On the minimum volume covering ellipsoid of ellipsoids

摘要

Let S denote the convex hull of m full- dimensional ellipsoids in 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 S and a (1 + delta) n-rounding of S. We extend the first-order algorithm of Kumar and Yildirim [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 R-n, which, in turn, is a modi. cation 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 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 S in asymptotically the same number of iterations as that required by the algorithm of Kumar and Yildirim 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 X subset of S with the property that the minimum volume covering ellipsoid of X provides a good approximation to the minimum volume covering ellipsoid of S. Furthermore, the size of the core set depends only on the dimension n and the approximation parameter epsilon, 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 R-n. We adopt the real number model of computation in our analysis.