# The cone of Z-transformations on the second order cone

## TL;DR

This paper explores the structural properties of the cone of Z-transformations on the second order cone, linking it to semidefinite and copositive cones, and illustrating how conic linear programs can be transformed.

## Contribution

It characterizes the dual cone of Z-transformations on the second order cone as slices of semidefinite and completely positive cones, providing new insights into conic programming.

## Key findings

- Dual cone is a slice of the semidefinite cone.
- Dual cone is a slice of the completely positive cone.
- Conic linear programs can be reduced between these cones.

## Abstract

In this paper, we describe the structural properties of the cone of $\mathcal{Z}$-transformations on the second order cone in terms of the semidefinite cone and copositive/completely positive cones induced by the second order cone and its boundary. In particular, we describe its dual as a slice of the semidefinite cone as well as a slice of the completely positive cone of the second order 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.