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Optimized Ellipse Packings in Regular Polygons Using Embedded Lagrange Multipliers

Frank J. Kampas (frank***at***physicistatlarge.com)
Ignacio Castillo (icastillo***at***wlu.ca)
Janos D. Pinter (Janos.D.Pinter***at***gmail.com)

Abstract: In this work, we present model development and numerical solution approaches to the general problem of packing a collection of ellipses into an optimized regular polygon. Our modeling and solution strategy is based on the concept of embedded Lagrange multipliers. This concept is applicable to a wide range of optimization problems in which explicit analytical expressions for the objective function and/or constraints are not available. Within this Lagrangian setting, we aim at minimizing the apothem of the regular polygon while preventing ellipse overlaps: our solution strategy proceeds towards meeting these two objectives simultaneously. To solve the ellipse packing models, we use the LGO solver system for global-local nonlinear optimization; for larger model instances, we use a nave combination of pure random start and local search. The numerical results presented demonstrate the applicability of our modeling and optimization approach to a broad class of difficult, highly non-convex ellipse packing problems, by consistently providing high quality feasible solutions in all model instances considered.

Keywords: General Ellipse Packings in Polygons; Model Development Using Embedded Lagrange Multipliers; Global and Local Nonlinear Optimization; LGO Solver Suite; Numerical Results

Category 1: Global Optimization

Category 2: Optimization Software and Modeling Systems

Category 3: Global Optimization (Applications )

Citation: Research Report, January 2018. Submitted for publication.

Download: [PDF]

Entry Submitted: 02/22/2016
Entry Accepted: 03/01/2016
Entry Last Modified: 01/15/2018

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