ILP constraint conditional on a value of a variable

Question

If $X_{ijklm}$ are Boolean Variables, where $i,j,k,l,m$ range from $1$ to $n$, then write an ILP constraint to ensure that for each value of $k$, either all the $jth$ variables are set to $0$ or all the $jth$ variables are set to $1$ irrespective of the values of $i,l$ and $m$ and there exists at least one value of $k$ for which all the variables are set to 1.

For example, when $k=1$, ensure all variables where $j=1,2,3 \ldots n$ are set to either all 0 or all 1 irrespective of the values of $i, l$ and $m$. Similarly, when $k=2$, ensure all variables where $j=1,2,3 \ldots n$ are set to either all 0 or all 1 irrespective of the values of $i, l$ and $m$. Additionally, ensure that there exists at least one $kth$ index for which all variables are set to $1$.

As a concrete enumerated case, consider $X_{ij1lm}$, in which case $k = 1$, in such a scenario, the set of variables for different values of j are $X_{i11lm}, X_{i21lm}, X_{i31lm}, \ldots, X_{in1lm}$. Similarly for $k = 2$, the set of variables for different values of j are $X_{i12lm}, X_{i22lm}, X_{i32lm}, \ldots, X_{in2lm}$. The ILP constraint should ensure either all these variables are set to 0 or set to 1 and there exists at least one value of $k$ for which all the variables are set to 1.

Answer 1

For each $i,k,l,m$, you want all $X_{ijklm}$ to take the same value for all $j$. Here are three alternative ways, in increasing order of efficiency:

1. Impose linear constraints $$X_{ijklm} = X_{ij'klm} \quad \text{for all $i,k,l,m,j < j'$}$$
2. Introduce a new variable $Y_{iklm}$ to represent the common value and impose linear constraints $$X_{ijklm} = Y_{iklm} \quad \text{for all $i,j,k,l,m$}$$
3. Omit the $j$ index and replace $X_{ijklm}$ with $X_{iklm}$

For the "there exists at least one value of $k$ for which all the variables are set to $1$" requirement, here are the corresponding constraints for the three alternative ways:

1. $$\sum_k X_{i1klm} \ge 1 \quad \text{for all $i,l,m$}$$
2. $$\sum_k Y_{iklm} \ge 1 \quad \text{for all $i,l,m$}$$
3. $$\sum_k X_{iklm} \ge 1 \quad \text{for all $i,l,m$}$$

Answer 2

$\sum_{j} x_{iklm}^j =ny_{iklm} \ \ \forall i,l,m,k $

$1 \le \sum_k y_{iklm} \ \ \forall i,l,m$

where $y_{iklm}$ is also binary