Out-of-the-Box Global Optimization for Packing Problems: New Models and Improved Solutions

Recent LLM-driven discoveries have renewed interest in geometric packing problems. In this paper, we study several classes of such packing problems through the lens of modern global nonlinear optimization. Starting from comparatively direct nonlinear formulations, we consider packing circles in squares and fixed-perimeter rectangles, packing circles into minimum-area ellipses, packing regular polygons into regular polygons, and packing Platonic solids into Platonic solids. For ellipse packing, we derive a novel containment formulation based on the S-lemma. For polygon and Platonic solid packing, we develop compact non-overlap formulations based on the Farkas lemma and study several natural modeling variants computationally. Using off-the-shelf global optimization solvers, namely FICO Xpress and SCIP, we obtain numerous new incumbent solutions as well as first solutions for previously unstudied variants without any problem-specific solver modifications beyond writing down the models. Our computational results analyze the impact of various formulation choices. Beyond the individual packing results, the paper illustrates a broader point. It provides further evidence that global nonlinear optimization has matured into an increasingly practical model-and-solve technology for highly nonconvex problems.