An Efficient Method to Verify the Inclusion of Ellipsoids

Abstract

We present a novel method for deciding whether a given n-dimensional ellipsoid contains another one (possibly with a different center). This method consists in constructing a particular concave function and deciding whether it has any value greater than -1 in a compact interval that is a subset of [0, 1]. This can be done efficiently by a bisection algorithm method that is guaranteed to stop in a finite number of iterations. The initialization of the method requires O(n3) floating-point operations and evaluating this function and its derivatives requires O(n). This can be also generalized to compute the smallest level set of a convex quadratic function containing a finite number of n-ellipsoids. In our benchmark with randomly generated ellipsoids, when compared with a classic method based on semidefinite programming, our algorithm performs 27 times faster for ellipsoids of dimension n = 3 and 2294 times faster for dimension n = 100. We illustrate the usefulness of this method with a problem motivated by control systems applications.