Extreme rays in polymake

Question

I am trying to find extreme rays of a polyhedral cone using polymake. My understanding is that in a cone, every feasible solution is also a ray and an extreme ray is a ray that cannot be written as the convex combination of two other rays. The methods RAYS and LINEARITY_SPACE give me some rays that generate the cone but I found an example where one of the rays can be written as the combination of two other rays. Are the results of RAYS not guaranteed to be extreme rays? If so, is there a method to get the extreme rays?

Example

declare $cone=new Cone<Rational>(INEQUALITIES=>[[1, 0, 1], [2, 1, 0]]);
# Note that all variables are unbounded.

print_constraints($cone);
Inequalities:
0: x0 + x2 >= 0
1: 2 x0 + x1 >= 0

print $cone->RAYS; 
1 -2 0
0 1 0

print $pq->LINEALITY_SPACE; 
-1 2 1

The first returned ray $(1, -2, 0)$ can be written as $(1, -2, -1) + (0, 0, 1)$ so it cannot be extreme.

Answer 1

Not all polyhedral cones possess extreme rays. Take for instance the half-space $x = 0$ in $\Bbb R^3$. There are no extreme rays, but there are lineality rays $(0, 1, 0)$, $(0,0,1)$. The pair $(0, 1, 0)$, $(0,0,1)$ isn't unique, in fact they can be rotated by any angle about $x$-axis, and you still get a valid lineality space. There exist polyhedral cones that can be decomposed into a Minkowski sum of lineality space and conic combination of a bunch of rays (say $R$), where the set $R$ is not unique. In such cases, you need to check what polymake will return. An example would be is $z\ge x$, $z\ge0$ in $(x,y,z)$ space. The concept of extreme rays is relevant mostly to pointed cones i.e cones that do not contain a line segment passing through the origin.