My question is about how usual/strange defining constraints compactly can be (in a scientific document). Formally, I define constraints $$f_{i,j,k}(x) \leq 0, \quad \forall i \in [I], \ \forall j \in [I], \ \forall k \in K(i,j)$$ where $K(i,j) := \{ k \ : \ A_k \cap (A_i \cap A_j) \neq \emptyset \}$. However, for some combinations of $(i,j)$, we have $A_{i} \cap A_j = \emptyset$ so we have $K(i,j) = \emptyset$. I have two options:
- Define the constraints for all $i \in [I]$ and for all $j,k$ in some (slightly more) complicated set parametrized by $i$, or,
- Just note that if we have some constraints where $k$ is not defined, we ignore these constraints (i.e., these constraints are not defined).
I was wondering, can I "get away" with the second option? I do not want to introduce a set for $(j,k)$, rather keep it as it is, and just ignore the constraints for $i,j$ where $K(i,j) = \emptyset$.
End-notes: $f$ is an arbitrary function, $x$ is a decision variable, $I$ is an abitrary natural number, e.g., $I = 5$, and $|I|$ denotes $1,\ldots, I$. The sets $A_i$ for $i \in [I]$ are not important in this context.