Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Convex integral functionals


Author: Nikolaos S. Papageorgiou
Journal: Trans. Amer. Math. Soc. 349 (1997), 1421-1436
MSC (1991): Primary 47H30, 49N15
DOI: https://doi.org/10.1090/S0002-9947-97-01478-5
MathSciNet review: 1321585
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study nonlinear integral functionals determined by normal convex integrands. First we obtain expressions for their convex conjugate, their $\varepsilon $-subdifferential $(\varepsilon \ge 0)$ and their $\varepsilon $-directional derivative. Then we derive a necessary and sufficient condition for the existence of an approximate solution for the continuous infimal convolution. We also obtain general conditions which guarantee the interchangeability of the conditional expectation and subdifferential operators. Finally we examine the conditional expectation of random sets.


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

  • 1. R. Aumann, Markets with a continuum of traders, Econometrica 32 (1964), 39-50. MR 30:2908
  • 2. -, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1-12. MR 32:2543
  • 3. R. Aumann and L. Shapley, Values of Nonatomic Games, Princeton Univ. Press, Princeton, 1974.
  • 4. J.-M. Bismut, Integrales convexes et probabilités, J. Math. Anal. Appl. 42 (1973), 639-673. MR 48:3083
  • 5. N. Bourbaki, Espaces Vectoriels Topologiques, Hermann, Paris, 1966. MR 34:3277
  • 6. C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Springer, Berlin, 1977. MR 57:7169
  • 7. J. Diestel and J. Uhl, Vector Measures, Math Surveys, Vol. 15, AMS, Providence, RI, 1977. MR 56:12216
  • 8. F. Hiai and H. Umegaki, Integrals, conditional expectations and martingales of multivalued functions, J. Multiv. Anal. 7 (1977), 149-182. MR 58:22463
  • 9. J.-B. Hiriart-Urruty, Thèse d'Etat, Chapter IV, Clermond-Ferrand, France, 1977.
  • 10. -, Lipschitz $r$-continuity of the approximate subdifferential of a convex function, Math Scand. 47 (1980), 123-134. MR 82c:58007
  • 11. A. Ioffe and V. Levin, Subdifferentials and convex functions, Trans. Moscow Math. Soc. 26 (1972), 1-72. MR 51:8817
  • 12. A. Ioffe and V. Tihomirov, The duality of convex functions and extremal problems, Russian Math. Surveys 23 (1968), 53-124. MR 44:5797
  • 13. -, Theory of Extremal Problems, North-Holland, Amsterdam, 1979. MR 80d:49001b
  • 14. H.-A. Klei, Thèse d'Etat, Université de Paris VI, Paris, 1985.
  • 15. N. Komuro, Basic properties of convex functions and convex integrands, Hokkaido Math. Jour. 18 (1989), 1-30. MR 90f:49008
  • 16. P.-J. Laurent, Approximation et Optimisation, Hermann, Paris, 1972. MR 57:6947
  • 17. V. Levin, The Lebesgue decomposition for functionals on the vector-function space $L^\infty _X$, Functional Anal. Appl. 8 (1974), 314-317. MR 57:850
  • 18. -, Convex integral functionals and the theory of lifting, Russian Math. Surveys 30 (1975), 115-178. MR 53:3698
  • 19. N. S. Papageorgiou, Convergence and representation theorems for set-valued random processes, J. Math. Anal. Appl. 150 (1990), 129-145. MR 92a:60117
  • 20. -, Weak convergence of random sets in Banach spaces, J. Math. Anal. Appl. 164 (1992), 571-589. MR 92m:49013
  • 21. R. T. Rockafellar, Integrals which are convex functionals I, Pacific J. Math. 24 (1968), 525-539. MR 38:4984
  • 22. -, Integrals which are convex functionals II, Pacific J. Math. 39 (1971), 439-469. MR 46:9710
  • 23. -, Convex integral functionals and duality, in Contributions to Nonlinear Functional Analysis, ed. by E. Zarantonello, Academic Press, New York, 1971, pp. 215-236. MR 52:11693
  • 24. -, Conjugate Duality and Optimization, Reg. Conf. Series in Appl. Math., vol. 16, SIAM, Philadelphia, 1973. MR 51:9811
  • 25. -, Integral functionals, normal integrands and measurable multifunctions, in Nonlinear Operators and the Calculus of Variations, ed. by J.-P. Gossez et al., Springer, New York, 1976, pp. 157-207. MR 58:23598
  • 26. -, Convex Analysis, Princeton Univ. Press, Princeton, 1970. MR 43:445
  • 27. R. T. Rockafellar and R. J.-B. Wets, On the interchange of subdifferentiation and conditional expectation for convex functionals, Stochastics 10 (1982), 173-182. MR 83j:49022
  • 28. M.-F. Saint-Beuve, On the extension of von Neumann-Aumann theorem, J. Funct. Anal. 17 (1974), 112-129. MR 51:10564
  • 29. L. Thibault, Esperance conditionelle d'integrandes semicontinus, Ann. Inst. H. Poincare 17 (1981), 337-350. MR 83a:60005
  • 30. M. Valadier, Integration de convexes fermes notamment d'epigraphes et $\inf $-convolution continue, RAIRO 4 (1970), 57-73. MR 43:3881
  • 31. D. Wagner, Survey of measurable selection theorems, SIAM J. Control. Optim. 15 (1977), 859-903. MR 58:6137

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 47H30, 49N15

Retrieve articles in all journals with MSC (1991): 47H30, 49N15


Additional Information

Nikolaos S. Papageorgiou
Affiliation: Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece

DOI: https://doi.org/10.1090/S0002-9947-97-01478-5
Keywords: Normal convex integrand, $\varepsilon$-subdifferential, multifunction, support function, singular functional, conditional expectation, Souslin space subdifferential
Received by editor(s): January 13, 1994
Received by editor(s) in revised form: January 23, 1995
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society