IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v81y2021i1d10.1007_s10898-020-00908-w.html
   My bibliography  Save this article

Using symbolic calculations to determine largest small polygons

Author

Listed:
  • Charles Audet

    (Polytechnique Montréal)

  • Pierre Hansen

    (HEC Montréal)

  • Dragutin Svrtan

    (University of Zagreb)

Abstract

A small polygon is a polygon of unit diameter. The question of finding the largest area of small n-gons has been answered for some values of n. Regular n-gons are optimal when n is odd and kites with unit length diagonals are optimal when $$n=4$$ n = 4 . For $$n=6$$ n = 6 , the largest area is a root of a degree 10 polynomial with integer coefficients and height 221360 (the height of a polynomial is the largest coefficient in absolute value). The present paper analyses the and octogonal cases, and under an axial symmetry conjecture, we propose a methodology that leads to a polynomial of degree 344 with integer coefficients that factorizes into a polynomial of degree 42 with height 23588130061203336356460301369344. A root of this last polynomial corresponds to the area of the largest small axially symmetrical octagon.

Suggested Citation

  • Charles Audet & Pierre Hansen & Dragutin Svrtan, 2021. "Using symbolic calculations to determine largest small polygons," Journal of Global Optimization, Springer, vol. 81(1), pages 261-268, September.
    • Handle: RePEc:spr:jglopt:v:81:y:2021:i:1:d:10.1007_s10898-020-00908-w
    DOI: 10.1007/s10898-020-00908-w
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-020-00908-w
    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/s10898-020-00908-w?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. Didier Henrion & Frédéric Messine, 2013. "Finding largest small polygons with GloptiPoly," Journal of Global Optimization, Springer, vol. 56(3), pages 1017-1028, July.
    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. Charles Audet & Frédéric Messine & Jordan Ninin, 2022. "Numerical certification of Pareto optimality for biobjective nonlinear problems," Journal of Global Optimization, Springer, vol. 83(4), pages 891-908, August.

    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. Liangyu Chen & Yaochen Xu & Zhenbing Zeng, 2017. "Searching approximate global optimal Heilbronn configurations of nine points in the unit square via GPGPU computing," Journal of Global Optimization, Springer, vol. 68(1), pages 147-167, May.

    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:jglopt:v:81:y:2021:i:1:d:10.1007_s10898-020-00908-w. 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.