On the Minimum Volume Covering Ellipsoid of Ellipsoids

We study the problem of computing a (1+ )-approximation to the minimum volume covering ellipsoid of a given set S of the convex hull of m full-dimensional ellipsoids in Rn. We extend the first-order algorithm of Kumar and Yıldırım that computes an approximation to the minimum volume covering ellipsoid of a finite set of points in Rn, which, in turn, is a modification of Khachiyan’s algorithm. For fixed > 0, we establish a polynomial-time complexity, which is linear in the number of ellipsoids m. In particular, the iteration complexity of our algorithm is identical to that for a set of m points. The main ingredient in our analysis is the extension of polynomialtime complexity of certain subroutines in the algorithm from a set of points to a set of ellipsoids. As a byproduct, our algorithm returns a finite “core” set X ⊆ S with the property that the minimum volume covering ellipsoid of X provides a good approximation to that of S. Furthermore, the size of X depends only on the dimension n and , but not on the number of ellipsoids m. We also discuss the extent to which our algorithm can be used to compute the minimum volume covering ellipsoid of the convex hull of other sets in Rn. We adopt the real number model of computation in our analysis.