P M Hill wrote:
Angela,
Starting with page 180, the definition of redundant, I find this hard to follow. In the context of a polyhedra, we have already extreme rays and we know these are not redundant. (But a ray which is defined by just a point, is only unique up to a positive multiple.)
Any other ray that is not extreme nor in the lineality space is redundant in the sense that it is the positive combination of extreme rays and lines.
The PPL is based on the double description of a Polyhedron and on the Chernikova's algorithm for "finding an irredundant set of vertices and rays of a given polyhedron defined by a mixed system of linear equations and inequalities" (Le Verge - A note on Chernikova's algorithm - page 1). Here does not appear any reference to extremal rays, but in the recursive construction of the set of rays starting from inequalities I found "The irredundant set Q' of extremal rays of the new cone ..." (Le Verge - page 6). I think that you're right: an extremal ray is not redundant...but I can't understand what Le Verge means talking about "irredundant set of extremal rays".
Moreover Le Verge give the definition of redundant vector in a set Q (page 4) but this seems to be definitions of extremal ray. Then on page 5 there are charaterizations of extremal ray not belonging to the lin. space and irredundant vector belonging to a subset of a polyhedral cone.
Then, on page 6, we have: " [...] If the adjacency property is not taken into account, it can be shown that the resulting set Q' constains all extremal rays...this set is not irredundant in general...".
Well, I'm a bit confused! Can an extremal ray be redundant? And if it can, does exist a definition for this? I found only characterization.
Ciao, Angela.