Ulrich Kulisch wrote:
[in: Re: P1788: PLEASE VOTE: M0013.04 & M0020.01]
Am 02.10.2010 18:17, schrieb Arnold Neumaier:
Corliss, George wrote:
Voting on Motion M0013.04 Comparison Relations ends on Friday,
October 8.
Current count: Yes: 13; No: 0; Required for quorum: 37
Voting on Motion M0020.01 Comparison Relations ends on Friday,
October 8.
Current count: Yes: 10; No: 2; Required for quorum: 37
The complexity pmomoted in these motions is not warranted by the
applications. After having challenged the use of these relations,
not a single application was pointed out.
This is not correct. Applications using <= were mentionend in a mail
by Juergen Wolff von Gudenberg.
The application to branch and bound is spurious:
For pure constraint satisfaction problems that are solved to completion,
the ordering used is completely immaterial, and using a stack no
ordering is needed at all.
But if the branch-and-bound cannot be completed for reasons of time or
space, or if used in a global optimization context, the <= ordering is
extremely inefficient, and dynamical orderings are needed.
Other comparisons were also mentioned to be needed by other mails.
Could you please collect for reference all the uses mentioned in
one single mail?
The <= relation is the connection link between the set definition of
the arithmetic operations for intervals and the explicit formulas
for computing the result of the operations by the bounds of the
operands, not just for real intervals but also for real interval
vectors and interval matrices and again for complex intervals and for
complex interval vectors and matrices.
This does not yet constitute an algorithmic use. it is a theoretical
result, and for this it suffices that the comparison is defined on the
theoretical level. The standard, however, is about what needs to be
implemented.
For the <= relation compatibility relations hold between the
algebraic structure and the order stucture similar to the real
numbers.. For instance:
1. If a <= b ==> a + c = b + c for all c. 2. If a <= b ==> -b <= -a.
3. If 0 <= a <= b and c >= 0 ==> ac <= bc and ca <= cb. Similar for
division.
Again, this is a theoretical result only.
I do not deny that the <= order has useful theoretical properties
that help in understanding the structure of intervals and interval
linear algebra.
But I haven't seen any algorithmic uses for these.
Thus my statement remains valid that the only comparison operations
useful in practice are
disjoint,
subset (containment),
interior (containment in the topological interior).
Arnold Neumaier