Exchanging the sum and the min in double sum

Question

I'm trying to know under which conditions the following equality holds: $$\underset{x\in \mathbb{R^n}}{\min}{\sum _{i=1}^n}{\sum _{j\in [[1,i]]}}f_{i,j}(x)={\sum _{i=1}^n}\underset{x\in \mathbb{R^n}}{\min}{\sum _{j\in [[1,i]]}}f_{i,j}(x),$$ where $f_{i,j}$ is a linear function with respect to $x$. I'm aware of the answers in Minimum of a sum of positive functions is the sum of the minimums of the functions but it seems that it cannot be applied to my case.

Thank you in advance.

Answer 1

There are really only two cases to consider, for linear functions $f(x)$:

• If $f(x) = 0$ is constant, clearly its minimum is $0$;
• If $f(x)$ is not the zero constant function, then its minimum does not exist (i.e. $\inf_x f(x) = -\infty$).

Since the sum of linear functions is linear, you should be able to use the above facts to answer your question.