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 semi-definite (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.

Automated summary

In this paper, we analyze self-dual polyhedral cones and prove properties related to their slack matrices. We demonstrate that self-duality is equivalent to the existence of a positive semi-definite slack matrix. Furthermore, we show that if the underlying cone is irreducible, the corresponding PSD slacks are not only doubly nonnegative matrices but also extreme rays of the cone of DNN matrices. Additionally, we find 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 discuss how semidefinite programming can be used to determine the existence of self-dual cones with specific combinatorial properties.