Skip to main content
SpringerLink
Log in

Global linear and quadratic one-step smoothing newton method for vertical linear complementarity problems

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

A one-step smoothing Newton method is proposed for solving the vertical linear complementarity problem based on the so-called aggregation function. The proposed algorithm has the following good features: (i) It solves only one linear system of equations and does only one line search at each iteration; (ii) It is well-defined for the vertical linear complementarity problem with vertical block P 0 matrix and any accumulation point of iteration sequence is its solution. Moreover, the iteration sequence is bounded for the vertical linear complementarity problem with vertical block P 0 +R 0 matrix; (iii) It has both global linear and local quadratic convergence without strict complementarity. Many existing smoothing Newton methods do not have the property (iii).

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.

Similar content being viewed by others

References

  1. Cottle R W, Pang J S, Stone R W.The Linear Complementarity Problem[M]. Boston: Academic Press, 1992.

    Google Scholar 

  2. Cottle R W, Dantzig G B. A generalization of the linear complementarity problem[J].Journal of Combinatorial Theory, 1970,8(1):79–90.

    MATH  MathSciNet  Google Scholar 

  3. Sun M. Monotonicity of mangsarian's iterative algorithm for generalized linear complementarity problems[J].J Math Anal Appl, 1989,144(2):474–485.

    Article  MATH  MathSciNet  Google Scholar 

  4. Gowda M S, Sznajder R. A generalization of the Nash equilibrium theorem on bimatrix games[J].Internat J Game Theory, 1996,25(1):1–12.

    Article  MATH  MathSciNet  Google Scholar 

  5. Ebiefung A A, Kostreva M M. The generalized Leontief input-output model and its application to the choice of the new technology[J].Ann Oper Res, 1993,44(1):161–172.

    Article  MATH  MathSciNet  Google Scholar 

  6. Fujisawa T, Kuh E S. Piecewise-linear theory of nonlinear networks[J].SIAM J Appl Math, 1972,22(1):307–328.

    Article  MATH  MathSciNet  Google Scholar 

  7. Peng J M, Lin Z H. A noninterior continuation method for generalized linear complementarity problems[J].Math Programming, 1999,86(2):533–563.

    Article  MATH  MathSciNet  Google Scholar 

  8. Qi H, Liao L Z. A smoothing Newton method for extended vertical linear complementarity problems [J].SIAM J Matrix Anal Appl, 1999,21(1):45–66.

    Article  MATH  MathSciNet  Google Scholar 

  9. Billups S C, Dirkse S P, Ferris M C. A Comparison of algorithms for large-scale mixed complementarity problems[J].Computational Optimization and Applications, 1997,7(1):3–25.

    Article  MATH  MathSciNet  Google Scholar 

  10. Qi L, Sun D, Zhou G. A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequality problems[J].Math Programming 2000,87 (1):1–35.

    MATH  MathSciNet  Google Scholar 

  11. Zhou G, Sun D, Qi L. Numerical experiences for a class of squared smoothing Newton methods for box constrained variational inequality problems[A]. In: Fukushima M, Qi L Eds.Reformulation-Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods[C]. Boston: Kluwer Academic Publishers, 1998,421–441.

    Google Scholar 

  12. Qi H. A regularized smoothing Newton method for box constrained variational inequality problems withP 0-functions[J].SIAM Journal on Optimization, 2000,10(1):315–330.

    MathSciNet  Google Scholar 

  13. Sun D. A regularization Newton method for solving nonlinear complementarity problems[J].Applied Mathematics and Optimization, 1999,35(1):315–339.

    Article  Google Scholar 

  14. Chen X, Qi L, Sun D. Global and superlinear convergence of the smoothing Newton method and its application to general box-constrained variational inequalities[J].Mathematics of Computation, 1998,67(2):519–540.

    Article  MATH  MathSciNet  Google Scholar 

  15. Kanzow C, Pieper H. Jacobian smoothing methods for nonlinear complementarity problems[J].SIAM Journal on Optimization, 1999,9(1):342–373.

    Article  MATH  MathSciNet  Google Scholar 

  16. LI X S. An aggregate function method for nonlinear programming[J].Science in China, Ser A, 1991,34(3):1467–1473.

    MATH  Google Scholar 

  17. Qi L, Sun J. A nonsmooth version of Newton's method[J].Math Programming, 1993,58(1): 353–367.

    Article  MATH  MathSciNet  Google Scholar 

  18. Qi L. Convergence analysis of some algorithms for solving nonsmooth equations[J].Mathematics of Operations Research, 1993,18(1):227–244.

    Article  MATH  MathSciNet  Google Scholar 

  19. Chen B, Xiu N. Superlinear noninterior one-step continuation method for monotone LCP in absence of strict complementarity[J].Journal of Optimization Theory and Applications, 2001,108(1):317–332.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Methematical Sciences, Tsinghua University, 100084, Beijing, P. R. China

    Zhang Li-ping Doctor (Lecturer)

  2. School of Traffic and Transportation, Northern Jiaotong University, 100044, Beijing, P. R. China

    Gao Zi-you

Authors
  1. Zhang Li-ping Doctor
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gao Zi-you
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by Zuphang Shi-sheng

Foundation items: the National Natural Science Foundation of China (10201001); the National Outstanding Young Investigator Grant (70225005)

Biography: Zuphang Li-ping (1970≈)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li-ping, Z., Zi-you, G. Global linear and quadratic one-step smoothing newton method for vertical linear complementarity problems. Appl Math Mech 24, 738–746 (2003). https://doi.org/10.1007/BF02437876

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02437876

Key words

Chinese Library Classification

2000 MR Subject Classification

Search

Navigation