Packing ovals in optimized regular polygons

We present a model development framework and numerical solution approach to the general problem-class of packing convex objects into optimized convex containers. Specifically, we discuss the problem of packing ovals (egg-shaped objects, defined here as generalized ellipses) into optimized regular polygons in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbb{R}}<^>{2} $$\end{document}. Our solution strategy is based on the use of embedded Lagrange multipliers, followed by nonlinear optimization. Credible numerical results are attained using randomized starting solutions, refined by a single call to a local optimization solver. We obtain visibly good quality packings for packing 4 to 10 ovals into regular polygons with 3 to 10 sides in all 224 test problems presented here. Our modeling and solution approach can be extended towards handling other difficult packing problems.