The literature and documentation in
operations research optimization is
enormous, going back to the 1950s. The
best of that literature is quite well
For combinatorial optimization, there is
George L. Nemhauser and Laurence A.
Wolsey, 'Integer and Combinatorial
Optimization', ISBN 0-471-35943-2, John
Wiley & Sons, Inc., New York, 1999.
The details of optimization are not always
trivial, and several fairly challenging
graduate applied math courses, complete
with non-trivial theorems and proofs, can
be needed for much depth in the subject.
I've had four such courses and taught one.
> I guess it's extremely difficult to
implement generic optimization solver,
A "solver"? Would be nice to have a
"solver". Lots of people talk about
giving a problem to a "solver". For
linear programming, maybe usually can do
Otherwise my experience is that asking for
a general purpose solver is, for now,
Instead, for the real problems I've had
success with, have to look carefully at
the problem and exploit special
structure particular to that problem.
Typically part of the work involves doing
some derivations using the math of
> Instead, for the real problems I've had success with, have to look carefully at the problem and exploit special structure particular to that problem.
I am in complete agreement.
One way I have thought about this is that, rather informally, "the mapping from problems to solutions is highly discontinuous". That is, suppose you have a problem statement, and you figure out a method to attack the problem and solve it in practice. Now, vary the problem statement slightly. There is a reasonable chance that the new problem is dramatically harder to solve, and you need a completely different body of theory to even think about it, let alone solve it.
For example, suppose you have a polynomial equation in a bunch of complex variables. This is easily solved numerically in practice to any degree of precision you care for. Let's reword the problem a little and require that you are only willing to accept integer-valued solutions. Now you're talking about Diophantine equations! Have fun!
In practice, if you're participating in some project to solve business / industrial problems using optimisation, it is pretty common to have the definition of the problem change slightly every few weeks or months, as you learn more about the domain, or as people change their minds, or as the business actually changes what it does. Often you need to figure out clever ways of exploiting the mathematical structure in a particular problem statement to figure out a tractable way of solving the problem. Best to hope that the new version of your problem doesn't suddenly break that structure while you're "sprinting" towards the next release!
IMHO it has been, since the
1950s, mostly this "sprinting"
that was in practice just too
difficult, that is, too costly
in time and money, for far too
large a fraction of organizations,
budgets, managers, etc. that
could have used optimization.
Now with current computing and
the Internet, a lot in applied
math, in principal powerful and
valuable for important real problems,
seems to be growing significantly
is much easier but, sadly,
too often still too challenging.
Of course part of the "sprinting"
challenge was solved by, say,
A Mathematical Programming Language
(AMPL) where can just type in
max z = f(x)
subject to g(x) >= 0
x in C
So, then, with the problem thusly
typed in, just need a solver!
Right: Maybe from the back side
of one of the moons of Pluto?
Gee, we know how to sort, know a
lot about sorting, in O( n log(n) ),
etc., so maybe we should have
solvers that would do as well
on optimization as we can do now
on sorting? Then quickly
we encounter the monster of the
question P versus NP.
So, without an algorithm that shows
that P = NP, we're back to
column generation here, Lagrangian
relaxation there, branch and bound
too often, some group theory
sometimes, etc. and "sprinting".
Yes, it was interesting
philosophically how central
and challenging P versus NP
is, but I'd rather have just
had a way to solve optimization
problems routinely, gotten
home in time for dinner, and spent
Sunday at a BBQ!
I of course meant the or-tools API docs. Overall I think it's nicely done and structured, but could be better.
Or perhaps you recommend some probability books before too Neveu?
Pity as it seemed like an excellent course.
I think the domain of solving problems by defining the constraints is super interesting. In my fantasy basketball example, I define the constraints for a valid roster (simple), then define how to score a roster, and less than a second later, I've got the optimal picks.
* designing utility networks (e.g. energy / telecommunications) to be cheap / robust
* routing vehicles / ships / parcels to serve all customers cheaply
* dispatching and arranging emergency services vehicles (e.g. ambulances) to best cover anticipated future demand
* managing inventory levels in blood banks. it goes off after a while! there are different types, some are substitutable. which types are better to keep in stock?
* figuring out the most profitable way to chop animals into different cuts of meat
can do that and is BSD licensed.
There are numerous free implementations listed here:
Was there something special you needed?
Translated: Hey guys! We might do something that might get shutdown next year but we are still cool!
If any Google devs read this, I do have one request: it would be awesome to see runtime analysis based on the theory / design of the provided code, with perhaps some explanation of why certain design decisions were made in the solution.
It's already an excellent educational tool, and this seems like an easy way to make it even better.