Hiriart-Urruty–Phelps-Like Formula for the Subdifferential of Integral Sums

Article
  • 15 Downloads

Abstract

We provide subdifferential calculus rules for continuous sums parametrized in measurable spaces that use the approximate subdifferentials of the data functions. As in Hiriart-Urruty and Phelps (J. Funct. Anal. 118: 154–166, 1993) given for finite sums, the resulting formulas hold without any conditions of continuity type on the involved functions. All this analysis is done in the setting of locally convex Suslin spaces.

Keywords

Integral sums Convex normal integrands ε-subdifferential Subdifferential calculus 

Mathematics Subject Classification (2010)

26B05 26J25 49H05 

Notes

Acknowledgements

We are very grateful to an anonymous referee for his/her suggestions and comments.

References

  1. 1.
    Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, vol. 580. Springer, Berlin Heidelberg (1977)CrossRefMATHGoogle Scholar
  2. 2.
    Correa, R., Hantoute, A., López, M.A.: Weaker conditions for subdifferential calculus of convex functions. J. Funct. Anal. 271, 1177–1212 (2016)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Diestel, J., Uhl, J.J. Jr.: Vector Measures. American Mathematical Society, Providence (1977)CrossRefMATHGoogle Scholar
  4. 4.
    Hantoute, A., Jourani, A.: Subdifferential calculus of integral functions without qualification conditions. Submitted (2017)Google Scholar
  5. 5.
    Correa, R., Hantoute, A., Jourani, A.: Characterizations of convex approximate subdifferential calculus in Banach spaces. Trans. Am. Math. Soc. 368, 4831–4854 (2016)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Hantoute, A., López, M.A., Zălinescu, C.: Subdifferential calculus rules in convex analysis: a unifying approach via pointwise supremum functions. SIAM J. Optim. 19, 863–882 (2008)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Hiai, F., Umegaki, H.: Integrals, conditional expectations, and martingales of multivalued functions. J. Multivar. Anal. 7, 149–182 (1977)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Hiriart-Urruty, J.-B., Phelps, R.R.: Subdifferential calculus using ε-subdifferentials. J. Funct. Anal. 118, 154–166 (1993)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Hiriart-Urruty, J.-B., Moussaoui, M., Seeger, A., Volle, M.: Subdifferential calculus without qualification conditions, using approximate subdifferentials: a survey. Nonlinear Anal. TMA 24, 1727–1754 (1995)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Ioffe, A.D., Tikhomirov, V.M.: On the minimization of integral functionals. Funkt. Analiz. 3, 61–70 (1969)MATHMathSciNetGoogle Scholar
  11. 11.
    Ioffe, A.D.: Three theorems on subdifferentiation of convex integral functionals. J. Convex Anal. 13, 759–772 (2006)MATHMathSciNetGoogle Scholar
  12. 12.
    Ioffe, A.D., Levin, V.L.: Subdifferentials of convex functions. Trudy Moskov. Mat. Obšč. 26, 3–73 (1972)MATHMathSciNetGoogle Scholar
  13. 13.
    Ioffe, A. D., Tihomirov, V. M.: Theory of Extremal Problems. Studies in Mathematics and its Applications, vol. 6. North-Holland Publishing Co., Amsterdam-New York (1979)Google Scholar
  14. 14.
    Jourani, A., Sanroma, M.: Variational sum of subdifferentials of convex functions. In: Garcia, C., Olive, C. (eds.) Proceedings of the IV Catalan Days of Applied Mathematics, pp. 71–79. Tarragona Press University, Tarragona (1998)Google Scholar
  15. 15.
    Levin, V.L.: Convex integral functionals and the theory of lifting. Russ. Math. Surv. 30, 119–184 (1975)CrossRefMATHGoogle Scholar
  16. 16.
    López, O., Thibault, L.: Sequential formula for subdifferential of integral sum of convex functions. J. Nonlinear Convex Anal. 9, 295–308 (2008)MATHMathSciNetGoogle Scholar
  17. 17.
    Mordukhovich, B.S., Sagara, N.: Subdifferentials of nonconvex integral functionals in Banach spaces with applications to stochastic dynamic programming. J. Convex Anal. (to appear) (2018)Google Scholar
  18. 18.
    Rockalellar, R.T.: Integrals which are convex functionals. Pac. J. Math. 24, 525–539 (1968)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Rockalellar, R.T.: Integrals which are convex functionals II. Pac. J. Math. 39, 439–469 (1971)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Rockalellar, R.T.: Integral functionals, normal integrands and measurable selections. In: Gossez, J.P., Dozo, E.J.L., Mawhin, J., Waelbroeck, L. (eds.) Nonlinear Operators and the Calculus of Variations. Lecture Notes in Mathematics, vol. 543, pp. 157–207. Springer (1976)Google Scholar
  21. 21.
    Thibault, L.: Sequential convex subdifferential calculus and sequential Lagrange multipliers. SIAM J. Control Optim. 35, 1434–1444 (1997)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Valle, C., Zălinescu, C.: Series of convex functions: subdifferential, conjugate and applications to entropy minimization. J. Convex Anal. 23, 1137–1160 (2016)MATHMathSciNetGoogle Scholar
  23. 23.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing Co. Pte. Ltd., Singapore (2002)CrossRefMATHGoogle Scholar
  24. 24.
    Zheng, X.Y.: A series of convex functions on a Banach space. Acta Math. Sinica 14, 77–84 (1998)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Centro de Modelamiento Matemático (UMI 2807 CNRS)Universidad de ChileSantiagoChile
  2. 2.Institut de Mathématiques de Bourgogne, UMR 5584 CNRSUniversité de Bourgogne Franche-ComtéDijon CédexFrance

Personalised recommendations