Lets assume $x$, $y$ are non negative continuous variables and $P$ an integer variable assuming either the value $1$ or the value $2$.
How could I possibly model the relation
If $x = y$ then $P = 2$ else $P = 1$ ?
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3$\begingroup$ Note that $P-1$ is binary and apply or.stackexchange.com/a/2632/500 with your $x-y$ as $x$, $0$ as $b$, and $P-1$ as $y$. $\endgroup$– RobPrattApr 6, 2022 at 21:12
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$\begingroup$ Hugh, I will need to introduce another two binaries? That is frustrating. Thank you very much Rob. $\endgroup$– ClementApr 6, 2022 at 21:35
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$\begingroup$ Well, you can eliminate one of the additional binary variables by substitution if you want. Also, if you instead wanted to enforce only $P=2 \implies x=y$, you wouldn't need any new variables. $\endgroup$– RobPrattApr 6, 2022 at 21:42
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1$\begingroup$ $(L_x - U_y)(2 - P) \le x -y \le (U_x - L_y)(2 - P)$, where $L$ and $U$ are lower and upper bounds on the variables. $\endgroup$– RobPrattApr 6, 2022 at 21:55
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1$\begingroup$ @RobPratt You should convert your comments to an answer. $\endgroup$– prubin ♦Apr 6, 2022 at 22:47
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1 Answer
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Note that $P−1$ is binary and apply the formulation in
In an integer program, how I can force a binary variable to equal 1 if some condition holds?
with your $x−y$ as $x$, $0$ as $b$, and $P−1$ as $y$.
If you instead wanted to enforce only $P=2 \implies x=y$, you wouldn't need any new variables. Just let $L$ and $U$ be lower and upper bounds on the variables, and impose $$(L_x−U_y)(2−P) \le x − y \le (U_x−L_y)(2−P).$$
