Skip to main content
SpringerLink
Log in

On the Coderivative of the Projection Operator onto the Second-order Cone

  • Published:
Set-Valued Analysis Aims and scope Submit manuscript

An Erratum to this article was published on 13 June 2009

Abstract

The limiting (Mordukhovich) coderivative of the metric projection onto the second-order cone \(\mathbb{R}^{n}\) is computed. This result is used to obtain a sufficient condition for the Aubin property of the solution map of a parameterized second-order cone complementarity problem and to derive necessary optimality conditions for a mathematical program with a second-order cone complementarity problem among the constraints.

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. Aubin, J.-P.: Lipschitz behaviour of solutions to convex minimization problems. Math. Oper. Res. 9, 87–111 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen, J.-S., Chen, X., Tseng, P.: Analysis of nonsmooth vector-valued functions associated with second-order cones. Math. Programming 101, 95–117 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, X.-D., Sun, D., Sun, J.: Complementarity functions and numerical experiments for second-order cone complementarity problems. Comput. Optim. Appl. 25, 39–56 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  5. Faraut, U., Korányi, A.: Analysis on Symmetric Cones. Oxford Mathematical Monographs. Oxford University Press, New York (1994)

    MATH  Google Scholar 

  6. Fukushima, M., Luo, Z.-Q., Tseng, P.: Smoothing functions for second-order cone complementarity problems. SIAM J. Optim. 12, 436–460 (2002)

    Article  MathSciNet  Google Scholar 

  7. Kanzow, C., Ferenzi, I., Fukushima, M.: Semismooth methods for linear and nonlinear second-order cone programs. Technical Report 2006-005, Department of Applied Mathematics and Physics, Kyoto University (April 2006, revised January 2007).

  8. Haslinger, J., Sassi, T.: Mixed finite element approximation of 3D contact problems with given friction: error analysis and numerical realization. Math. Model. Numer. Anal. 38, 563–578 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Koecher, M.: In: Brieg, A., Walcher, S. (eds.) The Minnesota Notes on Jordan Algebras and Their Applications. Springer, Berlin (1999)

    Google Scholar 

  10. Korányi, A.: Monotone functions on formally real Jordan algebras. Math. Ann. 269, 73–76 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  11. Liu, Y.-J., Zhang, L.-W.: Convergence of the augmented Lagrangian method for nonlinear optimization problems over second-order cones. J. Optim. Theory Appl. (2008, in press)

  12. Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Application of second-order cone programming. Linear Algebra Appl. 284, 193–228 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Mordukhovich, B.S.: Nonsmooth analysis with nonconvex generalized differentials and adjoint mappings. Dokl. Akad. Nauk SSSR 28, 976–979 (1984)

    MATH  MathSciNet  Google Scholar 

  14. Mordukhovich, B.S.: Approximation Methods in Problems of Optimization and Control (in Russian). Nauka, Moscow (1988)

    MATH  Google Scholar 

  15. Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. 183, 250–288 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  16. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, vol. 1. Springer, New York (2006)

    Google Scholar 

  17. Pang, J.S., Sun, D.F., Sun, J.: Semismooth homeomorphisms and strong stability of semidefinite and Lorentz cone complementarity problems. Math. Oper. Res. 28, 39–63 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Robinson, S.M.: Strongly regular generalized equation. Math. Oper. Res. 5, 43–62 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  19. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, New York (1998)

    MATH  Google Scholar 

  20. Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality and sensitivity. Math. Oper. Res. 25, 1–22 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  21. Shapiro, A.: On concepts of directional differentiability. J. Optim. Theory Appl. 66, 477–487 (1990)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, 18208, Prague, Czech Republic

    Jiří V. Outrata

  2. Department of Mathematics and Risk Management Institute, National University of Singapore, 2 Science Drive 2, Kent Ridge, Republic of Singapore

    Defeng Sun

Authors
  1. Jiří V. Outrata
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Defeng Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiří V. Outrata.

Additional information

The research of J.V. Outrata was supported by the grant IAA 100750802 of the Grant Agency of the Academy of Sciences of the Czech Republic.

The research of D. Sun was supported by the Academic Research Fund under Grant R-146-000-104-112 and the Risk Management Institute under Grants R-703-000-004-720 and R-703-000-004-646, National University of Singapore.

An erratum to this article can be found at http://dx.doi.org/10.1007/s11228-009-0115-2

Rights and permissions

Reprints and permissions

About this article

Cite this article

Outrata, J.V., Sun, D. On the Coderivative of the Projection Operator onto the Second-order Cone. Set-Valued Anal 16, 999–1014 (2008). https://doi.org/10.1007/s11228-008-0092-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-008-0092-x

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation