Elsevier

Discrete Applied Mathematics

Volume 155, Issue 13, 15 August 2007, Pages 1731-1744
Discrete Applied Mathematics

On Khachiyan's algorithm for the computation of minimum-volume enclosing ellipsoids

Dedicated to the memory of Leo Khachiyan
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Abstract

Given A{a1,,am}Rd whose affine hull is Rd, we study the problems of computing an approximate rounding of the convex hull of A and an approximation to the minimum-volume enclosing ellipsoid of A. In the case of centrally symmetric sets, we first establish that Khachiyan's barycentric coordinate descent (BCD) method is exactly the polar of the deepest cut ellipsoid method using two-sided symmetric cuts. This observation gives further insight into the efficient implementation of the BCD method. We then propose a variant algorithm which computes an approximate rounding of the convex hull of A, and which can also be used to compute an approximation to the minimum-volume enclosing ellipsoid of A. Our algorithm is a modification of the algorithm of Kumar and Yıldırım, which combines Khachiyan's BCD method with a simple initialization scheme to achieve a slightly improved polynomial complexity result, and which returns a small “core set.” We establish that our algorithm computes an approximate solution to the dual optimization formulation of the minimum-volume enclosing ellipsoid problem that satisfies a more complete set of approximate optimality conditions than either of the two previous algorithms. Furthermore, this added benefit is achieved without any increase in the improved asymptotic complexity bound of the algorithm of Kumar and Yıldırım or any increase in the bound on the size of the computed core set. In addition, the “dropping idea” used in our algorithm has the potential of computing smaller core sets in practice. We also discuss several possible variants of this dropping technique.

MSC

90C25
90C46
65K05

Keywords

Löwner ellipsoid
Core sets
Rounding of polytopes
Ellipsoid method
Approximation algorithms

Cited by (0)

1

This author was supported in part by NSF through grant DMS-0513337 and ONR through Grant N00014-02-1-0057.

2

This author was supported in part by NSF through CAREER grant DMI-0237415.