@Markiyan Hirnyk I used the GlobalOptimization package to find this point.
I did not have to supply a good intitial point in order for it to get that good a fit. In my opinion, the less specific detail that you are forced to supply, the better.
And so obtaining a particular fit is done better without requiring an close initial point. And better also if done with a wider range for the parameters than a narrow one (and better with not range than with any).
Indeed, having to supply a very judicious or fortuitous initial point goes against the spirit of having a global rather than a local optimizer. The best global solvers will succeed, without being told just where to look, even in the presence of many local minima.
Comparison of solvers is still hard to do comprehensively, because there will always be some problems that solver A can do but solver B cannot, and vice versa.
Using an objective procedure `obj` that simply added the squares of the reciprocals, I did the following and obtained a reproducible result in under a minute on an Intel i5 processor:
, a = 0..10, b = -10..10, c = 0..100, d = 0..7, e = 0..4
[0.00112973296049516327, [a = 2.6174720167848577,
b = -1.7195008832497265,
c = 2.309346760329378,
d = 1.5033851325407128,
e = 1.8458792860566562] ]
I would believe that increasing the timelimit and evaluationlimit to be very high, and possibly switching to the multistart method, might produce an even better residual. But (again), what impressed me here is that a good result was found quickly, without needed a tight range or initial point.