This work discusses the following general packing problem-class: given a finite collection of d-dimensional spheres with arbitrarily chosen radii, find the smallest sphere in R^d that contains the entire collection of these spheres in a non-overlapping arrangement. Generally speaking, analytical solution approaches cannot be expected to apply to this general problem-type, except for very small or certain specially structured sphere configurations. In order to find high-quality numerical (approximate) solutions, we propose a suitable combination of heuristic strategies with constrained global and local nonlinear optimization. We present numerical results for non-trivial model instance-classes of optimized sphere configurations with up to n = 50 spheres in dimensions d = 2,3,4. Our numerical results for an intensively studied model-class in R^2 are on average within 1% of the entire set of best known results, with new optimized (conjectured) packings for previously unexplored generalizations of the same model-class in R^d with d = 3,4. The results obtained also support the estimation of the optimized container sphere radii and of the packing fraction as functions of the model instance parameters n and 1/n, respectively.
Globally Optimized Finite Packings of Arbitrary Size Spheres in R^d. Technical report, submitted for publication.
View Globally Optimized Finite Packings of Arbitrary Size Spheres in R^d