An Efficient Method to Verify the Inclusion of Ellipsoids

TL;DR

This paper introduces a fast, efficient algorithm for verifying whether one ellipsoid contains another in n-dimensional space, outperforming traditional semidefinite programming methods significantly in speed.

Contribution

The paper presents a novel bisection-based method for ellipsoid inclusion verification that is faster and more scalable than existing semidefinite programming approaches.

Findings

27 times faster for 3D ellipsoids

2294 times faster for 100D ellipsoids

Effective in control theory applications

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(n^3) 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 in the control theory field.