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I would appreciate any help to solve the following task:
If $y=1$ then $x_i=1$ for at least $k$ of the possible indices $i\in\{1,\cdots,n\}$ where $k$ and $n$ are parameters, $x$ is a binary variable vector with $n$ elements, and $y$ is a binary variable.
You want to enforce $y=1 \implies \sum_i x_i \ge k$. You can do so by imposing linear big-M constraint $$k - \sum_i x_i \le (k - 0) (1 - y),$$ which simplifies to $$\sum_i x_i \ge k y.$$
