IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v119y2003i3d10.1023_bjota.0000006685.60019.3e.html
   My bibliography  Save this article

Algorithms with Adaptive Smoothing for Finite Minimax Problems

Author

Listed:
  • E. Polak

    (University of California)

  • J. O. Royset

    (University of California)

  • R. S. Womersley

    (University of New South Wales)

Abstract

We present a new feedback precision-adjustment rule for use with a smoothing technique and standard unconstrained minimization algorithms in the solution of finite minimax problems. Initially, the feedback rule keeps a precision parameter low, but allows it to grow as the number of iterations of the resulting algorithm goes to infinity. Consequently, the ill-conditioning usually associated with large precision parameters is considerably reduced, resulting in more efficient solution of finite minimax problems. The resulting algorithms are very simple to implement, and therefore are particularly suitable for use in situations where one cannot justify the investment of time needed to retrieve a specialized minimax code, install it on one's platform, learn how to use it, and convert data from other formats. Our numerical tests show that the algorithms are robust and quite effective, and that their performance is comparable to or better than that of other algorithms available in the Matlab environment.

Suggested Citation

  • E. Polak & J. O. Royset & R. S. Womersley, 2003. "Algorithms with Adaptive Smoothing for Finite Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 119(3), pages 459-484, December.
    • Handle: RePEc:spr:joptap:v:119:y:2003:i:3:d:10.1023_b:jota.0000006685.60019.3e
    DOI: 10.1023/B:JOTA.0000006685.60019.3e
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/B:JOTA.0000006685.60019.3e
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1023/B:JOTA.0000006685.60019.3e?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. F. Guerra Vazquez & H. Günzel & H.Th. Jongen, 2001. "On Logarithmic Smoothing of the Maximum Function," Annals of Operations Research, Springer, vol. 101(1), pages 209-220, January.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Rockafellar, R.T. & Royset, J.O., 2010. "On buffered failure probability in design and optimization of structures," Reliability Engineering and System Safety, Elsevier, vol. 95(5), pages 499-510.
    2. Helene Krieg & Tobias Seidel & Jan Schwientek & Karl-Heinz Küfer, 2022. "Solving continuous set covering problems by means of semi-infinite optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(1), pages 39-82, August.
    3. Juan Liang & Linke Hou & Xiaowu Li & Feng Pan & Taixia Cheng & Lin Wang, 2018. "Hybrid Second Order Method for Orthogonal Projection onto Parametric Curve in n -Dimensional Euclidean Space," Mathematics, MDPI, vol. 6(12), pages 1-23, December.
    4. B. Rustem & S. Žaković & P. Parpas, 2008. "Convergence of an Interior Point Algorithm for Continuous Minimax," Journal of Optimization Theory and Applications, Springer, vol. 136(1), pages 87-103, January.
    5. E. Y. Pee & J. O. Royset, 2011. "On Solving Large-Scale Finite Minimax Problems Using Exponential Smoothing," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 390-421, February.
    6. Zhengyong Zhou & Xiaoyang Dai, 2023. "An active set strategy to address the ill-conditioning of smoothing methods for solving finite linear minimax problems," Journal of Global Optimization, Springer, vol. 85(2), pages 421-439, February.
    7. Mohamed A. Tawhid & Ahmed F. Ali, 2016. "Simplex particle swarm optimization with arithmetical crossover for solving global optimization problems," OPSEARCH, Springer;Operational Research Society of India, vol. 53(4), pages 705-740, December.
    8. W. Hare & J. Nutini, 2013. "A derivative-free approximate gradient sampling algorithm for finite minimax problems," Computational Optimization and Applications, Springer, vol. 56(1), pages 1-38, September.
    9. E. Polak & J. O. Royset, 2003. "Algorithms for Finite and Semi-Infinite Min–Max–Min Problems Using Adaptive Smoothing Techniques," Journal of Optimization Theory and Applications, Springer, vol. 119(3), pages 421-457, December.
    10. Junxiang Li & Mingsong Cheng & Bo Yu & Shuting Zhang, 2015. "Group Update Method for Sparse Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 166(1), pages 257-277, July.
    11. Yi Chen & David Gao, 2016. "Global solutions to nonconvex optimization of 4th-order polynomial and log-sum-exp functions," Journal of Global Optimization, Springer, vol. 64(3), pages 417-431, March.
    12. Ryan Alimo & Pooriya Beyhaghi & Thomas R. Bewley, 2020. "Delaunay-based derivative-free optimization via global surrogates. Part III: nonconvex constraints," Journal of Global Optimization, Springer, vol. 77(4), pages 743-776, August.
    13. E. Polak & R. S. Womersley & H. X. Yin, 2008. "An Algorithm Based on Active Sets and Smoothing for Discretized Semi-Infinite Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 138(2), pages 311-328, August.
    14. J. O. Royset & E. Y. Pee, 2012. "Rate of Convergence Analysis of Discretization and Smoothing Algorithms for Semiinfinite Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 855-882, December.
    15. Johannes Royset, 2013. "On sample size control in sample average approximations for solving smooth stochastic programs," Computational Optimization and Applications, Springer, vol. 55(2), pages 265-309, June.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.

      Corrections

      All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:119:y:2003:i:3:d:10.1023_b:jota.0000006685.60019.3e. See general information about how to correct material in RePEc.

      If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

      If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

      If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

      For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

      Please note that corrections may take a couple of weeks to filter through the various RePEc services.

      IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.