[PPL-devel] thanks

[Angela Stazzone]

P M Hill wrote:

> Hi,
>
> > > Thanks Le Verge's paper and other goodies arrived this morning.
>
> [...]
>
> I have had a look through the papers you sent although I have not read
> them all carefully...
>
> Let me know if there is something I should look at and comment on today...

I think that it need to add the notion of duality in the  initial page of the
documentation, but I only find this definition for polytope (Wilde - A
library for doing polyhedra operations - publication interne 785 - december
1993 - page 12).
Can we use the same definition for general polyhedra, too?

>
>
> -----------------------------------------------------------------
>
> Previously, the meaning of the word "stable" on page 9 of LeVerge was in
> question.
> I found a definition of this in a book I borrowed from the library
> by Christopher Witzgall
> "Convexity and Optimization in Finite Dimensions"
> BUT I haven't managed to work out what "stable" really means!
> The book says (page 187/8):
>
> "We call the point X_0 \in K(f) stable if every linear (nonvertical)
> manifold N in R^{n+1} lying below [f] and satisfying X_0 \in \pi(N) can be
> separated from [f] by a nonvertical plane
>
> E:= {(X,z) | Y^T X - z = b} in R^{n+1}:
>
> Y^T X_1 - z_1 \leq b \leq Y^T X_2 - z_2
> for all (X_1,z_1) \in [f], (X_2,z_2) \in N."
>
>  (X,z) is my ascii representation for the column vector X
>                                                         z
> Also it says that
> "\pi is a projection from R^{n+1} to R^n in the direction of the z-axes"
> "f" is a convex function and
> "K(f) denotes the domain of finiteness of f:
> K(f) := {X | f(X) < +\infty}."
> I'm afraid that "lying below" has another complicated definition.
>  The definition relies on lots of other definitions (eg [f] denotes an
> "epigraph") and the book is not easy to read.
>
> (The author thinks all this is easy... eg
> "As is easily seen every linear manifold N covering M and lying below [f]
> is nonvertical.")
>
> I need help here...
> The stability concept was first invented by R T Rockafellar in 1963.
> It may be best to see how he explains it.
> Does anyone feels up to rewriting a specialised version of this definition
> just for its application to rational polyhedra?
>

??? I can't answer this question for two reasons:
- first I have not clear the meaning of "feels up" (sorry!)
- moreover I don't understand what does it means that a ray is stable (even
it seems to be
    very easy! ;-))

>
> -----------------------------------------------------------------
>
> In our devref.tex file it says:
>
> - The dimension of the \f$\mathop{\mathrm{lin. space}}\f$
>   is the number of irredundant lines.
>
> I still do not like talking about individual lines being "irredundant".
> I think it only has meaning in terms of an "irredundant set"
> Meaning that no vector can be removed without changing the system it is
> generating. I think that something like
>
> - The dimension of the \f$\mathop{\mathrm{lin. space}}\f$
>   is the rank of any set of lines that span the space.
>
> would be better.
>

Ok.

>
> -------------------------
>
> > Unfortunately (or fortunately, I don't know!) my work here is almost
> > finished. I mean that from Monday I will not work for University of
> > Parma.
>
> Sorry you are going! But all the best for the future.

Thanks a lot!

>
>
> > However I will read my mail every day and I will happy to help anyone of
> > you!
> >
> > I miss you much!
>
> Do keep in touch and tell us of all your adventures in the outside/real
> world.
>

Of course I will do.

>
> ciao,
>   Pat
>
> Ciao,

          Ange.

>

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