Computing the Signed Distance Between Overlapping Ellipsoids

Computing the signed distance between two ellipsoids is a convex optimization problem when the two ellipsoids have no intersection, but it becomes nonconvex when the ellipsoids overlap. Efficient algorithms for convex optimization problems are thus not guaranteed to find the correct signed distance between overlapping ellipsoids. In this paper, we first show that computing the signed distance is equivalent to minimizing the norm along the boundary of the Minkowski difference. We then derive an algorithm with running time $O(n^6)$, where $n$ is the dimension of the ellipsoids, that obtains a global minimizer on the boundary of the Minkowski difference and hence provides the exact signed distance. The algorithm first finds all the points that satisfy the Karush--Kuhn--Tucker (KKT) conditions, and then identifies a relevant KKT point with the smallest signed distance. The primary difficulty in computing the KKT points is that they are the solutions of two bivariate rational equations, whose poles are not kno...

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