9

Equivalently, you want to remove the smallest number of constraints so that the resulting problem is feasible. You can do this implicitly, without enumerating all Irreducible Infeasible Sets, by solving an auxiliary ILP problem. Introduce a binary variable $z_i$ for each constraint (including variable bounds as a special case) $$\sum_j a_{i,j} x_j \le b_i$$...


2

As already mentioned in the comments: If $𝑄_i$ is an integer variable, then $𝑄_i > 0$ is equivalent to $𝑄_i \geq 1$. Hence, you can use the indicator constraints offered by the most commercial solvers in case you don't know upper bounds, i.e. (x[i]==1) => (Q[i] >= 1). Note also prubin's comment on how one can calculate an upper bound: By the way,...


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