Abstract

In recent years there has been great interest in variational analysis of a class of nonsmooth functions called the minimal time function. In this paper we continue this line of research by providing new results on generalized differentiation of this class of functions, relaxing assumptions imposed on the functions and sets involved for the results. In particular, we focus on the singular subdifferential and the limiting subdifferential of this class of functions.

Author information

Contact details are reproduced from the original publication and may be historical.

Nguyen Mau Nam

Fariborz Maseeh Dept. of Mathematics and Statistics, Portland State University, PO Box 751, Portland, OR 97207, U.S.A.

mau.nam.nguyen@pdx.edu

Keywords

  • Minimal time function
  • epsilon-Fréchet subdifferential
  • limiting subdifferential
  • singular subdifferential
  • calmness

Mathematics Subject Classification

49J52, 49J53, 90C31

References

  1. H. H. Bauschke, P. L. Combettes: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York (2011).https://doi.org/10.1007/978-1-4419-9467-7Google Scholar
  2. D. P. Bertsekas, A. Nedić, A. E. Ozdaglar: Convex Analysis and Optimization, Athena Scientific, Belmont (2003).Google Scholar
  3. J. M. Borwein, Q. J. Zhu: Techniques of Variational Analysis, Springer, New York (2005).Google Scholar
  4. M. Bounkhel, L. Thibault: On various notions of regularity of sets in nonsmooth analysis, Nonlinear Analysis 48 (2002) 223–246.https://doi.org/10.1016/S0362-546X(00)00183-8Google Scholar
  5. J. V. Burke, M. C. Ferris, M. Qian: On the Clarke subdifferential of the distance function of a closed set, J. Math. Anal. Appl. 166 (1992) 199–213.https://doi.org/10.1016/0022-247X(92)90336-CGoogle Scholar
  6. F. H. Clarke: Nonsmooth Analysis and Optimization, John Wiley & Sons, New York (1983).Google Scholar
  7. F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, P. R. Wolenski: Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics 178, Springer, New York (1998).Google Scholar
  8. G. Colombo, P. R. Wolenski: The subgradient formula for the minimal time function in the case of constant dynamics in Hilbert space, J. Global Optim. 28 (2004) 269–282.https://doi.org/10.1023/B:JOGO.0000026460.10505.ddGoogle Scholar
  9. G. Colombo, P. R. Wolenski: Variational analysis for a class of minimal time functions in Hilbert spaces, J. Convex Analysis 11 (2004) 335–361.https://doi.org/10.68381/jca11021Google Scholar
  10. Y. He, K.F. Ng: Subdifferentials of a minimum time function in Banach spaces, J. Math. Anal. Appl. 321 (2006) 896–910.https://doi.org/10.1016/j.jmaa.2005.09.009Google Scholar
  11. J.-B. Hiriart-Urruty, C. Lemaréchal: Fundamentals of Convex Analysis, Springer, Berlin (2001).https://doi.org/10.1007/978-3-642-56468-0Google Scholar
  12. Y. Jiang, Y. He: Subdifferentials of a minimum time function in normed spaces, J. Math. Anal. Appl. 358 (2009) 410–418.https://doi.org/10.1016/j.jmaa.2009.05.016Google Scholar
  13. L. Meng, C. Li, J.-C. Yao: Limiting subdifferentials of perturbed distance functions in Banach spaces, Nonlinear Anal. 75(3) (2012) 1483–1495.https://doi.org/10.1016/j.na.2011.06.024Google Scholar
  14. B. S. Mordukhovich: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications, Springer, Berlin (2006).https://doi.org/10.1007/3-540-31246-3Google Scholar
  15. B. S. Mordukhovich, N. M. Nam: Subgradients of distance functions with applications to Lipschitzian stability, Math. Program. 104 (2005) 635–668.https://doi.org/10.1007/s10107-005-0632-1Google Scholar
  16. B. S. Mordukhovich, N. M. Nam: Subgradients of minimal time functions under minimal assumptions, J. Convex Analysis 18 (2011) 915–947.https://doi.org/10.68381/jca18058Google Scholar
  17. R. R. Phelps: Convex Functions, Monotone Operators and Differentiability, 2nd edition, Lecture Notes Math. 1364, Springer, Berlin (2003).Google Scholar
  18. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  19. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  20. Sh. Sun, Y. He: Exact characterization for subdifferentials of a special optimal value function, e-print (2017).https://doi.org/10.1007/s11590-017-1122-0Google Scholar
  21. Y. Zhang, Y. He, Y. Jiang: Subdifferentials of a perturbed minimal time function in normed spaces, Optim. Letters 8(6) (2014) 1921–1930.https://doi.org/10.1007/s11590-013-0689-3Google Scholar
  22. C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific, Singapore (2002).https://doi.org/10.1142/5021Google Scholar

Suggested citation

N. M. Nam, D. V. Cuong. “Subgradients of Minimal Time Functions without Calmness.” Journal of Convex Analysis 26 (2019), No. 1, 189–200. https://doi.org/10.68381/jca26011

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.