The variational geometry, projection expression and decomposition associated with ellipsoidal cones

Non-symmetric cones have long been mysterious to optimization researchers because of no unified analysis technique to handle these cones. Nonetheless, by looking into symmetric cones and non-symmetric cones, it is still possible to find relations between these kinds of cones. This paper tries an attempt to this aspect and focuses on an important class of convex cones, the ellipsoidal cone. There are two main reasons for it. The ellipsoidal cone not only includes the well known second-order cone, circular cone and elliptic cone as special cases, but also it can be converted to a second-order cone by a transformation and vice versa. With respect to the ellipsoidal cone, we characterize its dual cone, variational geometry, the projection mapping, and the decompositions. We believe these results may provide a fundamental approach on tackling with other unfamiliar non-symmetric cone optimization problems.