Skip to main content
SpringerLink
Log in

A dual algorithm for minimax problems

  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this paper, a dual algorithm, based on a smoothing function of Bertsekas (1982), is established for solving unconstrained minimax problems. It is proven that a sequence of points, generated by solving a sequence of unconstrained minimizers of the smoothing function with changing parametert, converges with Q-superlinear rate to a Kuhn-Tucker point locally under some mild conditions. The relationship between the condition number of the Hessian matrix of the smoothing function and the parameter is studied, which also validates the convergence theory. Finally the numerical results are reported to show the effectiveness of this algorithm.

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. D. P. Bertsekas,Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, 1982.

    MATH  Google Scholar 

  2. C. Charalambous,Nonlinear least pth optimization and nonlinear programming, Math. Programming12 (1977), 195–225.

    Article  MATH  MathSciNet  Google Scholar 

  3. C. Charalambous,Acceleration of the least pth algorithm for minimax optimization with engineering applications, Math. Programming19 (1979), 270–297.

    Article  MathSciNet  Google Scholar 

  4. G. Dipillo, L. Grippo and S. Lucidi,A smooth method for the finite minimax problem, Math. Programming60 (1993), 187–214.

    Article  MathSciNet  Google Scholar 

  5. J. Hald and K. Madsen,Combined LP and quasi-Newton methods for minimax optimization, Math. Programming20 (1981), 49–62.

    Article  MATH  MathSciNet  Google Scholar 

  6. S. P. Han and O. L. Mangasarian,Exact penalty function in nonlinear programming, Math. Programming17 (1979), 251–269.

    Article  MATH  MathSciNet  Google Scholar 

  7. X. S. Li,An entropy-based aggregate method for minimax optimization, Engineering Optimization18 (1992), 277–285.

    Article  Google Scholar 

  8. X. S. Li and S. C. Fang,On the entropic method for solving min-max problems with applications, Mathematical Methods of Operations Research46 (1997), 119–130.

    Article  MATH  MathSciNet  Google Scholar 

  9. E. Polak, J. E. Higgins and D. Q. Mayne,A barrier function method for minimax problems, Math. Programming54 (1992), 155–176.

    Article  MATH  MathSciNet  Google Scholar 

  10. E. Polak, D. Q. Mayne and J. E. Higgins,Superlinearly convergent algorithm for min-max problems, Journal of Optimization Theory and Applications69 (1991), 407–439.

    Article  MATH  MathSciNet  Google Scholar 

  11. R. A. Polyak,Smooth optimization methods for minimax problems, SIAM J. Control and Optimization26 (1988), 1274–1286.

    Article  MATH  MathSciNet  Google Scholar 

  12. R. A. Polyak,Nonlinear rescaling in discrete minimax, Nonsmooth/Nonconvex Mechanics: Modelling, Analysis, Numerical Methods, D. Gao, R. Ogden, G. Stavroulakis (eds.), Kluwer Academic Publisher, 2000 (with I. Griva, J. Sobieski).

  13. J. B. Rosen and S. Suzuki,Construction of Non-linear Programming Test Problem. Communication of the Association for Computing Machinery, 1965.

  14. A. B. Templeman and X. S. Li,A maximum entropy approach to constrained nonlinear programming, Engineering Optimization12 (1987), 191–205.

    Article  Google Scholar 

  15. J. H. Wilkinson,The algebraic eigenvalue problem, Oxford, Clarendon, 1965.

    MATH  Google Scholar 

  16. R. S. Womersley and R. Fletcher,An algorithm for composite nonsmooth optimization problems, Journal of Optimization Theory and Applications48 (1986), 493–523.

    Article  MATH  MathSciNet  Google Scholar 

  17. L. W. Zhang and H. W. Tang,A maximum entropy algorithm with parameters for solving minimax problem, Archives of Control Sciences6 (1997), 47–59.

    MATH  MathSciNet  Google Scholar 

  18. L. W. Zhang and S. X. He,The convergence of a dual algorithm for nonlinear programming, Korean J. Comput. & Appl. Math.7 (2000), 487–506.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Science, Wuhan University of Technology, 430070, Wuhan, P. R. China

    Suxiang He

Authors
  1. Suxiang He
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Suxiang He.

Additional information

Supported by the Doctorial Science Research Foundation of WHUT.

Suxiang He received her Ph. D at Dalian University of Technology in 2002. Since 2002, she has been studying and working at Wuhan University of Technology. Her research interests center on the theory of nonlinear programming and related application.

Rights and permissions

Reprints and permissions

About this article

Cite this article

He, S. A dual algorithm for minimax problems. JAMC 17, 401–418 (2005). https://doi.org/10.1007/BF02936065

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02936065

AMS Mathematics Subject Classification

Key words and phrases

Search

Navigation