On Wednesday, May 30, 2012 6:24:35 AM UTC-7, Nathann Cohen wrote:
>
> > Thanks. Can you recommend any good tool other than Sage (since it
> doesn't
> > have such interfaces yet) that would handle quadratic inequalities?
>
> Well, that's a problem I would be very glad to be able to answer
> myself.... But it highly depends on the characteristics of your
> equations... CPLEX can solve some instances (it is proprietary,
> though) depending on the matrix of constraints, but for general
> inequalities I have the same problem you have :-/
>
> Nathann
>
Is there any reason we shouldn't try to define a workflow in SAGE to tackle
this? I'm making this up, but I think the principal problem is that, for
practical evaluation, there really *isn't* such a thing a general system of
quadratic inequalities. By which I mean, the particular considerations are
the heart of the matter. Suppose we said, "Okay, there is no direct
solver, but how can we use SAGE to advance us as far as possible toward a
useful representation of such a system?"
I tumbled this problem around in my head for the last day, and I keep
thinking that the particularities of the constraints cannot --and should
not-- be avoided. That, as a general rule, seeking a direct solver might
be a bit of a misunderstanding. Like, we need to define our constraints
more carefully. Are we sure there are no differential relationships, first
of all!
@Nathann/Ruslan -- would it be useful for either of you if we worked out a
multipage Worksheet collecting relevant methods and providing some basic
ways to get from one representation to another?
For example, I think *whether or not there is any descriptive meaning, for
the particular system in question, to considering the constraints as
representing the geometric boundaries of conics, and the "solution" to be a
multidimensional region of unions and intersections, *could be a useful
question. It strikes me there is *not* one way to parameterize, and it
really is going to depend on how we want to represent the relationships
among the quadratic terms. For example, the list Ruslan started with,
pairwise components
x{a}x{b} + x{d}x{e} + ....
... if the other equations continue in this way, does not necessarily
represent a unique arrangement of generalized quadratic forms, even if the
total system has a finite solution set.
Or does it? (Afternoon musings not to be trusted!)
I can't get over feeling like the system is simply not adequately
described, and that a program which solves it for us is not *really* tackling
the problem. We are at the whim of the form of its output.
But, obviously I could choose a rational parameterization. Or unhook the
quadratics into linear pairs (like Ruslan did).
Is there really one way to solve this?
Can we build a useful tool for handling these problems, in a short period
of time, without trying to "crack" the solution to general solution spaces?
And for the love of God tell me if this is ridiculous and naive. I have no
frame of reference for how other people think about math. I have no one to
check with but me, to see if I've gone crazy.
--
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