why we don't calculate global minimum or maximum with looking the plot

Question

Sorry I'm kind of newbie on optimization. As you know we can plot the multivariate functions. example plot

So we can see the global minumum and maximum. so why we run gradient descent to calculate it ?

Answer 1

There are numerous problems with that approach:

• Considering a function of two variables (as in your example) or less, you can only plot a limited portion, which may restrict you to only being able to find local minima.
• A computer is ultimately going to plot your function in discrete steps, which might "obscure", or "smooth out" finer details of the function that you might be interested in.
• If there are more (potentially a lot more) than 2 variables, how are you planning to graph the function?
• There is a lot of nuance in optimization problems (convexity, compactness, lower semi-continuity, ...) which make realistic optimization problems much more difficult than what you are probably thinking of right now.

And probably many more that I don't know at the top of my head at the moment.