General Ellipse Packings in an Optimized Circle Using Embedded Lagrange Multipliers

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.