Packing Multidimensional Spheres in an Optimized Hyperbolic Container

The problem of packing multidimensional spheres in a container defined by a hyperbolic surface is introduced. The objective is to minimize the height of the hyperbolic container under non-overlapping and containment conditions for the spheres, considering minimal allowable distances between them. To the best of our knowledge, no mathematical models addressing optimized packing spheres in hyperbolic containers have been proposed before. Our approach is based on a space dimensionality reduction transformation. This transformation relies on projecting a multidimensional hyperboloid into a lower-dimensional space sequentially up to two-dimensional case. Employing the phi-function technique, packing spheres in the hyperbolic container is formulated as a nonlinear programming problem. The latter is solved using a model-based heuristic combined with a decomposition approach. Numerical results are presented for a wide range of parameters, i.e., space dimension, number of spheres, and metric characteristics of the hyperbolic container. The results demonstrate efficiency of the proposed modeling and solution approach highlighting new opportunities for packing problems within non-traditional geometries.