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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
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1 Answer

up vote 0 down vote accepted

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

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

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