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I have an MILP where we have $$ t_k = \sum_i P_i\cdot C_{ik} : P_i\ \in \{0,1\}, C_{ik} \in I^+ $$
and our model is constrained by the number of times $t_k$ is bigger than a certain value $T_k$.
$$ \left[\sum_k\left(t_k \ge T_k\right)\right] \ge N $$
where $N$ is the minimum number of constraints to be satisfied.
Can this problem be solved in MILP and how? I am new to this domain and any leads would be very helpful.
You can introduce a binary variable $x_k$ and linear constraints \begin{align} \sum_k x_k &\ge N\tag1\\ -t_k+T_k&\le M_k(1-x_k) &&\text{for $k\in K$}\tag2 \end{align} Here, the “big-M” constant $M_k$ is a small upper bound on $-t_k+T_k$. Because $t_k\ge 0$, you can take $M_k=T_k$, and the constraint simplifies to $t_k\ge T_k x_k$.
Constraint $(1)$ forces at least $N$ of the $x_k$ variables to be $1$. Constraint $(2)$ enforces the logical implication $x_k=1\implies t_k\ge T_k$.
