I am using NMinimize with the NelderMead method. I have a very good initial solution supplied via InitialPoints. If I turn off postprocessing ("PostProcess"->False) then the output is terrible, but if I turn it on, the output is very good. So clearly the postprocessing is doing all the work -- no surprise since my initial solution is good.
My understanding is that the postprocessing is some local search method. So I'd like to just use that. But how? Is it one of the NMinimize methods? The most plausible is RandomSearch, but when I choose this explicitly the running time is about 10 times what I was getting with NelderMead. So I suspect something different is going on.