Skip to main content
SpringerLink
Log in

A nonmonotone derivative-free algorithm for nonlinear complementarity problems based on the new generalized penalized Fischer–Burmeister merit function

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, we propose a new generalized penalized Fischer–Burmeister merit function, and show that the function possesses a system of favorite properties. Moreover, for the merit function, we establish the boundedness of level set under a weaker condition. We also propose a derivative-free algorithm for nonlinear complementarity problems with a nonmonotone line search. More specifically, we show that the proposed algorithm is globally convergent and has a locally linear convergence rate. Numerical comparisons are also made with the merit function used by Chen (J Comput Appl Math 232:455–471, 2009), which confirm the superior behaviour of the new merit function.

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. Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, New York (1992)

    MATH  Google Scholar 

  2. Harker, P.T., Pang, J.S.: Finite dimensional variational inequality and nonlinear complementarity problem: a survey of theory, algorithms and applications. Math. Program. 48, 161–220 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Mangasarian, O.L.: Equivalence of the complementarity problems to a system of nonlinear equations. SIAM J. Appl. Math. 31, 89–92 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  4. Yamashita, N., Fukushima, M.: Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems. Math. Program. 76, 469–491 (1997)

    MathSciNet  MATH  Google Scholar 

  5. Kanzow, C., Kleinmichel, H.: A new class of semismooth Newton method for nonlinear complementarity problems. Comput. Optim. Appl. 11, 227–251 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kanzow, C.: Some equation-based methods for the nonlinear complementarity problem. Optim. Method Softw. 3, 327–340 (1994)

    Article  Google Scholar 

  7. Chen, B.: Error bounds for R0-type and monotone nonlinear complementarity problems. J. Optim. Theory Appl. 108, 297–316 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, X.: Smoothing methods for complementarity problems and their applications: a survey. J. Oper. Res. Soc. Jpn. 43, 32–47 (2000)

    Article  MATH  Google Scholar 

  9. Mangasarian, O.L., Solodov, M.V.: A linearly convergent descent method for strongly monotone complementarity problems. Comput. Optim. Appl. 14, 5–16 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jiang, H.: Unconstrained minimization approaches to nonlinear complementarity problems. J. Global Optim. 9, 169–181 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Geiger, C., Kanzow, C.: On the resolution of monotone complementarity problems. Comput. Optim. Appl. 5, 155–173 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hu, S.L., Hang, Z.H., Chen, J.S.: Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems. J. Comput. Appl. Math. 230, 69–82 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, J.S., Gao, H.T., Pan, S.H.: An R-linearly convergent derivative-free algorithm for nonlinear complementarity problems based on the generalized Fischer–Burmeister merit function. J. Comput. Appl. Math. 232, 455–471 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Yamada, K., Yamashita, N., Fukushima, M.: A new derivative-free descent method for the nonlinear complementarity problems. In: Pillo, G.D., Giannessi, F. (eds.) Nonlinear Optimization and Related Topics, pp. 463–487. Kluwer Academic Publishers, Netherlands (2000)

    Google Scholar 

  15. Chen, B., Chen, X., Kanzow, C.: A penalized Fischer–Burmeister NCP-functions. Math. Program. 88, 211–216 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fischer, A.: Solution of monotone complementarity problems with Lipschitzian functions. Math. Program. 76, 513–532 (1997)

    MATH  Google Scholar 

  17. Chen, J.S., Pan, S.H.: A family of NCP-functions and a descent method for the nonlinear complementarity problem. Comput. Optim. Appl. 40, 389–404 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chen, J.S.: On some NCP-functions based on the generalized Fischer–Burmeister function. Asia-Pac. J. Oper. Res. 24, 401–420 (2007)

    Article  MathSciNet  Google Scholar 

  19. Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for Newton’s method. SIAM J. Numer. Anal. 23, 707–716 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Billups, S.C., Dirkse, S.P., Soares, M.C.: A comparison of algorithms for large scale mixed complementarity problems. Comput. Optim. Appl. 7, 3–25 (1977)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Xidian University, Xi’an, 710071, China

    Jianguang Zhu, Hongwei Liu & Changhe Liu

  2. Department of Applied Mathematics, Henan University of Science and Technology, Luoyang, China

    Changhe Liu

  3. School of Science, Xi’an University of Posts and Telecommunications, Xi’an, China

    Weijie Cong

Authors
  1. Jianguang Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hongwei Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Changhe Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Weijie Cong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianguang Zhu.

Additional information

This work is supported by the National Natural Science Foundation of China 61072144.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhu, J., Liu, H., Liu, C. et al. A nonmonotone derivative-free algorithm for nonlinear complementarity problems based on the new generalized penalized Fischer–Burmeister merit function. Numer Algor 58, 573–591 (2011). https://doi.org/10.1007/s11075-011-9471-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-011-9471-8

Keywords

Search

Navigation