Re: Non-linear least squares using piece-wise approximation....
- From: svlad <madsvlad@xxxxxxxxx>
- Date: Fri, 8 Feb 2008 10:51:08 -0800 (PST)
here you say the paths are unknown???
the question is: how to represent such an unknown path: with piecewise linear
functions this becomes even more involved (e.g. using SOS2)
my idea was to represent those paths by a finite set of unknown points
in space, interpolating these by a cubic spline, then to minimize the overall
goal with the x(i) = positioning variable and the points representing
the paths simultaneously
the modelling of the mutual exclusion constraints is already given in my previous
post:
say x(1), x(3) and x(7) cannot be nonzero simultaneously:
writing this as x(1)*x(7)=0
x(1)*x(3)=0
x(3)*x(7)=0
("complementarity constraints")
is all but a good idea: it not only introduces nasty nonlinearity but also
singularities in cases where more than one x(j) is zero.
better write this as
y(1)*x(1)+y(3)*x(3)+y(7)*x(7) in [0,1] (if you don't know the required outcome)
y(1)+y(2)+y(3)=1 one variable nonzero at most
y(1), y(3), y(7) either 0 or 1 (zero-one-variables)
but this brings you in the filed on mixed zero one nonlinear optimization.
possible, but hard
The paths are not "unknown", but rather I do not have an explicit
function for them. My plan was to sample points along the paths
generating piecewise linear approximations of them. So the position
for each point can be computed:
p'_i = sum{k=1,...,n} f_ik(x_k)
where each f_ik(x_k) has a piecewise linear representation.
It would seem that the real sticker is the mutex constraint. I was
wondering if it might be possible to approach the problem more
programmatically than mathematically? It depends on how the solver
algorithm works, but if I were able to detect when variable x_k became
non-zero, I could set the upper bounds of any variable mutex'd with it
to zero, thus excluding it from the possible solutions set. This only
works of course if the variables are activated serially. The BVLS code
used an active set strategy and worked in this manner.
Any thoughts?
.
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