Skip to main content
SpringerLink
Log in

Exact characterization for subdifferentials of a special optimal value function

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

For a closed set S and a bounded closed convex set U in a real normed vector space, we give exact subdifferential formulas of an optimal value function \(\mathrm {I}\!\Gamma _{S|U}\) whose definition is based on the Minkowski function of U. \(\mathrm {I}\!\Gamma _{S|U}\) covers distance function and indicator function as special cases. The main contribution is dropping two important assumptions of some main results in the literature.

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. Bounkhel, M.: On subdifferentials of a minimal time function in Hausdorff topological vector spaces. Appl. Anal. 93(8), 1761–1791 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Clark, F.H., Ledyaev, Y.S., Stern, R.J., et al.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)

    Google Scholar 

  3. Colombo, G., Wolenski, P.R.: The subgradient formula for the minimal time function in the case of constant dynamics in Hilbert space. J. Global Optim. 28(3–4), 269–282 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Colombo, G., Wolenski, P.R.: Variational analysis for a class of minimal time functions in Hilbert spaces. J. Convex Anal. 11(2), 335–361 (2004)

    MathSciNet  MATH  Google Scholar 

  5. De Blasi, F.S., Myjak, J.: On a generalized best approximation problem. J. Approx. Theory 94(1), 54–72 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. He, Y., Ng, K.F.: Subdifferentials of a minimum time function in Banach spaces. J. Math. Anal. Appl. 321(2), 896–910 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jiang, Y., He, Y.: Subdifferentials of a minimal time function in normed spaces. J. Math. Anal. Appl. 358(2), 410–418 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Li, C.: On well posed generalized best approximation problems. J. Approx. Theory 107(1), 96–108 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, C., Ni, R.: Derivatives of generalized distance functions and existence of generalized nearest points. J. Approx. Theory 115(1), 44–55 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mordukhovich, B.S., Nam, N.M.: Limiting subgradients of minimal time functions in Banach spaces. J. Global Optim. 46(4), 615–633 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Nam, N.M.: Subdifferential formulas for a class of non-convex infimal convolutions. Optimization 64(10), 2213–2222 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Nam, N.M., Villalobos, M.C., An, N.T.: Minimal time functions and the smallest intersecting ball problem with unbounded dynamics. J. Optim. Theory Appl. 154(3), 768–791 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Thibault, L.: On subdifferentials of optimal value functions. SIAM J. Control Optim. 29(5), 1019–1036 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhang, Y., He, Y., Jiang, Y.: Subdifferentials of a perturbed minimal time function in normed spaces. Optim. Lett. 8(6), 1921–1930 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant Nos. 11271274, 11461058), Scientific Research Fund of Sichuan Provincial Education Department (Grant Nos. 11ZB153, 11ZA180) and Scientific Research Fund of Sichuan Minzu College(Grant Nos. 13XYZB011, 12XYZB006).

Author information

Authors and Affiliations

  1. Department of Mathematics, Sichuan Normal University, Chengdu, 610066, Sichuan, China

    Shuqin Sun & Yiran He

  2. Department of Mathematics, Sichuan Minzu College, Kangding, 626001, Sichuan, China

    Shuqin Sun

Authors
  1. Shuqin Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yiran He
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Shuqin Sun or Yiran He.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sun, S., He, Y. Exact characterization for subdifferentials of a special optimal value function. Optim Lett 12, 519–534 (2018). https://doi.org/10.1007/s11590-017-1122-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-017-1122-0

Keywords

Search

Navigation