Largest dual ellipsoids inscribed in dual cones

Abstract

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 inscribed in K and K *, respectively, is maximized. Here the balls are defined using the two dual norms. When the cones are symmetric, that is self-dual and homogeneous, the solution arises directly from the Nesterov–Todd primal–dual scaling. This shows a desirable geometric property of this scaling in symmetric cone programming, namely that it induces primal/dual norms that maximize the product of the distances to the boundaries of the cones.