I'm trying to formulate a MILP with a variant of the standard integer cuts constraint (which excludes previously found integer solutions) by avoiding previously found combination of solutions.
For example, for a problem with four binary variables, if solution1 is {1,1,1,0} and solution2 is {1,0,1,1}, then the solution {1,1,1,1} should also be avoided.
I'd really appreciate any help here. Thanks!
The “no-good” cut to avoid $\hat{x}=(1,1,1,1)$ is $$\sum_{i=1}^4 x_i \le 3.$$ More generally, if $S_j=\{i:\hat{x}_i=j\}$ for $j\in\{0,1\}$, take $$\sum_{i\in S_0} x_i+\sum_{i\in S_1}(1-x_i)\ge 1.$$