Skip to main content
Springer Nature Link
Log in

A note on the connection between Chaney’s derivatives and epi-derivatives

  • Short Communication
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

An example is provided showing the necessity of a finiteness assumption in a result of the second author ensuring that the second-order Chaney derivative coincides with the second-order Rockafellar epi-derivative of a lower semicontinuous function.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Chaney, R.W.: On second derivatives for nonsmooth functions. Nonlinear Anal. Theory Methods Appl 9, 1189–1209 (1985)

    Google Scholar 

  2. Chaney, R.W.: Second-order directional derivatives for nonsmooth functions. J. Math. Anal. Appl. 128, 495–511 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chaney, R.W.: Second-order sufficient conditions in nonsmooth optimization. Math. Oper. Res 13, 660–673 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  5. Huang, L.R., Ng, K.F.: On second-order directional derivatives in nonsmooth optimization. In: Du, D.-Z., Qi, L., Womersley, R.S. (eds.) Recent advances in nonsmooth optimization, pp. 158–170. World Scientific Publishers, Singapore (1995)

    Google Scholar 

  6. Huang, L.R., Ng, K.F.: On some relations between Chaney’s generalized second-order directional derivative and that of Ben-Tal and Zowe. SIAM J. Control Optim. 34, 1220–1234 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Penot, J.-P.: A characterization of tangential regularity. Nonlinear Anal. Theory Methods Appl 5, 625–641 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  8. Penot, J.-P.: Second-order generalized derivatives: comparison of two types of epi-derivatives. In: Oettli, W., Pallaschke, D., (eds.) Advances in Optimization, Proceedings, Lambrecht FRG, 1991, Lecture Notes in Economics and Mathematical Systems, pp. 52–76. Springer Verlag, Berlin (1992) (1, 382)

  9. Penot, J.-P.: Second-order conditions for optimization problems with constraints. SIAM J. Control Optim. 37(1), 303–318 (1998)

    Article  MathSciNet  Google Scholar 

  10. Penot, J.-P.: Directionally limiting sub differentials and second-order optimality conditions. Optim. Lett. (2013). doi:10.1007/s11590-013-0663-0

  11. Penot, J.-P.: Calculus Without Derivatives. Graduate Texts in Math 266. Springer, New York (2013)

    Book  Google Scholar 

  12. Rockafellar, R.T.: First-and second-order epi-differentiability in nonlinear programming. Trans. Amer. Math. Soc. 307, 75–107 (1986)

    Article  MathSciNet  Google Scholar 

  13. Rockafellar, R.T.: Second-order optimality conditions in nonlinear programming obtained by way of pseudo-derivatives. Math. Oper. Res. 14, 462–484 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  14. Seeger, A.: Analyse du second ordre de problèmes non différentiables, Thesis, Univ Toulouse I (1986)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, South China Normal University, Guangzhou, China

    Liren Huang

  2. Sorbonne Universités, UPMC Univ Paris 6 UMR 7598 Laboratoire Jacques-Louis Lions, 75005 , Paris, France

    Jean-Paul Penot

Authors
  1. Liren Huang
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Jean-Paul Penot
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Jean-Paul Penot.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Huang, L., Penot, JP. A note on the connection between Chaney’s derivatives and epi-derivatives. Optim Lett 8, 2357–2360 (2014). https://doi.org/10.1007/s11590-014-0722-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-014-0722-1

Keywords