IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v121y2004i1d10.1023_bjota.0000026128.34294.77.html
   My bibliography  Save this article

A Newton Method for Linear Programming

Author

Listed:
  • O. L. Mangasarian

    (University of Wisconsin)

Abstract

A fast Newton method is proposed for solving linear programs with a very large (≈106) number of constraints and a moderate (≈102) number of variables. Such linear programs occur in data mining and machine learning. The proposed method is based on the apparently overlooked fact that the dual of an asymptotic exterior penalty formulation of a linear program provides an exact least 2-norm solution to the dual of the linear program for finite values of the penalty parameter but not for the primal linear program. Solving the dual problem for a finite value of the penalty parameter yields an exact least 2-norm solution to the dual, but not a primal solution unless the parameter approaches zero. However, the exact least 2-norm solution to the dual problem can be used to generate an accurate primal solution if m≥n and the primal solution is unique. Utilizing these facts, a fast globally convergent finitely terminating Newton method is proposed. A simple prototype of the method is given in eleven lines of MATLAB code. Encouraging computational results are presented such as the solution of a linear program with two million constraints that could not be solved by CPLEX 6.5 on the same machine.

Suggested Citation

  • O. L. Mangasarian, 2004. "A Newton Method for Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 121(1), pages 1-18, April.
  • Handle: RePEc:spr:joptap:v:121:y:2004:i:1:d:10.1023_b:jota.0000026128.34294.77
    DOI: 10.1023/B:JOTA.0000026128.34294.77
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/B:JOTA.0000026128.34294.77
    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.0000026128.34294.77?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. M. Ç. Pinar, 1997. "Piecewise-Linear Pathways to the Optimal Solution Set in Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 93(3), pages 619-634, June.
    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. Luke Winternitz & Stacey Nicholls & André Tits & Dianne O’Leary, 2012. "A constraint-reduced variant of Mehrotra’s predictor-corrector algorithm," Computational Optimization and Applications, Springer, vol. 51(3), pages 1001-1036, April.
    2. Hossein Moosaei & Milan Hladík, 2021. "On the Optimal Correction of Infeasible Systems of Linear Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 32-55, July.
    3. Ketabchi, Saeed & Behboodi-Kahoo, Malihe, 2015. "Augmented Lagrangian method within L-shaped method for stochastic linear programs," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 12-20.
    4. Saeed Ketabchi & Hossein Moosaei, 2012. "Minimum Norm Solution to the Absolute Value Equation in the Convex Case," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 1080-1087, September.
    5. Hossein Moosaei & Saeed Ketabchi & Milan Hladík, 2021. "Optimal correction of the absolute value equations," Journal of Global Optimization, Springer, vol. 79(3), pages 645-667, March.

    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:121:y:2004:i:1:d:10.1023_b:jota.0000026128.34294.77. 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.