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I have a binary variable $y$ and a set of binary variables $x_i$, where $i\in I$. My problem requires that $$\sum\limits_{i\in I}x_i = b.$$ What I want to formulate is the following implication: if $\sum\limits_{i\in \tilde{I}} x_i \leq b-1$ then $y=1$ where $\tilde{I}\subseteq I$, but I can't seem to figure out how. I have been able to find a formulation that says if $\sum\limits_{i\in \tilde{I}}x_i=m$ then $y=0$ by the inequality $\sum\limits_{i\in \tilde{I}}x_i + y \leq m$ but that is not exactly what I want. Any help is greatly appreciated!

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$x_i \le y$ for $i\in I \setminus \tilde{I}$

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  • $\begingroup$ Jesus Christ! Sometimes I really wonder where these brain-farts come from. Thank you very much! $\endgroup$
    – Djames
    Nov 1, 2019 at 12:30

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