Nelder–Mead has often not worked well for me in moderate to high dimensions. I'd recommend trying Powell's method if you want to quickly converge to a local optimum. If you're using scipy's wrappers, it's easy to swap between the two:

https://docs.scipy.org/doc/scipy/reference/optimize.html#loc...

For nastier optimization problems there are lots of other options, including evolutionary algorithms and Bayesian optimization:

https://facebookresearch.github.io/nevergrad/

https://github.com/facebook/Ax