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Qualification Conditions-Free Characterizations of the \(\varepsilon \)-Subdifferential of Convex Integral Functions

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Abstract

We provide formulae for the \(\varepsilon \)-subdifferential of the integral function \(I_f(x):=\int _T f(t,x) d\mu (t)\), where the integrand \(f:T\times X \rightarrow \overline{\mathbb {R}}\) is measurable in (t, x) and convex in x. The state variable lies in a locally convex space, possibly non-separable, while T is given a structure of a nonnegative complete \(\sigma \)-finite measure space \((T,\mathcal {A},\mu )\). The resulting characterizations are given in terms of the \(\varepsilon \)-subdifferential of the data functions involved in the integrand, f, without requiring any qualification conditions. We also derive new formulas when some usual continuity-type conditions are in force. These results are new even for the finite sum of convex functions and for the finite-dimensional setting.

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References

  1. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Modern Birkhäuser Classics. Birkhäuser Boston Inc, Boston, MA (2009)

    Book  Google Scholar 

  2. Aumann, R.J.: Integrals of set-valued functions. J. Math. Anal. Appl. 12, 1–12 (1965)

    Article  MathSciNet  Google Scholar 

  3. Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 3, 2nd edn. Springer, New York (2006). Theory and examples

    Book  Google Scholar 

  4. Borwein, J.M., Vanderwerff, J.D.: Convex Functions: Constructions, Characterizations and Counterexamples, Volume 109 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  5. Borwein, J.M., Yao, L.: Legendre-type integrands and convex integral functions. J. Convex Anal. 21(1), 261–288 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Bot, R.I., Grad, S.M., Wanka, G.: New constraint qualification and conjugate duality for composed convex optimization problems. J. Optim. Theory Appl. 135(2), 241–255 (2007)

    Article  MathSciNet  Google Scholar 

  7. Bourbaki, N.: Éléments de mathématique. VIII. Première partie: Les structures fondamentales de l’analyse. Livre III: Topologie générale. Chapitre IX: Utilisation des nombres réels en topologie générale. Actualités Sci. Ind., no 1045. Hermann et Cie., Paris (1948)

  8. Burachik, R.S., Jeyakumar, V.: A dual condition for the convex subdifferential sum formula with applications. J. Convex Anal. 12(2), 279–290 (2005)

    MathSciNet  MATH  Google Scholar 

  9. Castaing, C.: Sur les multi-applications mesurables. Rev. Fr. Informat. Rech. Opér. 1(1), 91–126 (1967)

    MathSciNet  MATH  Google Scholar 

  10. Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, vol. 580. Springer-Verlag, Berlin-New York (1977)

    Book  Google Scholar 

  11. Combari, C., Laghdir, M., Thibault, L.: On subdifferential calculus for convex functions defined on locally convex spaces. Ann. Sci. Math. Québec 23(1), 23–36 (1999)

    MathSciNet  MATH  Google Scholar 

  12. Correa, R., Hantoute, A., López, M.A.: Towards supremum-sum subdifferential calculus free of qualification conditions. SIAM J. Optim. 26(4), 2219–2234 (2016)

    Article  MathSciNet  Google Scholar 

  13. Correa, R., Hantoute, A., López, M.A.: Weaker conditions for subdifferential calculus of convex functions. J. Funct. Anal. 271(5), 1177–1212 (2016)

    Article  MathSciNet  Google Scholar 

  14. Correa, R., Hantoute, A., Pérez-Aros, P.: Sequential and exact formulae for the subdifferential of nonconvex integral functionals. ArXiv e-prints (2018). arXiv:2803.05521

  15. Correa, R., Hantoute, A., Pérez-Aros, P.: Characterizations of the subdifferential of convex integral functions under qualification conditions. J. Funct. Anal. 277(1), 227–254 (2019)

    Article  MathSciNet  Google Scholar 

  16. Diestel, J., Uhl, Jr. J.J.: Vector measures. American Mathematical Society, Providence, R.I. With a foreword by B. J. Pettis, Mathematical Surveys, No. 15 (1977)

  17. Hantoute, A., López, M.A., Zǎlinescu, C.: Subdifferential calculus rules in convex analysis: a unifying approach via pointwise supremum functions. SIAM J. Optim. 19(2), 863–882 (2008)

    Article  MathSciNet  Google Scholar 

  18. Hiriart-Urruty, J.-B., Lemaréchal, C.: Fundamentals of convex analysis. Grundlehren Text Editions. Springer-Verlag, Berlin, (2001). Abridged version of ıt Convex analysis and minimization algorithms. I [Springer, Berlin, 1993; MR1261420 (95m:90001)] and ıt II [ibid.; MR1295240 (95m:90002)]

  19. Hiriart-Urruty, J.-B., Moussaoui, M., Seeger, A., Volle, M.: Subdifferential calculus without qualification conditions, using approximate subdifferentials: a survey. Nonlinear Anal. 24(12), 1727–1754 (1995)

    Article  MathSciNet  Google Scholar 

  20. Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis. Volume 419 of Mathematics and its Applications, vol. I. Kluwer Academic Publishers, Dordrecht (1997)

    Google Scholar 

  21. Ioffe, A.D.: Survey of measurable selection theorems: Russian literature supplement. SIAM J. Control Optim. 16(5), 728–732 (1978)

    Article  MathSciNet  Google Scholar 

  22. Ioffe, A.D.: Three theorems on subdifferentiation of convex integral functionals. J. Convex Anal. 13(3–4), 759–772 (2006)

    MathSciNet  MATH  Google Scholar 

  23. Ioffe, A.D., Levin, V.L.: Subdifferentials of convex functions. Trudy Moskov. Mat. Obšč. 26, 3–73 (1972)

    MathSciNet  Google Scholar 

  24. Ioffe, A.D., Tikhomirov, V.M.: Duality of convex functions and extremum problems. Russ. Math. Surv. 23(6), 53 (1968)

    Article  Google Scholar 

  25. Laurent, P.-J.: Approximation et Optimisation. Hermann, Paris (1972)

    MATH  Google Scholar 

  26. Levin, V.L.: Convex integral functionals and the theory of lifting. Russ. Math. Surv. 30(2), 119 (1975)

    Article  Google Scholar 

  27. Lopez, O., Thibault, L.: Sequential formula for subdifferential of integral sum of convex functions. J. Nonlinear Convex Anal. 9(2), 295–308 (2008)

    MathSciNet  MATH  Google Scholar 

  28. Mordukhovich, B.S., Sagara, N.: Subdifferentials of nonconvex integral functionals in banach spaces with applications to stochastic dynamic programming. J. Convex Anal. 25(2), 643–673 (2018)

    MathSciNet  MATH  Google Scholar 

  29. Moreau, J.-J.: Fonctionnelles convexes. Séminaire Jean Leray 2, 1–108 (1967)

    Google Scholar 

  30. Pérez-Aros, P.: Formulae for the conjugate and the subdifferential of the supremum function. J. Optim. Theory Appl. 180(2), 397–427 (2019)

    Article  MathSciNet  Google Scholar 

  31. Pérez-Aros, P.: Subdifferential formulae for the supremum of an arbitrary family of functions. SIAM J. Optim. 29(2), 1714–1743 (2019)

    Article  MathSciNet  Google Scholar 

  32. Rockafellar, R.T.: Integrals which are convex functionals. Pac. J. Math. 24, 525–539 (1968)

    Article  MathSciNet  Google Scholar 

  33. Rockafellar, R.T.: Convex integral functionals and duality. In: Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971), pages 215–236. Academic Press, New York (1971)

  34. Rockafellar, R.T.: Integrals which are convex functionals. II. Pac. J. Math. 39, 439–469 (1971)

    Article  MathSciNet  Google Scholar 

  35. Rockafellar, R.T.: Integral Functionals, Normal Integrands and Measurable Selections. Lecture Notes in Mathematics, vol. 543. Springer, Berlin (1976)

    MATH  Google Scholar 

  36. Rockafellar, R.T.: Convex analysis. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, Reprint of the 1970 original, Princeton Paperbacks (1997)

  37. Rockafellar, R.T.: Variational Analysis, Volume 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin (1998)

    Google Scholar 

  38. Schwartz, L.: Radon measures on arbitrary topological spaces and cylindrical measures. Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, Tata Institute of Fundamental Research Studies in Mathematics, No. 6 (1973)

  39. Vallée, C., Zǎlinescu, C.: Series of convex functions: subdifferential, conjugate and applications to entropy minimization. J. Convex Anal. 23(4), 1137–1160 (2016)

    MathSciNet  MATH  Google Scholar 

  40. Wagner, D.H.: Survey of measurable selection theorems. SIAM J. Control Optim. 15(5), 859–903 (1977)

    Article  MathSciNet  Google Scholar 

  41. Wagner, D.H.: Survey of measurable selection theorems: an update. In: Measure theory, Oberwolfach 1979 (Proc. Conf., Oberwolfach, 1979), volume 794 of Lecture Notes in Math., pp. 176–219. Springer, Berlin-New York (1980)

  42. Zǎlinescu, C.: Convex analysis in general vector spaces. World Scientific Publishing Co. Inc., River Edge, NJ (2002)

    Book  Google Scholar 

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Acknowledgements

We very much appreciate the insightful suggestions and helpful comments of the referees, which have contributed to improving the current revision of the manuscript.

Funding

This work is partially supported by CONICYT Grants: Fondecyt projects N 1151003, 1190012, 1190110, and 1150909, Proyecto/Grant PIA AFB-170001, CONICYT-PCHA/doctorado Nacional/2014-21140621.

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Authors and Affiliations

  1. Universidad de O’Higgins, Rancagua, Chile

    Rafael Correa

  2. DIM-CMM of Universidad de Chile, Santiago, Chile

    Rafael Correa

  3. Center for Mathematical Modeling, Universidad de Chile, Santiago, Chile

    Abderrahim Hantoute

  4. Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins, Rancagua, Chile

    Pedro Pérez-Aros

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  1. Rafael Correa
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  2. Abderrahim Hantoute
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  3. Pedro Pérez-Aros
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Correspondence to Pedro Pérez-Aros.

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Correa, R., Hantoute, A. & Pérez-Aros, P. Qualification Conditions-Free Characterizations of the \(\varepsilon \)-Subdifferential of Convex Integral Functions. Appl Math Optim 83, 1709–1737 (2021). https://doi.org/10.1007/s00245-019-09604-y

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