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Finding largest small polygons with GloptiPoly

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  • Didier Henrion
  • Frédéric Messine

Abstract

A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices n. Many instances are already solved in the literature, namely for all odd n, and for n = 4, 6 and 8. Thus, for even n ≥ 10, instances of this problem remain open. Finding those largest small polygons can be formulated as nonconvex quadratic programming problems which can challenge state-of-the-art global optimization algorithms. We show that a recently developed technique for global polynomial optimization, based on a semidefinite programming approach to the generalized problem of moments and implemented in the public-domain Matlab package GloptiPoly, can successfully find largest small polygons for n = 10 and n = 12. Therefore this significantly improves existing results in the domain. When coupled with accurate convex conic solvers, GloptiPoly can provide numerical guarantees of global optimality, as well as rigorous guarantees relying on interval arithmetic. Copyright Springer Science+Business Media, LLC. 2013

Suggested Citation

  • Didier Henrion & Frédéric Messine, 2013. "Finding largest small polygons with GloptiPoly," Journal of Global Optimization, Springer, vol. 56(3), pages 1017-1028, July.
    • Handle: RePEc:spr:jglopt:v:56:y:2013:i:3:p:1017-1028
    DOI: 10.1007/s10898-011-9818-7
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    References listed on IDEAS

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    1. Klerk, Etienne de, 2010. "Exploiting special structure in semidefinite programming: A survey of theory and applications," European Journal of Operational Research, Elsevier, vol. 201(1), pages 1-10, February.
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    Cited by:

    1. 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.
    2. 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.

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