Below shows a simple example of how to specify "starting points" in NMinimize[]. The methods used by NMinimize[] usually need several starting points, at least n+1, where n is the number of variables.
NMinimize[{x^2 + (y - .5)^2, y >= 0 && y >= x + 1}, {x, y},
Method -> {Automatic, "InitialPoints" -> {{0, 1}, {0, 4}, {2, 4}}}]
I believe NMinimize[] will add extra points if needed. A specific method instead of Automatic may be used.