Inner approximation of convex cones via primal-dual ellipsoidal norms

Abstract

We study ellipsoids from the point of view of approximating convex sets. Our focus is on finding largest volume ellipsoids with specified centers which are contained in certain convex cones. After reviewing the related literature and establishing some fundamental mathematical techniques that will be useful, we derive such maximum volume ellipsoids for second order cones and the cones of symmetric positive semidefinite matrices. Then we move to the more challenging problem of finding a largest pair (in the sense of geometric mean of their radii) of primal-dual ellipsoids (in the sense of dual norms) with specified centers that are contained in their respective primal-dual pair of convex cones.