If I have two binary(0-1) decision variables: $P_1$ and $P_2$, and my constraint is that:
$P_2$ may be chosen if and only if $P_1$ is not chosen, is this equivalent to the formulation that: $$P_1 + P_2=1$$
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1$\begingroup$ "May be" sounds optional. Does the condition allow that neither variable is chosen? $\endgroup$Alex Kruckman– Alex Kruckman2021-07-11 15:10:11 +00:00Commented Jul 11, 2021 at 15:10
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1 Answer
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You can derive linear constraints via conjunctive normal form as follows. $$ P_2 \iff \lnot P_1 \\ (P_2 \implies \lnot P_1) \land (\lnot P_1 \implies P_2) \\ (\lnot P_2 \lor \lnot P_1) \land (P_1 \lor P_2) \\ (1 - P_2) + (1 - P_1) \ge 1 \land P_1 + P_2 \ge 1 \\ 1 \ge P_2 + P_1 \land P_1 + P_2 \ge 1 \\ P_1 + P_2 = 1 $$
