Non-linear regression fails to converge, but fit appears good I'm fitting a 4 parameter nonlinear regression model to multiple datasets, some of which fail to converge, however, the parameters output after a failure provide a fit that looks good, if not exceptional to my (and other's) eyes.
I've explored convergence criteria and they do converge eventually but the visual fit is terrible.
Is there any precedent for taking the visual fit ad ignoring convergence or are there some other things I can try?
I'm fitting a model of the form $\sqrt{c_1 x+c_2\exp(x)^y}/\sqrt{\exp(x)^y}+c_3$ where $y$ is a known constant, using nlminb in R.
 A: I will assume the values of all the variables and constants are such that there won't be problems with obtaining square roots of negative numbers.  Then
$$\frac{\sqrt{c_1 x + c_2 \exp(x)^y}}{\sqrt{\exp(x)^y}} + c_3 =\sqrt{c_2 + c_1 x \exp(-y x)} + c_3.$$
When $y \gt 0$ then eventually, for sufficiently large $x$, $\frac{c_1}{c_2} x \exp(-y x)$ gets small and the fit can be closely approximated as
$$(\sqrt{c_2} + c_3) + \frac{c_1}{2 c_2} x \exp(-y x).$$
Otherwise, when $y \lt 0$ and $c_1 \ne 0$, eventually (for largish positive $x$) the value of $c_2$ is inconsequential compared to the exponential and the fit simplifies to
$$\sqrt{c_1 x} \exp(-y x / 2) + c_3.$$
In either case there are redundant parameters: $\sqrt{c_2} + c_3$ can be kept constant while varying $c_2$ because $c_1$ can also be varied to keep $\frac{c_1}{2 c_2}$ constant.  In the second case $c_2$ can be freely varied throughout a wide range.
Therefore, in many circumstances you should expect not to be able to identify at least one of these parameters.  It sounds like your data are in one of these cases.  If so, you might consider eliminating one of the $c_i$ from the model altogether.
Alternatively, consider a reparameterization of the function in the form
$$\alpha \sqrt{1 + \beta y x \exp(-y x)} + \gamma - \alpha$$
($\alpha = \sqrt{c_2}$, $\beta = c_1/(c_2 y)$, $\gamma = c_3+\sqrt{c_2}$).
This might provide enough clues to the minimizer to help it converge.
