I do not claim any understanding of Nelder-Mead or anything like it, also called a downhill method of finding a minimum for a system of equations, which, if you keep up with these things, may be old news. But consider the possibility of the inverse, nearing the summit realizing that whatever your strategy and method, the climb has led you up the wrong peak. Or more to the point, in the valley, you cannot realize or see that the adjacent (or some very distant) valley is lower but not chosen because the initial solution set off in the wrong direction. The desired goal is now distant. Maybe in Step n, there could be a global check for other minima. Or maybe this accomplishes that trick.