[PPL-devel] Re: Documentation

[Angela Stazzone]

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.