Z-transformation, Lorentz cone, Semidefinite cone, Copositive cone, Completely positive cone
In this paper, the structural properties of the cone of $\calz$-transformations on the Lorentz cone are described in terms of the semidefinite cone and copositive/completely positive cones induced by the Lorentz cone and its boundary. In particular, its dual is described as a slice of the semidefinite cone as well as a slice of the completely positive cone of the Lorentz cone. This provides an example of an instance where a conic linear program on a completely positive cone is reduced to a problem on the semidefinite cone.
Nemeth, Sandor Zoltan and Gowda, Muddappa S..
"The cone of Z-transformations on Lorentz cone",
Electronic Journal of Linear Algebra,
Volume 35, pp. 387-393.