Self-dual polyhedral cones and their slack matrices

TL;DR

This paper investigates the properties of self-dual polyhedral cones and their slack matrices, revealing new insights into their structure, PSD slack matrices, and implications for semidefinite programming and polytopes.

Contribution

It establishes the equivalence between self-duality and positive semidefinite slack matrices, and characterizes the nature of these matrices for irreducible cones, including their extremal properties.

Findings

Self-duality is equivalent to the existence of a PSD slack matrix.

For irreducible cones, PSD slacks are extreme rays of the DNN cone.

PSD slacks generally do not belong to the cone of completely positive semidefinite matrices unless the cone is simplicial.

Abstract

We analyze self-dual polyhedral cones and prove several properties about their slack matrices. In particular, we show that self-duality is equivalent to the existence of a positive semidefinite (PSD) slack. Beyond that, we show that if the underlying cone is irreducible, then the corresponding PSD slacks are not only doubly nonnegative matrices (DNN) but are extreme rays of the cone of DNN matrices, which correspond to a family of extreme rays not previously described. More surprisingly, we show that, unless the cone is simplicial, PSD slacks not only fail to be completely positive matrices but they also lie outside the cone of completely positive semidefinite matrices. Finally, we show how one can use semidefinite programming to probe the existence of self-dual cones with given combinatorics. Our results are given for polyhedral cones but we also discuss some consequences for negatively self-polar polytopes.