does not easily become trapped in local minima. 

Essentially, the AL method employed seeks to minimize ω = R 2 iteratively with an increasing penalty for overlap between spheres. This can be seen as beginning with N permeable spheres, squeezing them together within a spherical boundary of radius R , and then iteratively decreasing their permeability such that they force the boundary outward. The AL method also does not guarantee a global minimum of (A1), though for large enough values of γ in certain problems, γ > γ$_{0}$ , a local saddle point of (A2) is guaranteed to exist [31]. For more detailed information on the augmented Lagrangian and conjugate gradient methods, we refer the reader to one of the many texts on the subject of nonlinear programming, such as [31]. 

The local minimum of (A1) found by the AL method for large enough γ is dependent not only on the algorithm parameters but also strongly on the initial conditions for x and λ . Accordingly, the procedure we use to find global minima begins with a variety of initial conditions for x , from disks positioned randomly according to the Poisson distribution inside or on the surface of a disk of a larger radius to disks positioned with centers on the sites of the triangular lattice. After each iteration of the AL method, a subset of the disk centers arranged as the local minimum in x are moved radially inward by a random amount not exceeding their initial distance from the origin. Immediately after, these same disks are rotated through a random angle such that they are no farther than distance unity from their previous position. This ‘shuffled’ configuration is used as the initial conditions for the first Nd variables of x in the next AL iteration, while the final variable of x is set to the greatest squared distance from the origin to any of the N disk centers. The initial λ at every iteration is set to zero to help keep the AL algorithm from becoming stuck in a local minimum. 

In our trials, generally no more than 50 iterative AL and shuffling steps were necessary to find a DLP optimal packing, even for large numbers of disks (for small numbers of disks, as few as 1 or 10 steps was often sufficient). To support the conjecture that the minima found are indeed global minima, we repeated each procedure of 50 iterations as many as 20 times, changing the governing parameters of the AL algorithm and employing the different initial conditions discussed above. For the vast majority of N in the packings presented in this work, the same least local minima were found in all or the majority of the 20 repetitions performed. 

