Skip to main content
SpringerLink
Log in

A smoothing Newton method for symmetric cone complementarity problems

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

Recently, there has been much interest in studying optimization problems over symmetric cones. This paper uses Euclidean Jordan algebras as a basic tool to construct a new smoothing function for symmetric cone complementarity problems. It is showed that this new function has similar structure and some good properties as the widely used symmetric perturbed Chen–Harker–Kanzow–Smale smooth function. In particularly, based on the function, we obtain global convergence and locally superlinear convergence of the smoothing Newton algorithm under two weaker assumptions respectively. Some numerical results for second-order cone complementarity problems are also reported.

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. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems.II, vol. I. Springer, New York (2003)

    Google Scholar 

  2. Fukushima, M.: Merit functions for variational inequality and complementarity problems. In: Di Pillo, G., Giannessi, F. (eds.) Nonlinear Optimization and Applications, pp. 155–170. Plenum Publishing Corporation, New York (1996)

  3. Sun, D., Qi, L.: On NCP-functions. Comput. Optim. Appl. 13, 201–220 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Kong, L.C., Xiu, N.H.: New smooth C-function for symmetric cone complementarity problems. Optim. Lett. 1, 391–400 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Liu, Y.J., Zhang, L.W., Liu, M.J.: Extension of smoothing functions to symmetric cones complementarity problems. Appl. Math. J. Chin. Univ. Ser. B. 22, 245–252 (2007)

    Article  Google Scholar 

  6. Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control. Optim. 15, 957–972 (1977)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  8. Huang, Z.H., Ni, T.: Smoothing algorithms for complementarity problems over symmetric cones. Comput. Optim. Appl. 45, 557–579 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Schmieta, S.H., Alizadeh, F.: Extension of primal-dual interior point algorithms to symmetric cones. Math. Program. 96, 409–438 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Yoshise, A.: Interior point trajectories and a homogeneous model for non-linear complementarity problems over symmetric cones. SIAM J. Optim. 17, 1129–1153 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Huang, Z.H., Han, J., Chen, Z.: Predictor-Corrector smoothing Newton method, based on a new smoothing function, for solving the nonlinear complementarity problem with a \(P_{0}\) function. J. Optim. Theory. App. 117, 39–68 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Huang, Z., Han, J., Xu, D., Zhang, L.: The noninterior continuation methods for solving the P0-function nonlinear complementarity problem. Sci. China. 44, 1107–1114 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Zhang, L.P., Gao, Z.Y.: Superlinear/quadratic one-step smoothing Newton method for P0-NCP without strict complementarity. Math. Meth. Oper. Res. 56, 231–241 (2002)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The project is supported by National Natural Science Foundation of China (Grant Nos. 11201074, 11071041) and The Scientific Research Special Fund Project of Fujian University (Grant No.JK2013060).

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Fujian Normal University, Fuzhou , 350007, People’s Republic of China

    Jia Tang & Changfeng Ma

Authors
  1. Jia Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Changfeng Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changfeng Ma.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tang, J., Ma, C. A smoothing Newton method for symmetric cone complementarity problems. Optim Lett 9, 225–244 (2015). https://doi.org/10.1007/s11590-013-0704-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-013-0704-8

Keywords

Search

Navigation