Local maximum != global maximum (usually).

I said global minimum, which can easily happen if you end up at local maximum, but you’ll never know unless you randomly search elsewhere (and potentially end up even lower).

A local maximum by definition cannot be a global minimum. A local maximum has to be higher than things around it, which are part of the global space.

Fair point. However, the local maximum doesn’t have to be far from the global minimum.

If the global maximum is 10, minimum is 1, you could easily end up in a region with local maximum 2, minimum 1.