Change the order of minimum in sums

Question

Is it possible to change the order of minimum as i did below:

$$\min_w \sum_x\sum_y f(x,y,w)^2 = \sum_x \min_w \sum_y f(x,y,w)^2 $$

If for all $x$ we have $$\min_w \sum_y f(x,y,w)^2,$$ then could I change the order of minimum?

Answer 1

No, it is not correct. In the first you have to choose one value of $w$ to use in every term of the sum. In the second you get to choose a different $w$ for every $x$. For example, take

$$\begin {array} {c|c|c|c} x&y&w&f(x,y,w)\\ \hline 0&0&0&0 \\ 0&0&1&10\\0&1&0&0\\0&1&1&10\\1&0&0&10\\1&0&1&0\\1&1&0&10\\1&1&1&0 \end {array}$$ Here you want $x$ to agree with $w$. The left side gives the same value regardless of $w$, which is $200$. The right side gives $0$.