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

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 new 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 algori...

• Type: Journal
• Year: 2007
• Where: DAM
• Authors: Michael J. Todd, E. Alper Yildirim
• Keywords: Algorithm, Approximate Rounding, DAM 2007, Minimum Volume