤ p < ∞, Can. Math. Bull. 41(1) (1998) 41–48.https://doi.org/10.4153/CMB-1998-008-5Google Scholar
  10. E. Giner: Lipschitzian properties of integral functionals on Lebesgue spaces Lp, 1 ≤ p < ∞, Set-Valued Anal. 15 (2007) 125–138.https://doi.org/10.1007/s11228-006-0031-7Google Scholar
  11. E. Giner: Subdifferential regularity and characterizations of the Clarke subgradients of integral functionals, J. Nonlinear Convex Anal. 9(1) (2008) 25–36.Google Scholar
  12. E. Giner: A useful underestimate for the convergence of integral functionals, submitted.Google Scholar
  13. E. Giner: A useful underestimate for the convergence of integral functionals, arXiv:1506.06005 [math.OC] (2015).Google Scholar
  14. F. Hiai, H. Umegaki: Integrals, conditional expectations and martingales of multivalued functions, J. Multivariate Anal. 7 (1977) 149–182.https://doi.org/10.1016/0047-259X(77)90037-9Google Scholar
  15. A. D. Ioffe, V. L. Levin: Subdifferentials of convex functionals, Tr. Mosk. Mat. Obshch. 26 (1972) 3–73.Google Scholar
  16. A. D. Ioffe, V. M. Tihomirov: Theory of Extremal Problems, North-Holland, Amsterdam (1979).Google Scholar
  17. A. D. Ioffe: Absolutely continuous subgradients of non convex integral functionals, Nonlinear Anal. 11(2) (1987) 245–257.https://doi.org/10.1016/0362-546X(87)90102-7Google Scholar
  18. C. M. Marle: Mesures et Probabilités, Hermann, Paris (1974).Google Scholar
  19. B. Mordukhovich: Variational Analysis and Generalized Differentiation I, Springer, Berlin (2006).https://doi.org/10.1007/3-540-31246-3Google Scholar
  20. B. Mordukhovich: Variational Analysis and Generalized Differentiation II, Springer, Berlin (2006).https://doi.org/10.1007/3-540-31246-3Google Scholar
  21. J. J. Moreau: Fonctionnelles convexes, Séminaire Jean Leray 2 (1966–67) 1–108.Google Scholar
  22. J. P. Penot: Calculus whithout Derivatives, Graduate Texts in Mathematics 266, Springer, New York (2013).https://doi.org/10.1007/978-1-4614-4538-8Google Scholar
  23. J. P. Penot: Image space approach and subdifferentials of integral functionals, Optimization 60(1–3) (2011) 69–87.https://doi.org/10.1080/02331934.2010.505651Google Scholar
  24. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  25. R. T. Rockafellar: Integral functionals, normal integrands and measurable selections, in: Nonlinear Operators and the Calculus of Variations (Bruxelles, 1975), J. P. Gossez et al. (ed.), Lect. Notes in Math. 543, Springer, Berlin (1976) 157–206.https://doi.org/10.1007/BFb0079944Google Scholar
  26. R. T. Rockafellar: Directionally Lipschitzian functions and subdifferential calculus, Proc. Lond. Math. Soc., III. Ser. 39 (1979) 331–355.https://doi.org/10.1112/plms/s3-39.2.331Google Scholar
  27. R. T. Rockafellar: Generalized directional derivatives and subgradients of nonconvex functions, Can. J. Math. 32(2) (1980) 257–280.https://doi.org/10.4153/CJM-1980-020-7Google Scholar
  28. R. T. Rockafellar, J. B. Wets: Variational Analysis, Springer, New York (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  29. R. L. Schiling: Measures, Integrals and Martingales, Cambridge University Press, Cambridge (2005).https://doi.org/10.1017/CBO9780511810886Google Scholar
  30. L. Thibault: Quelques propriétés des sous différentiels de fonctions réelles localement lipschitziennes définies sur un espace de Banach séparable, C. R. Acad. Sci., Paris, Sér. A 282 (1976) 507–510.Google Scholar

Suggested citation

E. Giner. “Clarke and Limiting Subdifferentials of Integral Functionals.” Journal of Convex Analysis 24 (2017), No. 2, 661–678. https://doi.org/10.68381/jca24041

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