Is the solver guaranteed not to land in a local minima/maxima?
The solver generates a relaxed lower bound that indicates how far they could be from the global optimal solution. The moment that the lower bound improves enough to match a path they can guarantee that it's the global optimum
(I don't know)
But I would guess the answer is "no".
If you can prove, as they claim, that you have an algorithm that gives you the optimal solution (aside from the obvious, brute-forced one), you might be one stone throw away to make an argument for some P == NP, that would be HUGE.
But it seems that some people get offended when you tell them their perpetual motion machines are not real.