Optimized ellipse packings in regular polygons

  • Frank J. Kampas
  • Ignacio CastilloEmail author
  • János D. Pintér
Original Paper


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.


General ellipse packings in regular polygons Model development using embedded Lagrange multipliers Global and local nonlinear optimization LGO solver suite Random starting points and local search by Ipopt Numerical results 



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Physicist at Large Consulting LLCBryn MawrUSA
  2. 2.Lazaridis School of Business and EconomicsWilfrid Laurier UniversityWaterlooCanada
  3. 3.Department of Industrial and Systems EngineeringLehigh UniversityBethlehemUSA

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