## On the minimum volume covering ellipsoid of ellipsoids

##### Author

YIldırım, E. A.

##### Date

2006##### Source Title

SIAM Journal on Optimization

##### Print ISSN

1052-6234

##### Electronic ISSN

1095-7189

##### Publisher

Society for Industrial and Applied Mathematics

##### Volume

17

##### Issue

3

##### Pages

621 - 641

##### Language

English

##### Type

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Show full item record##### Abstract

Let S denote the convex hull of m full-dimensional ellipsoids in ℝn. Given ε > 0 and δ > 0, we study the problems of computing a (1 + ε)-approximation to the minimum volume covering ellipsoid of S and a (1 + δ)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 ℝn, 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 + δ)n-rounding of 5. For fixed ε > 0 and δ > 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 χ ⊆ 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 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 ℝn. We adopt the real number model of computation in our analysis.

##### Keywords

Approximation algorithmsMinimum volume covering ellipsoids

Löowner ellipsoids

Core sets

Rounding of convex sets