John
Sorry. In your earlier post it was the IBM vector processor that you
wanted to know more about. The additional information about integer
linear programming was for the information of those that were not as
familiar with it as you.
If I was up to speed in J perhaps I could give a J example--if anyone
can please jump in.
A standard linear problem (not in J )is given- by equations:
Max μ = α,χ
Αχ=β
χ≥Ο
Where Α is an m x n shaped matrix, α and χ are vectors of length n
and β is a vector of length m and Ο is 0 vector.
It is easy to see this can be expressed in J or APL.
There are many situations where it does not make sense for these
variables to assume other than INTEGER values such as how many
machines do you need, how many people should you assign to a task to
produce the shortest schedule, or how many houses should you build,
how many stocks should you buy/sell to yield the largest profit.
Unlike the case for general LP problems which are almost universally
solved by one algorithm there are (were?) large numbers of Integer LP
algorithms none of which were universally successful in solving LARGE
problems. The primary limitation is the number of integer variables
so if there are more than few hundred variables they cannot be solved
except in special cases.
Anyway, some methods to approach such problems involve building a
simplex plateau as for general problems but with only integer
elements and this property must be maintained at each iteration.
INTEGER not RATIONAL, but if you are forced at some step to say
divide 111 by 333 you might want to keep the result as 111r333 and go
on until a later step where you can get back into an Integer. You
are best to consult the literature as this is getting quite long and
I don't seem to be making it clear.
Perhaps you can find discussions of why the performance of Gormory's
algorithms which were some of the most effective were completely
unpredictable.
I had a discussion recently with a Canadian economist who was
completely dismissive of Computer Analysts and on further discussion
it may have been due to the inability of any that he had worked with
to grasp these details. I was thinking in the back of my mind how Dr
Iverson had told me that his initial motivation for arrays in APL
were due to his work with Leontief. I wondered if J would be able to
improve these results. Anyway, I am not sure if any of this suggests
an answer to your question.
http://www.news.harvard.edu/gazette/1999/02.11/leontief.html
On 27-Jun-06, at 8:28 PM, John Randall wrote:
rational numbers (which are defined as equivalence
classes), I still don't see the difference between representatives.
On 27-Jun-06, at 4:42 PM, John Randall wrote:
Could you explain this a little? I know something about mixed and
pure
integer linear programming, but nothing about IBM's vector processor.
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