Skip to main content
Log in

New extremal principles with applications to stochastic and semi-infinite programming

  • Full Length Paper
  • Series B
  • Published:
Mathematical Programming Submit manuscript

Abstract

This paper develops new extremal principles of variational analysis that are motivated by applications to constrained problems of stochastic programming and semi-infinite programming without smoothness and/or convexity assumptions. These extremal principles concern measurable set-valued mappings/multifunctions with values in finite-dimensional spaces and are established in both approximate and exact forms. The obtained principles are instrumental to derive via variational approaches integral representations and upper estimates of regular and limiting normals cones to essential intersections of sets defined by measurable multifunctions, which are in turn crucial for novel applications to stochastic and semi-infinite programming.

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

Access this article

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. Bauschke, H.H., Borwein, J.M., Li, W.: Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization. Math. Program. 86, 135–160 (1999)

    Article  MathSciNet  Google Scholar 

  2. Cánovas, M.J., López, M.A., Mordukhovich, B.S., Parra, J.: Variational analysis in semi-infinite and infinite programming, I: stability of linear inequality systems of feasiable solutions. SIAM J. Optim. 20, 1504–1526 (2009)

    Article  MathSciNet  Google Scholar 

  3. Cánovas, M.J., López, M.A., Mordukhovich, B.S., Parra, J.: Variational analysis in semi-infinite and infinite programming, II: necessary optimality conditions. SIAM J. Optim. 20, 2788–2806 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Ernst, E., Théra, M.: Boundary half-strips and the strong CHIP. SIAM J. Optim. 18, 834–852 (2007)

    Article  MathSciNet  Google Scholar 

  6. Goberna, M.A., López, M.A.: Post-Optimal Analysis in Linear Semi-Infinite Optimization. Springer, New York (2014)

    Book  Google Scholar 

  7. Hantoute, A., Henrion, R., Pérez-Aros, P.: Subdifferential characterization of probability functions under Gaussian distribution. Math. Program. B (2018). https://doi.org/10.1007/s10107-018-1237-9

    Article  MATH  Google Scholar 

  8. Kelley, J.L.: General topology. Springer, New York (1975)

    MATH  Google Scholar 

  9. Kruger, A.Y., López, M.A.: Stationarity and regularity of infinite collections of sets. J. Optim. Theory Appl. 154, 339–369 (2012)

    Article  MathSciNet  Google Scholar 

  10. Kruger, A.Y., Mordukhovich, B.S.: Extremal points of sets and the Euiler equation in nomsmooth optimization problems. Dokl. Akad. Nauk BSSR 24, 684–687 (1980). (in Russian)

    MathSciNet  MATH  Google Scholar 

  11. Li, C., Ng, K.F., Pong, T.K.: The SECQ, linear regularity, and the strong CHIP for an infinite system of closed convex sets in normed linear spaces. SIAM J. Optim. 18, 643–665 (2008)

    Article  MathSciNet  Google Scholar 

  12. Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. 183, 250–288 (1994)

    Article  MathSciNet  Google Scholar 

  13. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications. Springer, Berlin (2006)

    Google Scholar 

  14. Mordukhovich, B.S.: Variational Analysis and Applications. Springer, Cham (2018)

    Book  Google Scholar 

  15. Mordukhovich, B.S., Nghia, T.T.A.: Constraint qualifications and optimality conditions in nonlinear semi-infinite and infinite programming. Math. Program. 139, 271–300 (2013)

    Article  MathSciNet  Google Scholar 

  16. Mordukhovich, B.S., Nghia, T.T.A.: Subdifferentials of nonconvex supremum functions and their applications to semi-infinite and infinite programs with Lipschitzian data. SIAM J. Optim. 23, 406–431 (2013)

    Article  MathSciNet  Google Scholar 

  17. Mordukhovich, B.S., Nghia, T.T.A.: Nonsmooth cone-constrained optimization with applications to semi-infinite programming. Math. Oper. Res. 39, 301–337 (2014)

    Article  MathSciNet  Google Scholar 

  18. Mordukhovich, B.S., Phan, H.M.: Tangential extremal principle for finite and infnite systems, I: basic theory. Math. Program. 136, 31–63 (2012)

    Article  MathSciNet  Google Scholar 

  19. Mordukhovich, B.S., Phan, H.M.: Tangential extremal principle for finite and infnite systems, II: applications to semi-infnite and multiobjective optimization. Math. Program. 136, 31–63 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

    Book  Google Scholar 

  22. Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming, 2nd edn. SIAM Publications, Philadelphia, PA (2014)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to three anonymous referees for their numerous comments and remarks that helped us to essentially improve the original presentation.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Boris S. Mordukhovich.

Additional information

Dedicated to Marco López in honor of his 70th birthday, with great respect.

Publisher's Note

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

Research of the Boris S. Mordukhovich was partly supported by the USA National Science Foundation under Grants DMS-1512846 and DMS-1808978, by the USA Air Force Office of Scientific Research under Grant #15RT04, and by the Australian Research Council under Discovery Project DP-190100555. Pedro Pérez-Aros was partially supported by: CONICYT Grants: Fondecyt Regular 1190110 and Fondecyt Regular 1200283 and Programa Regional Mathamsud 20-Math-08 CODE: MATH190003.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mordukhovich, B.S., Pérez-Aros, P. New extremal principles with applications to stochastic and semi-infinite programming. Math. Program. 189, 527–553 (2021). https://doi.org/10.1007/s10107-020-01548-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-020-01548-4

Keywords

Mathematics Subject Classification

Navigation