[abetusk]

So isn't the magic here convexity? Convexity is such a huge constraint that I would guess convergence would be hard to avoid.

[jleahy]

Tell that to Newton’s method, which can fail to converge for convex functions.

[kergonath]

There is no way of ensuring that you have found a global minimum if the function is not convex, or if you don’t make equivalent assumptions, in the general case (if the domain of the function is continuous, and/or infinite).

If you don’t make assumptions, you would need to consider every single point of the domain, and there is an infinite number of them. The job gets easier the more assumptions you make about the continuous, differentiable, and monotonous character of the function.