Skip to main content
SpringerLink
Log in

Generalized Differentiation of a Class of Normal Cone Operators

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper investigates generalized differentiation of normal cone operators to parametric smooth-boundary sets in Asplund spaces. We obtain formulas for computing the Fréchet and Mordukhovich coderivatives of such normal cone operators. We also give several examples to illustrate how the formulas can be used in practical calculations and applications.

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. Robinson, S.M.: Generalized equations and their solutions, I. Basic theory. Math. Program. Stud. 10, 128–141 (1979)

    Article  MATH  Google Scholar 

  2. Lee, G.M., Yen, N.D.: Coderivatives of a Karush–Kuhn–Tucker point set map and applications. Nonlinear Anal. (2013). Accepted for publication

  3. Qui, N.T., Yen, N.D.: A class of linear generalized equations. SIAM J. Optim. (2013). Accepted for publication

  4. Qui, N.T.: Stability for trust-region methods via generalized differentiation. J. Glob. Optim. (2013). doi:10.1007/s10898-013-0086-6

    MATH  Google Scholar 

  5. Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. MPS-SIAM Series on Optimization. SIAM, Philadelphia (2000)

    Book  MATH  Google Scholar 

  6. Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Optimal control of the sweeping process. Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms 19, 117–159 (2012)

    MATH  MathSciNet  Google Scholar 

  7. Mordukhovich, B.S.: Discrete approximations and refined Euler–Lagrange conditions for differential inclusions. SIAM J. Control Optim. 33, 882–915 (1995)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  9. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. II: Applications. Springer, Berlin (2006)

    Google Scholar 

  10. Dontchev, A.L., Rockafellar, R.T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6, 1087–1105 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Yao, J.-C., Yen, N.D.: Coderivative calculation related to a parametric affine variational inequality. Part 1: Basic calculations. Acta Math. Vietnam. 34, 157–172 (2009)

    MATH  MathSciNet  Google Scholar 

  12. Yao, J.-C., Yen, N.D.: Coderivative calculation related to a parametric affine variational inequality. Part 2. Appl. Pac. J. Optim. 5, 493–506 (2009)

    MATH  MathSciNet  Google Scholar 

  13. Henrion, R., Mordukhovich, B.S., Nam, N.M.: Second-order analysis of polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities. SIAM J. Optim. 20, 2199–2227 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Nam, N.M.: Coderivatives of normal cone mappings and Lipschitzian stability of parametric variational inequalities. Nonlinear Anal. 73, 2271–2282 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Qui, N.T.: Upper and lower estimates for a Fréchet normal cone. Acta Math. Vietnam. 36, 601–610 (2011)

    MATH  MathSciNet  Google Scholar 

  16. Qui, N.T.: Linearly perturbed polyhedral normal cone mappings and applications. Nonlinear Anal. 74, 1676–1689 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. Qui, N.T.: New results on linearly perturbed polyhedral normal cone mappings. J. Math. Anal. Appl. 381, 352–364 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Qui, N.T.: Nonlinear perturbations of polyhedral normal cone mappings and affine variational inequalities. J. Optim. Theory Appl. 153, 98–122 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Mordukhovich, B.S., Outrata, J.V.: On second-order subdifferentials and their applications. SIAM J. Optim. 12, 139–169 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. Henrion, R., Outrata, J., Surowiec, T.: Analysis of M-stationary points to an EPEC modeling oligopolistic competition in an electricity spot market. ESAIM Control Optim. Calc. Var. 18, 295–317 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. Yao, J.-C., Yen, N.D.: Parametric variational system with a smooth-boundary constraint set. In: Burachik, R.S., Yao, J.-C. (eds.) Variational Analysis and Generalized Differentiation in Optimization and Control. Series Optimization and Its Applications, vol. 47, pp. 205–221. Springer, Berlin (2010). In honor of B.S. Mordukhovich

    Chapter  Google Scholar 

  22. Qui, N.T.: Variational inequalities over Euclidean balls. Math. Methods Oper. Res. (2013). doi:10.1007/s00186-013-0442-9

    MATH  MathSciNet  Google Scholar 

  23. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  24. Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Dordrecht (2009)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02–2011.01. The author is indebted to Prof. Nguyen Dong Yen for helpful discussions, and the two anonymous referees for their valuable remarks and detailed suggestions that have greatly improved the original version.

Author information

Authors and Affiliations

  1. College of Information and Communication Technology, Can Tho University, 1 Ly Tu Trong, Can Tho, Vietnam

    Nguyen Thanh Qui

Authors
  1. Nguyen Thanh Qui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nguyen Thanh Qui.

Additional information

Communicated by Viorel Barbu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Qui, N.T. Generalized Differentiation of a Class of Normal Cone Operators. J Optim Theory Appl 161, 398–429 (2014). https://doi.org/10.1007/s10957-013-0427-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-013-0427-0

Keywords

Search

Navigation