I've found a question regarding this notation on another board. I can't get it myself and decided to ask here. I don't know the source of the notation.
Usually under the $\max$ sign we have a definition of the set of the maximum search for the $\max$ argument. Here we have two indexed $x$. Is it "for each pair of such $x$"?
Furthermore, on the inside function there's an iteration over the same indices. Is it a poor choice of indices or I don't get something?
How this notation should be read? Is it widely used?
$$ \max_{(x_i),(x_{ij})} \left\{ -\sum_{i\in I} f_i x_i + \sum_{j\in J}\sum_{i\in I} p_{ij} x_{ij} \right\}. $$
Looks like two different sets of decision variables, both of which are called $x$. I think it would be clearer as follows: $$\max_{x, y} \left\{ -\sum_{i\in I} f_i x_i + \sum_{i\in I} \sum_{j\in J}p_{ij} y_{ij} \right\}.$$ Alternatively, you could combine the two $\sum_{i\in I}$: $$\max_{x, y} \left\{\sum_{i\in I}\left(-f_i x_i + \sum_{j\in J}p_{ij} y_{ij}\right) \right\}.$$