Non-linear least squares using piece-wise approximation....

Hi,

I have the following problem:

p'_i = f_i1(x1) + f_i2(x2) + f_i3(x3) .... f_in(xn)

Where each f_i(x) is an unknown non-linear function. Each p has a
desired target value and I wish to find the values of x that minimize
the distance between each p and its target position (i.e. in a least-
squares sense). Additionally, the variables x need to be bounded and I
need to enforce mutual exclusivity between certain x variables (i.e.
within a set of variables only one can be non-zero).

My idea was to approximate each f_i(x) with a sampled piecewise linear
curve. Typically, I would expect to have a roughly a 40x50 matrix.
Depending on how finely I sample the nonlinear curves, this could turn
into a system of size, 40 x 500 to 40 x 5000.

What's the best way to solve this problem? Given the piecewise
approximation of the nonlinear curves, I was thinking that a LP/MILP
solution using SOS2 might be the answer, but a few LP folks I've heard
from don't think that is the best approach.

Any help would be greatly appreciated.
.

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