IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v136y2008i1d10.1007_s10957-007-9290-1.html
   My bibliography  Save this article

Convergence of an Interior Point Algorithm for Continuous Minimax

Author

Listed:
  • B. Rustem

    (Imperial College)

  • S. Žaković

    (Imperial College)

  • P. Parpas

    (Imperial College)

Abstract

We propose an algorithm for the constrained continuous minimax problem. The algorithm uses a quasi-Newton search direction, based on subgradient information, conditional on maximizers. The initial problem is transformed to an equivalent equality constrained problem, where the logarithmic barrier function is used to ensure feasibility. In the case of multiple maximizers, the algorithm adopts semi-infinite programming iterations toward epiconvergence. Satisfaction of the equality constraints is ensured by an adaptive quadratic penalty function. The algorithm is augmented by a discrete minimax procedure to compute the semi-infinite programming steps and ensure overall progress when required by the adaptive penalty procedure. Progress toward the solution is maintained using merit functions.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:136:y:2008:i:1:d:10.1007_s10957-007-9290-1
    DOI: 10.1007/s10957-007-9290-1
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-007-9290-1
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-007-9290-1?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. I. Akrotirianakis & B. Rustem, 2005. "Globally Convergent Interior-Point Algorithm for Nonlinear Programming," Journal of Optimization Theory and Applications, Springer, vol. 125(3), pages 497-521, June.
    2. 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.
    3. 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.
    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. P. Parpas & B. Rustem, 2009. "An Algorithm for the Global Optimization of a Class of Continuous Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 141(2), pages 461-473, May.
    2. Vicente J. Bolós & Rafael Benítez & Vicente Coll-Serrano, 2023. "Continuous models combining slacks-based measures of efficiency and super-efficiency," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 31(2), pages 363-391, 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.
    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. Hui-juan Xiong & Bo Yu, 2010. "An aggregate deformation homotopy method for min-max-min problems with max-min constraints," Computational Optimization and Applications, Springer, vol. 47(3), pages 501-527, November.
    3. 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.
    4. E. Obasanjo & G. Tzallas-Regas & B. Rustem, 2010. "An Interior-Point Algorithm for Nonlinear Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 144(2), pages 291-318, February.
    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. 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.
    7. Adil Bagirov & Asef Ganjehlou, 2008. "An approximate subgradient algorithm for unconstrained nonsmooth, nonconvex optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(2), pages 187-206, April.
    8. 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.
    9. 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.
    10. Yan-Chao Liang & Gui-Hua Lin, 2012. "Stationarity Conditions and Their Reformulations for Mathematical Programs with Vertical Complementarity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 54-70, July.
    11. 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.
    12. Johannes O. Royset, 2016. "Preface," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 713-718, June.
    13. 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.
    14. Alfred Auslender & Miguel A. Goberna & Marco A. López, 2009. "Penalty and Smoothing Methods for Convex Semi-Infinite Programming," Mathematics of Operations Research, INFORMS, vol. 34(2), pages 303-319, May.
    15. Z. Akbari & R. Yousefpour & M. Reza Peyghami, 2015. "A New Nonsmooth Trust Region Algorithm for Locally Lipschitz Unconstrained Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 733-754, March.
    16. 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.
    17. Hatim Djelassi & Alexander Mitsos, 2021. "Global Solution of Semi-infinite Programs with Existence Constraints," Journal of Optimization Theory and Applications, Springer, vol. 188(3), pages 863-881, March.
    18. 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.
    19. A. M. Bagirov & B. Karasözen & M. Sezer, 2008. "Discrete Gradient Method: Derivative-Free Method for Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 137(2), pages 317-334, May.
    20. Zhou Sheng & Gonglin Yuan, 2018. "An effective adaptive trust region algorithm for nonsmooth minimization," Computational Optimization and Applications, Springer, vol. 71(1), pages 251-271, September.

    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:136:y:2008:i:1:d:10.1007_s10957-007-9290-1. 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.