I am trying to determine if the following holds.
$\max_{i\in I}\max_{a_j \in P_j}\{\sum_j a_{ij}x_j - b_i\}=\max_{a_j \in P_j}\max_{i\in I}\{\sum_j a_{ij}x_j - b_i\}$
$P_j$ is a closed convex set, $I$ is an index set (finite), $b_i$ is a known parameter, and $x_j$ is a nonnegative variable. Also, I define $a_j=(a_{1j},a_{2j},...,a_{mj})$ i.e. the column vectors of $A$.
This holds independently of the details of the function being maximized. Just like quantifiers of the same type commute, extremizations of the same type commute, and for the same reason: they can be combined into a single quantifier/extremization, which in this case would be
$$ \max_{i\in I,a_j\in P_j}\left\{\sum_ja_{ij}x_j-b_i\right\}\;. $$