Disk Packing in a Square: A New Global Optimization Approach

We present a new computational approach to the problem of placing n identical non overlapping disks in the unit square in such a way that their radius is maximized. The problem has been studied in a large number of papers, both from a theoretical and from a computational point of view. In this paper we conjecture that the problem possesses a so-called funneling landscape, a feature which is commonly found in molecular conformation problems. Based upon this conjecture we develop a stochastic search algorithm which displays excellent numerical performance. Thanks to this algorithm we could improve over previously known putative optima in the range n ≤ 130 in as many as 32 instances, the smallest of which is n = 53. First experiments in the related problem of packing equal spheres in the unit cube led us to an improvement for n = 28 spheres.