I do not know the answers to most of these but I'll comment on what I can.
From what little I recall of Levenberg-Marquardt, steps in a Newton (I believe) direction are either accepted or rejected based on objective function behavior. I would guess the ratio parameter is involved in that determination. Is L-M best for curve fitting? I would have to guess it depends on the objective function. It is really a method for explicit sum-of-squares formulations and often works quite well for those. But that does not equate to being always best-in-class.
For NMinimize
, our experience is that Method->"DifferentialEvolution"
tends to be slow but often gives best results. But bear in mind "often"!="always".
The post-process step will use FindMinimum
in an effort to polish the result found by the main step, whatever method was used. This often helps insofar as the main methods are better suited for putting one in the right ballpark, whereas the local FindMinimum
methods are better for fast convergence once in the right ballpark. That said, once in a while the post-process step will actually deliver a significant improvement when it manages to find a region the global main method in play had missed.