Advertisement

Optimization Letters

, Volume 12, Issue 3, pp 519–534 | Cite as

Exact characterization for subdifferentials of a special optimal value function

  • Shuqin Sun
  • Yiran He
Original Paper
  • 125 Downloads

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.

Keywords

Subdifferential Normal cone Optimal value function 

Notes

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).

References

  1. 1.
    Bounkhel, M.: On subdifferentials of a minimal time function in Hausdorff topological vector spaces. Appl. Anal. 93(8), 1761–1791 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Clark, F.H., Ledyaev, Y.S., Stern, R.J., et al.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)Google Scholar
  3. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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)MathSciNetzbMATHGoogle Scholar
  5. 5.
    De Blasi, F.S., Myjak, J.: On a generalized best approximation problem. J. Approx. Theory 94(1), 54–72 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    He, Y., Ng, K.F.: Subdifferentials of a minimum time function in Banach spaces. J. Math. Anal. Appl. 321(2), 896–910 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Jiang, Y., He, Y.: Subdifferentials of a minimal time function in normed spaces. J. Math. Anal. Appl. 358(2), 410–418 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Li, C.: On well posed generalized best approximation problems. J. Approx. Theory 107(1), 96–108 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Li, C., Ni, R.: Derivatives of generalized distance functions and existence of generalized nearest points. J. Approx. Theory 115(1), 44–55 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mordukhovich, B.S., Nam, N.M.: Limiting subgradients of minimal time functions in Banach spaces. J. Global Optim. 46(4), 615–633 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Nam, N.M.: Subdifferential formulas for a class of non-convex infimal convolutions. Optimization 64(10), 2213–2222 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Thibault, L.: On subdifferentials of optimal value functions. SIAM J. Control Optim. 29(5), 1019–1036 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Zhang, Y., He, Y., Jiang, Y.: Subdifferentials of a perturbed minimal time function in normed spaces. Optim. Lett. 8(6), 1921–1930 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsSichuan Normal UniversityChengduChina
  2. 2.Department of MathematicsSichuan Minzu CollegeKangdingChina

Personalised recommendations