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General Ellipse Packings in an Optimized Circle Using Embedded Lagrange Multipliers

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

Abstract: The general ellipse packing problem is to find a non-overlapping arrangement of 𝑛 ellipses with (in principle) arbitrary size and orientation parameters inside a given type of container set. Here we consider the general ellipse packing problem with respect to an optimized circle container with minimal radius. Following the review of selected topical literature, we introduce a new model formulation approach based on using embedded Lagrange multipliers. This optimization model is implemented using the computing system Mathematica: we present illustrative numerical results using the LGO global-local optimization software package linked to Mathematica. Our study demonstrates the applicability of the embedded Lagrange multipliers based modeling approach combined with global optimization tools to solve challenging ellipse packing problems.

Keywords: General ellipse packing problem; Optimized circular container; Model formulation in Mathematica using embedded Lagrange multipliers; Numerical global optimization; LGO global-local optimization software linked to Mathematica; Illustrative numerical results.

Category 1: Global Optimization

Category 2: Optimization Software and Modeling Systems

Category 3: Applications -- Science and Engineering

Citation: Kampas, F.J., Pinter, J.D., Castillo, I., General Ellipse Packings in an Optimized Circle Using Embedded Lagrange Multipliers. (Submitted for publication January 2016)

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Entry Submitted: 01/19/2016
Entry Accepted: 01/19/2016
Entry Last Modified: 01/19/2016

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