Abstract

The paper is devoted to calculation of subdifferential of the integral functional ∫Tf(t,x)dμ(t)\int_Tf(t,x)d\mu(t) with ff being a normal convex integrand. The first theorem gives a precise formula for the case when xx is finite dimensional and some qualification condition, which reduce to the general qualification condition for the sum rule of convex analysis, is satisfied. The two other theorems deal with infinite dimensional case and the absence of any qualification conditions and offer some approximations, one for the case when the space of xx is reflexive and separable and the other, when it is just separable.

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Keywords

  • Convex function
  • normal convex integrand
  • subdifferential
  • normal cone
  • fuzzy calculus

Mathematics Subject Classification

90C25, 52A41, 26E15

References

  1. H. Attouch, J.-B. Baillon, M. Théra: Variational sum of monotone operators, J. Convex Analysis 1 (1994) 1–29.https://doi.org/10.68381/jca01001Google Scholar
  2. J. M. Borwein, A. D. Ioffe: Proximal analysis in smooth spaces, Set-Valued Anal. 4 (1996) 1–24.https://doi.org/10.1007/BF00419371Google Scholar
  3. J. M. Borwein, Q. J. Zhu: Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity, SIAM J. Control Optimization 34 (1996) 1576–1991.https://doi.org/10.1137/S0363012994268801Google Scholar
  4. S. Fitzpatrick, S. Simon: The conjugates, compositions and marginals of convex functions, J. Convex Analysis 8 (2001) 423–446.https://doi.org/10.68381/jca08023Google Scholar
  5. J.-B. Hiriart-Urruty, M. Moussaoui, A. Seeger, M. Volle: Subdifferential calculus without qualification conditions, using approximate subdifferentials: A survey, Nonlinear Anal., Theory Methods Appl. 24 (1995) 1727–1754.https://doi.org/10.1016/0362-546X(94)00221-3Google Scholar
  6. J.-B. Hiriart-Urruty, R. R. Phelps: Suddifferential calculus using ε-subdifferentials, J. Funct. Analysis 118 (1993) 154–166.https://doi.org/10.1006/jfan.1993.1141Google Scholar
  7. A. D. Ioffe: Calculus of Dini subdifferentials of functions and contingent coderivatives of set-valued maps, Nonlinear Anal., Theory Methods Appl. 5 (1984) 517–539.https://doi.org/10.1016/0362-546X(84)90091-9Google Scholar
  8. A. D. Ioffe: A variational problem associated with a model of welfare economics with a measure space of agents, Adv. Math. Economics 8 (2006) 297–314.https://doi.org/10.1007/4-431-30899-7_12Google Scholar
  9. A. D. Ioffe, V. L. Levin: Subdifferentials of convex functions, Tr. Mosk. Mat. Obshch. 26 (1972) 3–73.Google Scholar
  10. A. D Ioffe, V. M. Tikhomirov: On minimization of integral functionals, Funct. Anal. Appl. 3 (1969) 218–227.https://doi.org/10.1007/BF01676623Google Scholar
  11. A. D. Ioffe, V. M. Tikhomirov: Theory of Extremal Problems, Nauka, Moscow (1974), (in Russian); English translation: North Holland, Amsterdam (1979).Google Scholar
  12. F. Jules, M. Lassonde: Formulas for subdifferentials of sums of convex functions, J. Convex Analysis 9 (2002) 519–533.https://doi.org/10.68381/jca09030Google Scholar
  13. V. L. Levin: Convex Analysis in Spaces of Measurable Functions and its Application to Mathematics and Economics, Nauka, Moscow (1985), (in Russian).Google Scholar
  14. J.-P. Penot: Subdifferential calculus without qualification assumptions, J. Convex Analysis 3 (1996) 207–219.https://doi.org/10.68381/jca03014Google Scholar
  15. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  16. L. Thibault: A generalized sequential formula for subdifferentials of sums of convex functions defined on Banach spaces, in: Recent Developments in Optimization, R. Durier (ed.) et al., Lecture Notes in Economics and Math. Systems 429, Springer, Berlin (1995) 340–345.https://doi.org/10.1007/978-3-642-46823-0_25Google Scholar
  17. L. Thibault: Sequential convex subdifferential calculus and sequential Lagrange multipliers, SIAM J. Control Optimization 35 (1997) 1434–1444.https://doi.org/10.1137/S0363012995287714Google Scholar
  18. L. Thibault: Limiting convex subdifferential calculus with applications to integration and maximal monotonicity of subdifferential, in: Constructive, Experimental, and Nonlinear Analysis, M. Théra (ed.), CMS Conf. Proc. 27, Amer. Math. Soc., Providence (2000) 279–289.Google Scholar
  19. M. Valadier: Sous-differentiels d’une borne supérieure et d’une somme continue de fonctions convexes, C. R. Acad. Sci., Paris, Sér. A 268 (1969) 39–42.Google Scholar

Suggested citation

A. Ioffe. “Three Theorems on Subdifferentiation of Convex Integral Functionals.” Journal of Convex Analysis 13 (2006), No. 3/4, 759–772. https://doi.org/10.68381/jca13057

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Published by Heldermann Verlag, 2006. Rights now held by Banach Press LLC.