Largest dual ellipsoids inscribed in dual cones

Suppose x̄ and s̄ lie in the interiors of a cone K and its dual K∗ respectively. We seek dual ellipsoidal norms such that the product of the radii of the largest inscribed balls centered at x̄ and s̄ and incribed in K and K∗ respectively is maximized. Here the balls are defined using the two dual norms. We provide a solution when the cones are symmetric, that is self-dual and homogeneous. This provides a geometric justification for the Nesterov-Todd primal-dual scaling in symmetric cone programming. School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York 14853, USA ([email protected]). This author was supported in part by NSF through grant DMS-0209457 and ONR through grant N00014-02-1-0057.