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constraints, requires that none of the centers of the spheres move within distance unity of the origin, i.e., that none of the N spheres overlap the fixed central sphere. The second category, also containing N constraints, requires that the difference between the squared distance from the origin to the centers of each of the N spheres and the objective function, simply the independent variable ω , be less than or equal to zero. Since ω is minimized, this condition sets ω equal to R $^{2}$, i.e., the greatest of the squared distances from the origin to the any of the N sphere centers. The third category, containing $_{N}$C$_{2}$ constraints, requires that none of the N spheres overlap. 

An augmented Lagrangian (AL) method is employed to attempt solutions to the problem A1. In brief, an AL method is an iterative process designed to minimize a function that is the Lagrangian of (A1) augmented with a quadratic penalty function. The augmented Lagrangian is written, 

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where the { λ$_{l}$ } are the Lagrange multipliers, γ is the penalty parameter (often denoted by ρ ), and the c$_{l}$ ( x ) (often denoted by g$_{l}$ ( x )) are the m = 2 N + $_{N}$C$_{2}$ inequality constraints. The function (A2) is minimized iteratively over x , yielding ˆ x$_{p}$ a local minimum of L$_{γ}$ ( x , λ$_{p}$ ), with λ$_{p}$ fixed. For each successive ˆ x$_{p}$ that is found, a new estimate for λ$_{p}$ is made based on the violated constraints, i.e., the constraints in (A1) with positive values. Concurrently, the penalty parameter γ is increased by a pre-specified multiple if the cumulative squared violation, or total error, is not a set amount smaller than the previous total error. Eventually, γ reaches a value such that the total error is smaller than a specified (small) tolerance, at which point the algorithm terminates. 

In the version of the AL method we use to find putative DLP optimal packings, the function L$_{γ}$ ( x , λ ) in each iteration is minimized using a conjugate gradient method with a directional minimizing algorithm that employs cubic interpolation. Though the conjugate gradient method can only guarantee a global minimum when the function to be minimized is quadratic, as L$_{γ}$ ( x , λ ) is quartic in x , the method is nonetheless efficient. It is important to note also that a directional minimizing algorithm employing cubic interpolation does not produce step sizes that would guarantee a global minimum through the conjugate gradient method even for a quadratic function; this feature is key in producing an AL algorithm that 

