[SciPy-User] [ANN] Guaranteed solution of nonlinear equation(s)

[Dmitrey]

   --- Исходное сообщение ---
   От кого: "Yury V. Zaytsev" <yury at shurup.com>
   Кому: "Dmitrey" <tmp50 at ukr.net>
   Дата: 25 мая 2011, 15:55:41
   Тема: Re: [SciPy-User] [ANN] Guaranteed solution of nonlinear
   equation(s)



     Hi Dmitrey,
     
     On Wed, 2011-05-25 at 10:36 +0300, Dmitrey wrote:
     
     >         1) Are there any scientific publications detailing the algorithm that
     >         can be referenced from a paper?
     >         
     > not yet.
     
     Please keep us posted when it happens!
     
     >         2) Sorry, I am not sure if I've got it right from the examples. 
     >         
     >         Is it true that your implementation of interalg is impossible to use
     >         outside of OpenOpt the way one would, for example, use fmin from SciPy?
     > 
     > From all your questions, it seems you haven't read interalg webpage.
     
     Yes, sorry, it's a shame, I was distracted by the benchmarks page :-/
     
     It would be helpful as well if it would mention the size of the problems
     that interalg is suitable for. I have > 1000 variables and a scalar
     non-linear objective function (which is proven to be concave, though).
        

   It depends on hardware involved, so I cannot formulate it precisely.
   Currently interalg will hardly handle problem of such big nVars, if
   exist any variable that is present at least for twice. I tested it on
   problems with 1...100 variables.
   For that problem you've mentioned I would consider using local solvers
   ralg or gsubg (http://openopt.org/ralg, http://openopt.org/gsubg)

      Also, are the box bound constraints necessary for the algorithm to work
     or the support for infinite constraints could be added later?
        

   Only finite values are required (but you could use rather big ones)

      > As it is mentioned in      http://openopt.org/interalg      , only FuncDesigner
     > models can be handled (because interval analysis is required).
     
     Ok, now it's clear. I have read the FuncDesigner documentation page
     again and I think that now I understand it a little bit better.
     
     Will it be correct if I summarize that in order to use interalg:
     
     the objective function has to be constructed using FuncDesigner as an
     oofunc, where every instance that depends on the optimization variables
     (oovars) has to be an oofunc constructed from other oofuncs or oovars
     using only numbers, FuncDesigner mathematical functions (you named
     below) and Python for loops / ifThenElse expressions 
        

   ifThenElse expressions can't be handled (interval arithmetics is
   unimplemented for them yet), but I intend to implement FuncDesigner
   theta-function
   theta(x) = 0 if x <= 0 else 1, and you can use the func instead of
   ifThenElse, e.g. instead of
   ifThenElse(a**2+b**2<c**2+4, a**4, b**3) you could use interval
   arithmetics on
   ff = theta(a**2+b**2 - c**2-4)
   f3 =   ff * b**3 + (1-ff) * a**4

     
     etc.?
     
     Can you estimate the memory overhead of using an oovar / oofunc as
     opposed to a NumPy array? I.e., I think I would need like 3 x 5 000 000
     oovars in size and few hundreds individual oovars for my function.
        

   I guess I haven't understood your question.

      When I use a NumPy float arrays for storage and Cython to implement my
     for loops in C, it only needs few hundred megabytes of RAM and works
     quite fast (x100 times faster than pure Python + NumPy vectorized
     operations).
     
     Therefore, I imagine that without JIT this is hopeless? Does OpenOpt
     work with PyPy as of yet?
        

   I don't know

      > However, you could easily compare scipy optimize fmin and other
     > openopt-connected solvers with interalg.
     
     ... which is, of course, very nice, except that for this you have first
     to implement your model in FuncDesigner :-(
     
     > Currently only these funcs are supported:
     > +, -, *, /, pow (**), sin, cos, arcsin, arccos, arctan, sinh, cosh,
     > exp, sqrt, abs, log, log2, log10, floor, ceil
     > Future plans: 1-D splines , min, max
     > Also, any monotone func R->R (or with known "critical points", where
     > order of monotonity changes) can be easily connected. 
     
     Ok, I see, right now I need expi (as in SciPy) in addition to that. Is
     there some documentation on how to connect my own functions it to
     OpenOpt, 
        

   did you meant to FuncDesigner?
   You can use
   myoofun = oofun(scipy.special.expi, input, d=lambda x: exp(x)/x)
   See http://openopt.org/FuncDesignerDoc#Creating_special_oofuns
   But connecting interval analysis feature is a little bit harder.

     so that they work like the supplied ones with regards to
     interval analysis and automatic differentiation that is needed of
     interalg?
        

   The function seems to decrease in (-inf, 0) and grow up  in (0, inf),
   am I right? If yes, it could be quite easily connected, however, the
   func is not defined in zero, that brings some inconveniences.

      Thanks!
     
     -- 
     Sincerely yours,
     Yury V. Zaytsev
     
     
   
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