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Optimized ellipse packings in regular polygons

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Abstract

We present model development and numerical solution approaches to the problem of packing a general set of ellipses without overlaps into an optimized polygon. Specifically, for a given set of ellipses, and a chosen integer m ≥ 3, we minimize the apothem of the regular m-polygon container. Our modeling and solution strategy is based on the concept of embedded Lagrange multipliers. To solve models with up to n ≤ 10 ellipses, we use the LGO solver suite for global–local nonlinear optimization. In order to reduce increasing runtimes, for model instances with 10 ≤ n ≤ 20 ellipses, we apply local search launching the Ipopt solver from selected random starting points. The numerical results demonstrate the applicability of our modeling and optimization approach to a broad class of highly non-convex ellipse packing problems, by consistently returning good quality feasible solutions in all (231) illustrative model instances considered here.

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Authors and Affiliations

  1. Physicist at Large Consulting LLC, Bryn Mawr, PA, USA

    Frank J. Kampas

  2. Lazaridis School of Business and Economics, Wilfrid Laurier University, Waterloo, ON, Canada

    Ignacio Castillo

  3. Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, USA

    János D. Pintér

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  1. Frank J. Kampas
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  2. Ignacio Castillo
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  3. János D. Pintér
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Correspondence to Ignacio Castillo.

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Kampas, F.J., Castillo, I. & Pintér, J.D. Optimized ellipse packings in regular polygons. Optim Lett 13, 1583–1613 (2019). https://doi.org/10.1007/s11590-019-01423-y

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