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Let $i$ and $j$ be two binary variables.

How can I express $i \neq j$ as a linear constraint?

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    $\begingroup$ $i+j=1$ ........... $\endgroup$ Jan 15 '21 at 22:32
  • $\begingroup$ in fact, i want to prevent both to be 1 at the same time. how can I do this? $\endgroup$ Jan 15 '21 at 22:36
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@user3680510 gave the correct answer in a comment. Here's a way to derive it via conjunctive normal form: $$ i \not= j \\ (i \implies \lnot j) \land (\lnot i \implies j) \\ (\lnot i \lor \lnot j) \land (i \lor j) \\ (1 - i + 1 - j \ge 1) \land (i + j \ge 1) \\ (i + j \le 1) \land (i + j \ge 1) \\ i + j = 1 $$


To prevent both to be 1 at the same time: $$ \lnot(i \land j) \\ \lnot i \lor \lnot j \\ 1 - i + 1 - j \ge 1 \\ i + j \le 1 $$

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  • $\begingroup$ in fact, i want to prevent both to be 1 at the same time. how can I do this? $\endgroup$ Jan 15 '21 at 22:40
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    $\begingroup$ I updated my answer. $\endgroup$
    – RobPratt
    Jan 15 '21 at 22:41

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