my problem at hand: I am using scipy curve_fit to fit a curve (https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html) but in many occasions, the parameters estimated for such curve refer to one of the local many "local" minima and not "global" minimum. Now this is to be expected given how curve_fit was designed. Still, I really need my global minimum.
In order to find it, my initial hunch would be to multiply initial starting points, run multiple curve_fit instances and choose the one with the lowest fit error but I would suffer from a number of biases in my personal initial guess estimates (also potentially the number of combinations could be considerable and this would be detrimental to performance).
Do you happen to know better, faster and/or methodologically sounder methods on how to proceed? (they do not need to pass for least squares, I can build ad hoc stuff if necessary)
There are a couple possible approaches. One would be to do a "brute force" search through your parameter space to find candidate starting points for the local solver in curve_fit. Another would be to use a global solver such as differential evolution. For sure, both of these can be much slower than a single curve_fit, but they do aim at finding "global minima". Within scipy.optimize, these methods are brute and differential_evolution, respectively. It should be noted neither of these is actually global optimizers, as they both require upper and lower bounds for the search space of all parameters. Still, within those boundaries, they do attempt to find the best result, not just a local minimum close to your starting values.
A common approach is to use brute with medium-sized steps for each parameter, then take the best ten of those and use Levenberg-Marquardt (from leastsq, as used in curve_fit) starting from each of these. Similarly, you can use differential_evolution and then refine.
You might find lmfit (https://lmfit.github.io/lmfit-py) helpful, as it allows you to set up the model once and switch between solvers, including
brute, differential_evolution, and leastsq. Lmfit also makes it easy to fix some parameters or place upper/lower bounds on some parameters. It also provides a higher-level interface to model building and curve-fitting, and methods to explore the confidence intervals of parameters.