IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v287y2020i2d10.1007_s10479-019-03186-2.html
   My bibliography  Save this article

A linear programming primer: from Fourier to Karmarkar

Author

Listed:
  • Atlanta Chakraborty

    (Indian Institute of Science)

  • Vijay Chandru

    (Indian Institute of Science)

  • M. R. Rao

    (Indian School of Business)

Abstract

The story of linear programming is one with all the elements of a grand historical drama. The original idea of testing if a polyhedron is non-empty by using a variable elimination to project down one dimension at a time until a tautology emerges dates back to a paper by Fourier in 1823. This gets re-invented in the 1930s by Motzkin. The real interest in linear programming happens during World War II when mathematicians ponder best ways of utilising resources at a time when they are constrained. The problem of optimising a linear function over a set of linear inequalities becomes the focus of the effort. Dantzig’s Simplex Method is announced and the Rand Corporation becomes a hot bed of computational mathematics. The range of applications of this modelling approach grows and the powerful machinery of numerical analysis and numerical linear algebra becomes a major driver for the advancement of computing machines. In the 1970s, constructs of theoretical computer science indicate that linear programming may in fact define the frontier of tractable problems that can be solved effectively on large instances. This raised a series of questions and answers: Is the Simplex Method a polynomial-time method and if not can we construct novel polynomial time methods, etc. And that is how the Ellipsoid Method from the Soviet Union and the Interior Point Method from Bell Labs make their way into this story as the heroics of Khachiyan and Karmarkar. We have called this paper a primer on linear programming since it only gives the reader a quick narrative of the grand historical drama. Hopefully it motivates a young reader to delve deeper and add another chapter.

Suggested Citation

  • Atlanta Chakraborty & Vijay Chandru & M. R. Rao, 2020. "A linear programming primer: from Fourier to Karmarkar," Annals of Operations Research, Springer, vol. 287(2), pages 593-616, April.
    • Handle: RePEc:spr:annopr:v:287:y:2020:i:2:d:10.1007_s10479-019-03186-2
    DOI: 10.1007/s10479-019-03186-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-019-03186-2
    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/s10479-019-03186-2?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. Robert E. Bixby, 1994. "Commentary—Progress in Linear Programming," INFORMS Journal on Computing, INFORMS, vol. 6(1), pages 15-22, February.
    2. Michael J. Todd, 1994. "Commentary—Theory and Practice for Interior-Point Methods," INFORMS Journal on Computing, INFORMS, vol. 6(1), pages 28-31, February.
    3. Nimrod Megiddo, 1991. "On Finding Primal- and Dual-Optimal Bases," INFORMS Journal on Computing, INFORMS, vol. 3(1), pages 63-65, February.
    4. I. Adler & S. Cosares, 1991. "A Strongly Polynomial Algorithm for a Special Class of Linear Programs," Operations Research, INFORMS, vol. 39(6), pages 955-960, 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. In Lee, 2020. "Internet of Things (IoT) Cybersecurity: Literature Review and IoT Cyber Risk Management," Future Internet, MDPI, vol. 12(9), pages 1-21, September.

    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. Erling D. Andersen, 1999. "On Exploiting Problem Structure in a Basis Identification Procedure for Linear Programming," INFORMS Journal on Computing, INFORMS, vol. 11(1), pages 95-103, February.
    2. Samuel Bismuth & Ivan Bliznets & Erel Segal-Halevi, 2019. "Fair Division with Bounded Sharing: Binary and Non-Degenerate Valuations," Papers 1912.00459, arXiv.org, revised Jul 2024.
    3. Alper Atamtürk & Andrés Gómez, 2017. "Maximizing a Class of Utility Functions Over the Vertices of a Polytope," Operations Research, INFORMS, vol. 65(2), pages 433-445, March-Apr.
    4. Yann Briheche & Frederic Barbaresco & Fouad Bennis & Damien Chablat, 2018. "Theoretical Complexity of Grid Cover Problems Used in Radar Applications," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 1086-1106, December.
    5. Yinyu Ye, 2011. "The Simplex and Policy-Iteration Methods Are Strongly Polynomial for the Markov Decision Problem with a Fixed Discount Rate," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 593-603, November.
    6. Robert E. Bixby, 2002. "Solving Real-World Linear Programs: A Decade and More of Progress," Operations Research, INFORMS, vol. 50(1), pages 3-15, February.
    7. László A. Végh, 2017. "A Strongly Polynomial Algorithm for Generalized Flow Maximization," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 179-211, January.

    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:annopr:v:287:y:2020:i:2:d:10.1007_s10479-019-03186-2. 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.