On the Minimum Volume Covering Ellipsoid of Ellipsoids

• Published in: SIAM journal on optimization Vol. 17; no. 3; pp. 621 - 641
• Main Author: Yildirim, E. Alper
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2006
• Subjects: Algorithms; Approximation; By products; Optimization
• ISSN: 1052-6234; 1095-7189
• DOI: 10.1137/050622560

Let ${\cal S}$ denote the convex hull of $m$ full-dimensional ellipsoids in $\mathbb{R}^n$. Given $\epsilon > 0$ and $\delta > 0$, we study the problems of computing a $(1+ \epsilon)$-approximation to the minimum volume covering ellipsoid of ${\cal S}$ and a $(1 + \delta)n$-rounding of ${\cal S}$. We extend the first-order algorithm of Kumar and [J. Optim. Theory Appl., 126 (2005), pp. 1-21] that computes an approximation to the minimum volume covering ellipsoid of a finite set of points in $\mathbb{R}^n$, which, in turn, is a modification of Khachiyan's algorithm [L. G. Khachiyan, Math. Oper. Res., 21 (1996), pp. 307-320]. Our algorithm can also compute a $(1 +\delta)n$-rounding of ${\cal S}$. For fixed $\epsilon > 0$ and $\delta > 0$, we establish polynomial-time complexity results for the respective problems, each of which is linear in the number of ellipsoids $m$. In particular, our algorithm can approximate the minimum volume covering ellipsoid of ${\cal S}$ in asymptotically the same number of iterations as that required by the algorithm of Kumar and to approximate the minimum volume covering ellipsoid of a set of $m$ points. The main ingredient in our analysis is the extension of polynomial-time 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 ${\cal X} \subseteq {\cal S}$ with the property that the minimum volume covering ellipsoid of χ provides a good approximation to the minimum volume covering ellipsoid of ${\cal S}$. Furthermore, the size of the core set depends only on the dimension $n$ and the approximation parameter ε, but not on the number of ellipsoids $m$. We also discuss the extent to which our algorithm can be used to compute an approximate minimum volume covering ellipsoid and an approximate $n$-rounding of the convex hull of other sets in $\mathbb{R}^n$. We adopt the real number model of computation in our analysis.