Skip to main content
SpringerLink
Log in

Exact Formulae for Coderivatives of Normal Cone Mappings to Perturbed Polyhedral Convex Sets

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

Abstract

In this paper, without using any regularity assumptions, we derive a new exact formula for computing the Fréchet coderivative and an exact formula for the Mordukhovich coderivative of normal cone mappings to perturbed polyhedral convex sets. Our development establishes generalizations and complements of the existing results on the topic. An example to illustrate formulae is given.

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. 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  MathSciNet  MATH  Google Scholar 

  2. 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  MathSciNet  MATH  Google Scholar 

  3. Ban, L., Mordukhovich, B.S., Song, W.: Lipschitzian stability of parametric variational inequalities over generalized polyhedra in Banach spaces. Nonlinear Anal. 74, 441–461 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Trang, N.T.Q.: Lipschitzian stability of parametric variational inequalities over perturbed polyhedral convex sets. Optim. Lett. (2011). doi:10.1007/s11590-011-0299-x

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Henrion, R., Outrata, J.: On calculating the normal cone to a finite union of convex polyhedra. Optimization 57, 57–78 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Henrion, R., Outrata, J., Surowiec, T.: On the co-derivative of normal cone mappings to inequality systems. Nonlinear Anal. 71, 1213–1226 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lu, S.: Variational conditions under the constant rank constraint qualification. Math. Oper. Res. 35, 120–139 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lu, S., Robinson, S.M.: Variational inequalities over perturbed polyhedral convex sets. Math. Oper. Res. 33, 689–711 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  15. Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the Grant NSC 99-2115-M-037-002-MY3 (Taiwan) and was supported by the project “Joint research and training on Variational Analysis and Optimization Theory, with oriented applications in some technological areas” (Vietnam–USA). The authors are indebted to the anonymous referees for careful reading and valuable comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, Hanoi Pedagogical University No. 2, Vinh Phuc, Vietnam

    N. Q. Huy

  2. Center for General Education, Kaohsiung Medical University, Kaohsiung, 807, Taiwan

    J.-C. Yao

  3. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan

    J.-C. Yao

Authors
  1. N. Q. Huy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J.-C. Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J.-C. Yao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Huy, N.Q., Yao, JC. Exact Formulae for Coderivatives of Normal Cone Mappings to Perturbed Polyhedral Convex Sets. J Optim Theory Appl 157, 25–43 (2013). https://doi.org/10.1007/s10957-012-0157-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-012-0157-8

Keywords

Search

Navigation