Damped Arrow–Hurwicz algorithm for sphere packing

We consider algorithms that, from an arbitrarily sampling of N spheres (possibly overlapping), find a close packed configuration without overlapping. These problems can be formulated as minimization problems with non-convex constraints. For such packing problems, we observe that the classical iterative Arrow–Hurwicz algorithm does not converge. We derive a novel algorithm from a multi-step variant of the Arrow–Hurwicz scheme with damping. We compare this algorithm with classical algorithms belonging to the class of linearly constrained Lagrangian methods and show that it performs better. We provide an analysis of the convergence of these algorithms in the simple case of two spheres in one spatial dimension. Finally, we investigate the behaviour of our algorithm when the number of spheres is large in two and three spatial dimensions.