I am trying to fit some raw data using a function of the form
$f(r) = \sum_{i=1}^{K} d_kS_k(n_k,\alpha_k,r)$
where
$S_k(n_k,\alpha_k,r) = \frac{\alpha_k ^{n_k+3}}{(n_k+2)!}r^{n_k}\exp(-\alpha_kr)$
(note that the actual fit function is far more complex).
Now, for each $k$, this has two non-linear parameters ($\alpha_k$ and $n_k$) and one linear parameters ($d_k$). I am considering various methods for non-linear optimization (genetic algorithms, conjugate gradient method, etc.) but I am wondering how I should work with the linear functions.
Should I include them in the non-linear optimization (use them as a variables when I take the gradient for CGM, etc)? Should I optimize the non-linear parameters completely on $\alpha$ and $n$ and then perform a linear fit with $d$? Should I mix and match - perform one sequence of conjugate gradient steps and then perform a linear fit, rinse and repeat?
NOTE: If it helps at all, $d_k$ is a real number, $n_k$ is a positive integer, $\alpha_k$ is a positive real.