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There are also Optimization and Nonconvex , which seem like umbrella packages and I am not sure what methods to use inside these packages. Any help on these?
So I usually fallback to NLopt.jl, it is an interface around the old NLopt library (written in C/FORTRAN/C++). It is not super hard to use but it is more bare bones than the alternatives you mentioned, however it has dozens of optimization methods and options, great documentation and it is super fast. I am sure it would work great with your problem if you are willing to spend the time to tweak its configuration option.