Re: NaI's as decorated empty sets
Arnold Neumaier wrote:
> Nate Hayes wrote
> [in: Interval hardware and existing practice]:
>
>> In my view, the problem with decorated intervals is they don't
>> appear to provide any benefit or function that can't be achieved
>> more simply and efficiently with a few standardized NaIs, which
>> would retrofit very nicely into existing hardware.
>>
>> I think those in favour of decorated intervals need to show why they
>> are absolutely necessary; even if this can be done, it should also
>> be explained why we CAN'T also have NaIs for the branch-and-bound
>> algorithms.
>
> The Paving C picture for Alexandre Goldsztjn's example
> sqrt(x^2 +y^2-9) > 4
> proves that the spurious inner circle is _never_ eliminated when using
> NaI's, since any box with rational endpoints containing an irrational
> point on this circle will evaluate to NaI.
>
> Thus the decorated mode in place of NaI is absolutely necessary
> for any branch and bound algorithm that leaves he user the choice
> of how to write a given constraint. I don't want to be deprived of
> this choice.
>
>
> The NaI with payload is essentially equivalent to propagating only
> the decoration while throwing away the interval. This saves space
> but wastes information. In some cases, it may lead to a saving.
>
> But it leads to a loss in all cases where both the interval and the
> decoration contain useful information (e.g. in any interval Newton
> method that wants to verify existence and uniqueness).
>
> Thus the user should have a choice what to use.
>
>
> This leads to an interesting possibility:
> Perhaps we should treat NaI's as decorated empty sets!
>
> This would make their processing fast, people who don't need them
> can throw away all decorations and only need to propagate the
> bare intervals, while those who can/need employ decorations
> and/or NaI's can do so, too.
>
>
>
> Arnold Neumaier
If I understand properly, this is a very neat idea.
It would basically mean users have one of three options:
-- use the whole decorated interval,
-- throw away the interval portion, or
-- throw away the decoration portion.
Is that about correct?
Nate Hayes