  We study several versions of the alternating direction method of multipliers
(ADMM) for solving the convex problem of finding the distance between two
ellipsoids and the nonconvex problem of finding the distance between the
boundaries of two ellipsoids. In the convex case we present the ADMM with and
without automatic penalty updates and demonstrate via numerical experiments on
problems of various dimensions that our methods significantly outperform all
other existing methods for finding the distance between ellipsoids. In the
nonconvex case we propose a heuristic rule for updating the penalty parameter
and a heuristic restarting procedure (a heuristic choice of a new starting for
point for the second run of the algorithm). The restarting procedure was
verified numerically with the use of a global method based on KKT optimality
conditions. The results of numerical experiments on various test problems
showed that this procedure always allows one to find a globally optimal
solution in the nonconvex case. Furthermore, the numerical experiments also
demonstrated that our version of the ADMM significantly outperforms existing
methods for finding the distance between the boundaries of ellipsoids on
problems of moderate and high dimensions.
