Hello everyone,
this post is somewhat related to the thread on multidimensional
minimization, but not quite, so I decided to start a clean thread on
this one.
I am sure the problem of "constraining some values during
minimization" is well known and often encountered, but the issue is only
vaguely covered in most articles which employ some methods for this.
Consider the following problem: the function being minimized, f(x),
where x is the parameter vector, has several minima. One does not know
which one is global, but that is a standard problem. However, the
parameter space must be limited: there exist functions g0(x), ...,
gN(x), which have upper/lower limits of acceptability. For example,
g0<=g(x)<=g1 - i.e. a certain value which is a function of x must lie
within the experimental error.
One way of constructing a "constraint" (given no analytic solution is
possible) is to construct a dual chi squared function: C = chi squared +
constraint. Given chi squared is polynomially bounded, for example, the
constraint for "g(x) = g0" can look like this: constraint = pow
(P*[g(x)-g0],2)*exp(pow(P*[g(x)-g0],2).
Such a constraint is smooth, and the derivative is smooth, thus it
should not spoil the convergence properties of CG methods. Moreover, it
contains an adjustable "strength" parameter P, which allows to adjust
how strong it is relative to the current chi squared, and what dominates
the "total" chi squared fed to the fit.
The problem is, it is far from transparent how it influences each
individual parameter in parameter space, and without any possibility of
an analytic solution, I do not see how to "map" the constraint over the
whole parameter space per parameter and determine which areas of that
space will be suitable to constrain each parameter individually, instead
of by such a general function which "influences all together, but you
don't really know how".
However, this method works for me quite well - convergence without
constraints or with constraints does not change significantly.
Has anyone tried other ways of implementing this?
Thanks!
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