Evaluating a function using a densely spaced grid and plotting it does work. This is brute-force search. You will see the global minima immediately in the way you describe, provided your grid is dense enough to capture all local variation.

It's just that when the function is implemented on the computer, evaluating so many points takes a long time, and using a more sophisticated optimization algorithm that exploits information like the gradient is almost always faster. In physical reality all the points already exist, so if they can be observed cheaply the brute force approach works well.

Edit: Your question was good. Asking superficially-naive questions like that is often a fruitful starting point for coming up with new tricks to solve seemingly-intractable problems.

Thanks!

It does feels to me that we do some sort of sampling, definitely is not a naive grid search.

Also I find it easier to find the minima in specific directions (up, down, left, right) rather than let’s say a 42 degree one. So some sort of priors are probably used to improve sample efficiency.