Skip to main content
SpringerLink
Log in

Subdifferentials of optimal value functions under metric qualification conditions

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this paper, by revisiting intersection rules for normal cones, we give formulas for estimating or computing the Fréchet/Mordukhovich/Moreau–Rockafellar subdifferentials of optimal value functions of constrained parametric optimization problems under metric qualification conditions. The results are then applied to derive chain rules for composite functions in both convex and nonconvex situations. Illustrative examples and comparisons to existing results, including those of Mordukhovich and Shao (Trans Amer Math Soc 348:1235–1280, 1996), Mordukhovich et al. (Math Program Ser B 116:369–396, 2009) and of An and Jourani (J Optim Theory Appl 192:82–96, 2022), are also addressed.

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

Notes

  1. According to Ioffe [15, p. 534] the notations “A” and “G” stands respectively for “analytic” and “geometric”.

References

  1. An, D.T.V., Jourani, A.: Subdifferentials of the marginal functions in parametric convex optimization via intersection formulas. J. Optim. Theory Appl. 192, 82–96 (2022)

    Article  MathSciNet  Google Scholar 

  2. An, D.T.V., Yao, J.-C., Yen, N.D.: Differential stability of a class of convex optimal control problems. Appl. Math. Optim. 81, 1–22 (2020)

    Article  MathSciNet  Google Scholar 

  3. An, D.T.V., Yen, N.D.: Differential stability of convex optimization problems under inclusion constraints. Appl. Anal. 94, 108–128 (2015)

    Article  MathSciNet  Google Scholar 

  4. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  Google Scholar 

  5. Borwein, J., Lucet, Y., Mordukhovich, B.: Compactly epi-Lipschitzian convex sets and functions in normed spaces. J. Convex Anal. 2, 375–393 (2000)

    MathSciNet  Google Scholar 

  6. Borwein, J.M., Strójwas, H.M.: Tangential approximations. Nonlinear Anal. 9, 1347–1366 (1985)

    Article  MathSciNet  Google Scholar 

  7. Burke, J.V., Ferris, M.C., Qian, M.: On the Clarke subdifferential of the distance function of a closed set. J. Math. Anal. Appl. 166, 199–213 (1992)

    Article  MathSciNet  Google Scholar 

  8. Clarke, F.H.: A new approach to Lagrange multipliers. Math. Oper. Res. 1, 165–174 (1976)

    Article  MathSciNet  Google Scholar 

  9. Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)

    Book  Google Scholar 

  10. Cui, Y., Pang, J.-S.: Modern Nonconvex Nondifferentiable Optimization. SIAM, Philadelphia (2022)

    Google Scholar 

  11. Henrion, R., Jourani, A.: Subdifferential conditions for calmness of convex constraints. SIAM J. Optim. 13, 520–534 (2002)

    Article  MathSciNet  Google Scholar 

  12. Huong, V.T., Yao, J.-C., Yen, N.D.: Differentiability properties of a parametric consumer problem. J. Nonlinear Convex Anal. 19, 1217–1245 (2018)

    MathSciNet  Google Scholar 

  13. Ioffe, A.D.: Proximal analysis and approximate subdifferentials. J. London Math. Soc. 41, 175–192 (1989)

    MathSciNet  Google Scholar 

  14. Ioffe, A.D.: Approximate subdifferentials and applications III: The metric theory. Mathematika 36, 1–38 (1989)

    Article  MathSciNet  Google Scholar 

  15. Ioffe, A.D.: Metric regularity and subdifferential calculus. Russ. Math. Surv. 55, 501–558 (2000)

    Article  MathSciNet  Google Scholar 

  16. Ioffe, A.D., Penot, J.-P.: Subdifferentials of performance functions and calculus of set-valued mappings. Serdica Math. J. 22, 359–384 (1996)

    MathSciNet  Google Scholar 

  17. Ivanov, G.E., Thibault, L.: Well-posedness and subdifferentials of optimal value and infimal convolution. Set-Valued Var. Anal. 27, 841–861 (2019)

    Article  MathSciNet  Google Scholar 

  18. Jourani, A.: Intersection formulae and the marginal function in Banach spaces. J. Math. Anal. Appl. 192, 867–891 (1995)

    Article  MathSciNet  Google Scholar 

  19. Jourani, A., Thibault, L.: Metric regularity and subdifferential calculus in Banach spaces. Set-Valued Anal. 3, 87–100 (1995)

    Article  MathSciNet  Google Scholar 

  20. Kruger, A.Y.: On Fréchet subdifferentials. J. Math. Sci. 116, 3325–3358 (2003)

    Article  MathSciNet  Google Scholar 

  21. Luc, D.T.: Theory of Vector Optimization. Springer, New York (1989)

    Book  Google Scholar 

  22. Mordukhovich, B.S.: Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40, 960–969 (1976)

    Article  MathSciNet  Google Scholar 

  23. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. Volume I: Basic Theory. Springer, Berlin (2006)

  24. Mordukhovich, B.S.: Variational Analysis and Applications. Springer, Switzerland (2018)

    Book  Google Scholar 

  25. Mordukhovich, B.S, Nam, N.M.: Convex Analysis and Beyond: Volume I: Basic Theory. Springer Series in Operations Research and Financial Engineering (2022)

  26. Mordukhovich, B.S., Nam, N.M., Rector, R.B., Tran, T.: Variational geometric approach to generalized differential and conjugate calculi in convex analysis. Set-Valued Var. Anal. 25, 731–755 (2017)

    Article  MathSciNet  Google Scholar 

  27. Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Subgradients of marginal functions in parametric mathematical programming. Math. Program. Ser. B 116, 369–396 (2009)

    Article  MathSciNet  Google Scholar 

  28. Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund spaces. Trans. Amer. Math. Soc. 348, 1235–1280 (1996)

    Article  MathSciNet  Google Scholar 

  29. Ngai, H.V., Théra, M.: Metric inequality, subdifferential calculus and applications. Set-Valued Var. Anal. 9, 187–216 (2001)

    Article  MathSciNet  Google Scholar 

  30. Penot, J.-P.: Cooperative behavior of functions, sets and relations. Math. Methods Oper. Res. 48, 229–246 (1998)

    Article  MathSciNet  Google Scholar 

  31. Penot, J.-P.: Calculus Without Derivatives. Graduate Texts In Mathematics 266, Springer, New York (2013)

  32. Qui, N.T., Wachsmuth, D.: Subgradients of marginal functions in parametric control problems of partial differential equations. SIAM J. Optim. 30, 1724–1755 (2020)

    Article  MathSciNet  Google Scholar 

  33. Rockafellar, R.T.: Directionally Lipschitzian functions and subdifferential calculus. Proc. London Math. Soc. 39, 331–355 (1979)

    Article  MathSciNet  Google Scholar 

  34. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1997)

    Google Scholar 

  35. Schirotzek, W.: Nonsmooth Analysis. Springer, Berlin (2007)

    Book  Google Scholar 

  36. Toan, N.T.: Differential stability of discrete optimal control problems with mixed constraints. Positivity 25, 1229–1254 (2021)

    Article  MathSciNet  Google Scholar 

  37. Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

    Book  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referee and the handling Associate Editor for their very careful readings and valuable suggestions which improved the presentation of this manuscript.

Funding

Hong-Kun Xu was supported in part by National Natural Science Foundation of China (Grant Number U1811461) and by Australian Research Council (Grant Number DP200100124).

Author information

Authors and Affiliations

  1. School of Management, Hangzhou Dianzi University, Hangzhou, 310018, China

    Vu Thi Huong & Duong Thi Viet An

  2. Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, 10072, Vietnam

    Vu Thi Huong

  3. Department of Mathematics and Informatics, Thai Nguyen University of Sciences, Thai Nguyen, 250000, Vietnam

    Duong Thi Viet An

  4. School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China

    Hong-Kun Xu

  5. College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China

    Hong-Kun Xu

Authors
  1. Vu Thi Huong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Duong Thi Viet An
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hong-Kun Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally, read and approved the final manuscript.

Corresponding author

Correspondence to Hong-Kun Xu.

Ethics declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

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

Dedicated to Professor Nguyen Dong Yen on the occasion of his 65th birthday.

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

Huong, V.T., An, D.T.V. & Xu, HK. Subdifferentials of optimal value functions under metric qualification conditions. J Glob Optim 88, 253–283 (2024). https://doi.org/10.1007/s10898-023-01304-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-023-01304-w

Keywords

Mathematics Subject Classification

Search

Navigation