Completely Positive and Copositive Cones over Symmetric Cones: Approximations and Facial Structure

This thesis studies approximations and the facial structure of completely positive cones and their duals, copositive cones, over symmetric cones from the optimization aspect.In the part on the approximations, we first provide approximation hierarchies for copositive cones over general and specific symmetric cones.These hierarchies are described by positive semidefinite constraints.Additionally, we compare them with existing ones.Through numerical experiments, we demonstrate that by combining the proposed approximation hierarchies with the existing ones, we can evaluate the optimal values of copositive programming problems more accurately and efficiently.We second examine various extensions of doubly nonnegative cones.Inner-approximation hierarchies for generalized (in the sense that the cones underlying completely positive and copositive cones are beyond nonnegative orthants) copositive cones are utilized to obtain two generalized doubly nonnegative cones from the doubly nonnegative cones.We conduct theoretical and numerical comparisons to assess the relaxation strengths of the generalized doubly nonnegative cones over a direct product of a nonnegative orthant and second-order or positive semidefinite cones.These comparisons also include an analysis of an existing generalized doubly nonnegative cone.Through numerical experiments, we demonstrate that the three generalized doubly nonnegative cones provide significantly tighter bounds for generalized completely positive programming problems than positive semidefinite cones.In the part on the facial structure, we first investigate two values decided by the facial structure of closed convex cones: the length of the longest chain of faces and the distance to polyhedrality.We prove that the two lengths of completely positive cones over nonnegative orthants, doubly nonnegative cones, and their duals are the worst cases possible.We second show that copositive cones over symmetric cones are never facially exposed when the underlying symmetric cone has dimension at least 2. We do so by explicitly exhibiting a non-exposed extreme ray.Our result extends the known fact that copositive cones over nonnegative orthants are not facially exposed in general.