How to compute the argmin over vectors?

Question

Can I just differentiate to find the argmin if I'm looking for a vector and if yes which differentiation is used for such questions as below? $f$ is differentiable. $v_0 = argmin_{v\in \mathbb{R^n}} f(v)$ And further can anyone give me a source with useful differentiation rules if I have vectors, e.g. how to differentiate $\|v-x\|_2^2.$

Answer 1

Your question is not too clear. So I make some assumptions.

Yes, you can use ArgMin to get the argument of the minimum.

Concerning your vectors, I assume that one vector b is a given constant and the other a is a position vector, With this, the derivatives are:

a = {x, y};
b = {x0, y0};
D[(a - b).(a - b), x]
(* 2 (x - x0 *)

D[(a - b).(a - b), y]
(* 2 (y - y0) *)