Skip to main content
SpringerLink
Log in

Random Multifunctions as Set Minimizers of Infinitely Many Differentiable Random Functions

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

Under mild assumptions, we prove that any random multifunction can be represented as the set of minimizers of an infinitely many differentiable normal integrand, which preserves the convexity of the random multifunction. This result is an extended random version of work done by Azagra and Ferrera (Proc Am Math Soc 130(12):3687–3692, 2002). We provide several applications of this result to the approximation of random multifunctions and integrands. The paper ends with a characterization of the set of integrable selections of a measurable multifunction as the set of minimizers of an infinitely many differentiable integral function.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aubin, J.-P., Ekeland, I.: Applied Nonlinear Analysis. Dover Publications, Inc., Mineola, NY, Reprint of the 1984 original (2006)

  2. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Mod. Birkhäuser Class. Birkhäuser Boston, Inc., Boston, MA, Reprint of the 1990 edition (2009)

  3. Aussel, D., Lalitha, C.S. (eds.): Generalized Nash Equilibrium Problems, Bilevel Programming and MPEC. Forum Interdiscip. Math, Springer, Singapore (2017)

    MATH  Google Scholar 

  4. Aussel, D., Svensson, A.: Is pessimistic bilevel programming a special case of a mathematical program with complementarity constraints? J. Optim. Theory Appl. 181(2), 504–520 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Azagra, D., Ferrera, J.: Every closed convex set is the set of minimizers of some \(C^\infty \)-smooth convex function. Proc. Am. Math. Soc. 130(12), 3687–3692 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Balder, E.J.: Necessary and sufficient conditions for \(L_1\)-strong-weak lower semicontinuity of integral functionals. Nonlinear Anal. 11(12), 1399–1404 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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

  8. 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  MATH  Google Scholar 

  9. Correa, R., Hantoute, A., Pérez-Aros, P.: Subdifferential calculus rules for possibly nonconvex integral functions. SIAM J. Control Optim. 58(1), 462–484 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  10. Correa, R., Hantoute, A., Pérez-Aros, P.: Qualification conditions-free characterizations of the \(\varepsilon \)-subdifferential of convex integral functions. Appl. Math. Optim. 83(3), 1709–1737 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dempe, S., Dutta, J.: Is bilevel programming a special case of a mathematical program with complementarity constraints? Math. Program. 131(1–2, Ser. A), 37–48 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Giner, E.: On the Clarke subdifferential of an integral functional on \(L_p, 1\le p<\infty \). Canad. Math. Bull. 41(1), 41–48 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Giner, E.: Clarke and limiting subdifferentials of integral functionals. J. Convex Anal. 24(2), 661–678 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Hájek, Petr: Smooth functions on \(c_0\). Isr. J. Math. 104, 17–27 (1998)

    Article  MATH  Google Scholar 

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

  16. Jourani, A., Thibault, L.: Noncoincidence of approximate and limiting subdifferentials of integral functionals. SIAM J. Control Optim. 49(4), 1435–1453 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mordukhovich, B., Pérez-Aros, P.: Generalized Leibniz rules and Lipschitzian stability for expected-integral mappings. SIAM J. Optim. 31(4), 3212–3246 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation II. Grundlehren Math, vol. 331. Wiss. Springer, Berlin (2006)

  19. Mordukhovich, B.S., Pérez-Aros, P.: Generalized sequential differential calculus for expected-integral functionals. Set-Valued Var. Anal. 29(3), 621–644 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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 

  21. Papageorgiou, N.S., Kyritsi-Yiallourou, S.: Handbook of Applied Analysis. Adv, vol. 19. Mech. Math. Springer, New York (2009)

  22. Prékopa, A.: Stochastic Programming. Mathematics and Its Applications, vol. 324. Kluwer Academic Publishers Group, Dordrecht (1995)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Grundlehren Math, vol. 317. Wiss. Springer, Berlin (1998)

  26. Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming, volume 9 of MOS-SIAM Ser. Optim. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, second edition, 2014. Modeling and theory

  27. van den Berg, J.B., Hetebrij, W., Rink, B.: The parameterization method for center manifolds. J. Differ. Equ. 269(3), 2132–2184 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  28. Veselý, L., Zajíček, L.: Delta-convex mappings between Banach spaces and applications. Dissertationes Math. (Rozprawy Mat.) 289, 52 (1989)

Download references

Acknowledgements

The authors wish to thank the referee for providing several helpful suggestions. This work was partially supported by ANID-Chile under grants Fondecyt Regular 1190110, Fondecyt Regular 1200283, Fondecyt Regular 1220886, Fondecyt de Exploración 13220097 and CMM BASAL funds for center of excellence FB210005.

Author information

Authors and Affiliations

  1. Departamento de Ingeniería Matemática, Universidad de Chile, Santiago, Chile

    Juan Guillermo Garrido

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

    Juan Guillermo Garrido, Pedro Pérez-Aros & Emilio Vilches

Authors
  1. Juan Guillermo Garrido
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pedro Pérez-Aros
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Emilio Vilches
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emilio Vilches.

Additional information

Communicated by Lionel Thibault.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Garrido, J.G., Pérez-Aros, P. & Vilches, E. Random Multifunctions as Set Minimizers of Infinitely Many Differentiable Random Functions. J Optim Theory Appl 198, 86–110 (2023). https://doi.org/10.1007/s10957-023-02240-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-023-02240-1

Keywords

Mathematics Subject Classification

Search

Navigation