In a simplified version of the ILP problem I am trying to formulate, I have the following:
- A set of elements $A_{i,j} \in \mathcal{A}$.
- Each element $A_{i,j} \in \mathcal{A}$ has an associated parameter $a_{i,j} > 1$.
- A binary decision variable $x_{i,j}$ corresponding to an element $A_{i,j} \in \mathcal{A}$ ($x_{i,j}=1$ iff $A_{i,j}$ is selected).
- A binary decision variable $y_{i,j;k,l}$ corresponding to two elements $A_{i,j}$ and $A_{k,l}$ in $\mathcal{A}$, such that $y_{i,j;k,l} = 1$ iff both elements $A_{i,j}$ and $A_{k,l}$ are selected.
I have the following constraints: \begin{equation} \sum_{A_{i,j}, A_{i,k} \in \mathcal{A}} a_{i,j}a_{i,k} y_{i,j;i,k} \geq \sum_{A_{i,j} \in \mathcal{A}} a_{i,j}(a_{i,j}-1) x_{i,j} \end{equation}
\begin{equation} a_{i,j}x_{i,j} + a_{k,j}x_{k,j} \geq 10, \forall A_{i,j}, A_{k,j} \in \mathcal{A} \end{equation}
My questions regarding the above constraints concern the elements in $\mathcal{A}$ over which the summations and the $\forall$ operate:
- On the LHS of the first constraint, is it clearly conveyed that the first summation is over every pair of elements in $\mathcal{A}$ such that their first index has the same value $i$ (the second index can be different or the same)? Is this a conventional and common way to indicate the intended meaning or should I use a different notation/approach, e.g. $(A_{i,j}, A_{i,k}) \in \mathcal{A}$, or is my notation adequate/understandable/correct? (please note that I need to use the set $\mathcal{A}$ in the summation bound).
- On the RHS of the first constraint, is it clear that the summation is independent of the LHS summation, even though they both operate over $\mathcal{A}$?
- In the second constraint, similar to my first question, is it clearly conveyed that the constraint is over every pair of elements in $\mathcal{A}$ such that their second index has the same value $j$ (the first index can be different or the same)?
I think it is clear enough if you provide text commentary describing what is going on (as you did in this question). The issue is not really notational correctness or clarity so much as the fact that it's easy to misread subscripts.