Abstract

We present a subdifferential analysis for a general concept of infinite sum f:=∑i∈Ifif:=\sum_{i\in I}f_{i} of arbitrary collections of convex functions fif_{i}, called Lebesgue infinite sum. Since this problem cannot be addressed, at least directly, through classical arguments from the theory of normal convex integrands, we perform a reduction analysis showing that the ε\varepsilon-subdifferential of ff reduces to that of countable/finite subsums via appropriate lower limit and closure processes. Then, the usual calculus rules of (countable) integral functions give rise to characterizations of the ε\varepsilon-subdifferential of ff, which are written exclusively by means of ε\varepsilon-subdifferentials of the data fif_{i}. The resulting characterizations do not assume any qualification or boundedness condition.

Author information

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

Abderrahim Hantoute

Departamento de Matemáticas, Universidad de Alicante, Spain

hantoute@ua.es

José Vicente-Pérez

Departamento de Matemáticas, Universidad de Alicante, Spain

jose.vicente@ua.es

Keywords

  • Lebesgue infinite sum
  • convex functions
  • subdifferential calculus

Mathematics Subject Classification

26B05, 26J25, 49H05

References

  1. R. I. Boţ, V. Jeyakumar, G. Y. Li: Robust duality in parametric convex optimization, Set-Valued Var. Analysis 21 (2013) 177–189.https://doi.org/10.1007/s11228-012-0219-yGoogle Scholar
  2. R. Correa, A. Hantoute, M. A. López: Fundamentals of Convex Analysis and Optimization: A Supremum Function Approach, Springer Series in Operations Research and Financial Engineering, Springer, Cham (2023).https://doi.org/10.1007/978-3-031-29551-5Google Scholar
  3. R. Correa, A. Hantoute, P. Pérez-Aros: Characterizations of the subdifferential of convex integral functions under qualification conditions, J. Funct. Analysis 277 (2019) 227–254.https://doi.org/10.1016/j.jfa.2019.02.007Google Scholar
  4. R. Correa, A. Hantoute, P. Pérez-Aros: Qualification conditions-free characterizations of the ε-subdifferential of convex integral functions, Appl. Math. Optim. 83 (2021) 1709–1737.https://doi.org/10.1007/s00245-019-09604-yGoogle Scholar
  5. N. Dinh, M. A. Goberna, M. Volle: Primal-dual optimization conditions for the robust sum of functions with applications, Appl. Math. Optim. 80 (2019) 643–664.https://doi.org/10.1007/s00245-019-09596-9Google Scholar
  6. N. Dinh, M. A. Goberna, M. Volle: Duality for the robust sum of functions, Set-Valued Var. Analysis 28 (2020) 41–60.https://doi.org/10.1007/s11228-019-00515-2Google Scholar
  7. A. Hantoute, A. Jourani: Hiriart-Urruty-Phelps-like formula for the subdifferential of integral sums, Vietnam J. Math. 46 (2018) 391–405.https://doi.org/10.1007/s10013-018-0291-1Google Scholar
  8. A. Hantoute, A. Jourani: Subdifferential and conjugate calculus of integral functions with and without qualification conditions, J. Convex Analysis 30/1 (2023) 17–45.https://doi.org/10.68381/jca30003Google Scholar
  9. A. Hantoute, M. A. López, C. Zălinescu: Subdifferential calculus rules in convex analysis: A unifying approach via pointwise supremum functions, SIAM J. Optim. 19 (2008) 863–882.https://doi.org/10.1137/070700413Google Scholar
  10. J.-B. Hiriart-Urruty, M. Moussaoui, A. Seeger, M. Volle: Subdifferential calculus without qualification conditions, using approximate subdifferentials: a survey, Nonlinear Anal. Theory Meth. Appl. 24 (1995) 1727–1754.https://doi.org/10.1016/0362-546X(94)00221-3Google Scholar
  11. A. D. Ioffe: Three theorems on subdifferentiation of convex integral functionals, J. Convex Analysis 13/3-4 (2006) 759–772.https://doi.org/10.68381/jca13057Google Scholar
  12. V. L. Levin: Convex integral functionals and lifting theory (Russian), Uspehi Mat. Nauk 30 (182) (1975) 115–178.https://doi.org/10.1070/RM1975v030n02ABEH001408Google Scholar
  13. G. Li, K. F. Ng: On extension of Fenchel duality and its application, SIAM J. Optim. 19 (2008) 1489–1509.https://doi.org/10.1137/080716803Google Scholar
  14. O. Lopez, L. Thibault: Sequential formula for subdifferential of integral sum of convex functions, J. Nonlinear Convex Analysis 9 (2008) 295–308.Google Scholar
  15. D. T. Luc, M. Volle: Duality for optimization problems with infinite sums, SIAM J. Optim. 29 (2019) 1819–1843.https://doi.org/10.1137/18M117950XGoogle Scholar
  16. 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
  17. C. Vallée, C. Zălinescu: Series of convex functions: subdifferential, conjugate and applications to entropy minimization, J. Convex Analysis 23/4 (2016) 1137–1160.https://doi.org/10.68381/jca23042Google Scholar
  18. C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific, Singapore (2002).https://doi.org/10.1142/5021Google Scholar
  19. X. Y. Zheng: A series of convex functions on a Banach space, Acta Math. Sin. 4 (1998) 19–28.Google Scholar
  20. X. Y. Zheng, K. F. Ng: Error bound moduli for conic convex systems on Banach spaces, Math. Oper. Res. 29 (2004) 213–228.https://doi.org/10.1287/moor.1030.0088Google Scholar

Suggested citation

A. Hantoute, A. Jourani, J. Vicente-Pérez. “Lebesgue Infinite Sums of Convex Functions: Subdifferential Calculus.” Journal of Convex Analysis 30 (2023), No. 3, 1053–1072. https://doi.org/10.68381/jca30048

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2023. Rights now held by Banach Press LLC.