Re: Motion P1788/M0013.04 - Comparisons
Nate Hayes wrote:
Arnold Neumaier wrote:
Nate Hayes wrote:
Arnold Neumaier wrote:
Nate Hayes wrote:
John Pryce wrote:
On 18 Sep 2010, at 23:15, Nate Hayes wrote:
I speak against this.
Ulrich's interior is better.
Note that the topological interior, i.e., "proper subset," is
already expressed efficiently in terms of Ulrich's relations for
intervals A,B:
( A \subset B ) and not ( A == B )
Doesn't that make [2,3] interior to [1,3]?
I don't see Entire should be interior to Entire.
Well, it seems weird to me too, but there it is. You're an expert
on quantified statements. Isn't it inescapable from the definition
"B is a neighbourhood of each a in A" (eqn1)?
As you are so fond of quoting from George Corliss: if it even seems
weird to you -- a seasoned mathemetician -- then "God help the
casual user!"
The thing about definitions grounded in standard theory (and this
theory has been around for roughly 100 years) is that, compared
with ad-hoc definitions, you KNOW they can't lead to
inconsistencies -- assuming math itself is consistent.
This is a reason I think P1788 might want to investigate sticking
to compact intervals, instead.... which if you remember was one of
my first choices.
Unbounded intervals are essential for applications to general global
optimization and nonlinear systems solving, where for complex models
one often doesn't know in advance bounds on all variables.
Yes, being able to initialize such unknown variables appropriately is
important for these applications. I agree. I don't suggest P1788
should not provide a mechanism for this.
Keep in mind, Ian McKintosh has suggested replacing unbounded closed
intervals with "overflown" compact intervals. In practice, there is
not really a difference between the two. For example, +INF and -INF
are replaced by +OVR and -OVR in all compuations, where OVR is some
unknown finite real number larger than the magnitude of the largest
floating-point number. In this way, intervals such as [1,+OVR] remain
compact intervals and are not unbounded like the interval [1,+INF].
Arithmetic operations on the endpoints of compact overflown intervals
is the same as in Motion 5, i.e., (-OVR) + (-OVR) = -OVR, 0 * OVR =
0, etc.
But then OVR has a different meaning in different places, which is
unacceptable from a mathematical point of view. It is just INF in
duisguise, and it hides mathematical propoerties of the limit under
the carpet. One cannot use it in a mathematical argument....
To argue for existence, one _needs_ in certain case the unbounded case.
Mathematical programming had once a tradition for using big M (which
is about the same as your OVR), and it is now deprecated (though still
used by a few who don't like unbounded variables) since it behaves
poorly not only mathematically but also algorithmically.
I would like to see examples specific to intervals and OVR why this
should be the case.
An example where boundedness is not needed but the interior property
seems essential is the existence theorem 5.1.7 from my book
''Interval methods for systems of equations''. In dimenison >1, it is
unknown whether the result holts without assuming interiorness in the
topological sense.
I tried to turn this into a definite example, but the only easily
tractable case is dimension 1, and there simple conntainment can be
proved sufficient for the conclusion. In higher dimensions, no such
proof is knon, hence one currently needs the standard interior to
be able to use the theorem computationally.
An example where global optimization needs unbeounded intervals is
for constraint propagation, for example to deduce automatically
from the inequalities x_i>=0, sum(x_i<=1 that each x_i in [0,1].
Of course, one can rewrite the given inequalities as x_i in [0,M]
for some unknown large M, but that this is unwise even for approximate
calculations is well-known -- many variants of the simplex methods
lead to numerical instability in more complex such examples.
So this approach seems to retain all the important advantages of
unbounded intervals without actually introducing the complications
that arise from them.
Could you please point to a real complication of the implementation
that Inf in place of OVR produces?
For me it mainly looks like a change of names messing with
the conceptual meaning of intervals but introducing no visible gain.
Computationally, in IEEE arithmetic you still need to treat OVR as
some float and only Inf qualifies for that.
To the extent I can see, it has big potential
to simplify P1788 a great deal, without hurting or compromising the
important needs of global optimization and nonlinear programming
application you mention.
It is an artificial device without a proper mathematical support.
There are very good reasons why mathematics has developed all these
topological concepts, sincve they are _exactly_ right in the
circumstances.
No doubt.
But IEEE 1788 is a standard for computing as much as it is a standard
for mathematics. Therefore it requires an interdisciplinary approach.
I think it is a pity you should be so quick to judge the possible
outcome of such an investigation.
A standard should not require new research but take stock of what
exists already in a good state, and select from that.
The worst thing about this suggestion is that it compromises
Motion 3 by weakening the conceptual correspondence of intervals
with definite sets, for the sake of at best a very slight gain
and at the cost of other difficulties introduced.
Arnold Neumaier