Self-dual polyhedral cones and their slack matrices

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 DNN matrices, which correspond to a family of extreme rays not previously described. This leads to a curious consequence for 5x5 DNN matrices: the extreme rays that are not rank 1 must come from slack matrices of self-dual cones over a pentagon. 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. Our results are given for polyhedral cones but we also discuss some consequences for negatively self-polar polytopes.