Algorithm, find local/global minima, function of 2 variables

Question

Let us have a function of 2 variables:

 z=f(x,y) = ....

Can you advise me any suitable method (simply algorithmizable, fast convergence) to calculate the the local extreme on some intervals or the global extreme?

Thanks for your help.

Answer 1

Gradient Descent is a wise choice for finding local minima for functions, assuming you can calculate the gradient.

Depending on the specific domain - sometimes there are other solutions as well.
For example, for Linear-Least-Squares (which is used for regression in the field of machine learning) , you can find local (and global, the function in this case is convex) - you can use normal equations

EDIT: As suggested in comments: If you don't have any information on the function, you might be able to use a hill climbing algorithm where you sample the candidates where to advance (you need to take a sample because there are infinite number of directions if the function is of real numbers) - and chose the most promising one.
You can also try to extract the derivatives numerically using numerical differentiation, and use gradient descent.

Answer 2

You might also look into simulated annealing if you like the idea of algorithms driven by ideas from thermodynamics and metallurgy.

Or perhaps you'd rather look at genetic algorithms, because you like the current explosion of knowledge in biology.