An LMI description for the cone of Lorentz-positive maps II

Abstract

Let L _n be the n-dimensional second-order cone. A linear map from ℝ ^m to ℝ ⁿ is called positive if the image of L _m under this map is contained in L _n . For any pair (n, m) of dimensions, the set of positive maps forms a convex cone. We construct a linear matrix inequality of size (n − 1)(m − 1) that describes this cone.

Keywords:

• second-order cone
• semidefinite cone

AMS Subject Classifications::

• 15A48
• 15A69
• 90C22

Notes

1. In the literature, the term regular cone might also be used in other contexts. Sometimes the cones we call regular here are called proper, but proper cone might also have different meanings. We will stick to the notation used in the conic programming literature.

2. We would like to thank an anonymous referee who pointed out the necessity of including closure in the formulation of the proposition. The original version 2, Proposition 2.9] is false as stated. However, as the cones K and the projections of K* used in the proofs are always closed, the results of 2 are not affected.