# exchanging operators in max-max function

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 $A$.

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What are $a_j$ and $a_{ij}$? How are they related? – joriki Nov 7 '12 at 0:22
Sorry, I neglected to define it: $a_j=(a_{1j},a_{2j},...,a_{mj})$, the columns of A – holger3000 Nov 7 '12 at 0:47

$$\max_{i\in I,a_j\in P_j}\left\{\sum_ja_{ij}x_j-b_i\right\}\;.$$