I am working on a Dantzig-Wolfe decomposition and one of my subproblems is a polyhedral cone. Until recently, I thought the Dantzig-Wolfe decomposition would add new columns based on extreme rays of the cone and terminate eventually because the number of extreme rays is finite. Then I learned that a cone doesn't necessarily have extreme rays if it is not pointed and that it could be generated by a set of non-extreme rays instead. As the set of generators is not unique and the set of (extreme and non-extreme) rays is infinite, I'm worried that the Dantzig-Wolfe decomposition would lose its termination guarantee.

My question is about how polymake deals with this: are there any guarantees about the rays returned in Cone::RAYS when the cone has no extreme rays? Is there a property that distinguishes those rays from other sets of rays that would generate the cone?