When the minimization of a sum is equal to the sum of the minimization?

Question

I would like to know when is allowed to exchange a $\min$ operator with a summation of certain function. For example, supposing to have a function of two variables $f(x,y)$:

$\min_x \sum_{k=1}^{K} \inf_y |y| + 1/2 (a_k^{T}x-b_k-y)^{2} $

When can I write $\min_x \sum_{k=1}^{K} \inf_y =\min_x \inf_y \sum_{k} $ ?

Thank you.