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Well, this is actually already sort of possible, but we just havn't explained how to do it anywhere. Look at the main loop for most of the algorithms here https://github.com/JuliaNLSolvers/Optim.jl/blob/master/src/multivariate/optimize/optimize.jl#L22 . We can make an explicit version like the on you show. Each next call would just iterate on the loop shown in the code I linked to.
Using iterator interface, we can make the optimizer like a white box,
This kind of interface is very welcome is debugging, tunning optimizer et. al.
Please consider ~
A python version is of white box optimizer is https://github.com/BRML/climin
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