I have a continuous variable $z, {-1 < z < 1},$ and a binary variable $w$.
How do I write a conditional constraint which guarantees for $z < 0$, $w = 1$,
and for $z \ge 0$, $w = 0$?
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1$\begingroup$ Technically, you cannot do this. The best you can do is either to allow both values of $w$ when $z=0$ (as in the answer by ytsao) or to change the domain of $z$ to $[-1, -\epsilon] \cup [0, 1]$ for some small but strictly positive value of $\epsilon.$ $\endgroup$– prubin ♦Commented Jul 15 at 15:57
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$\begingroup$ See cs.stackexchange.com/q/69531/755 and cs.stackexchange.com/q/67163/755 $\endgroup$– D.W.Commented Jul 15 at 21:43
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2 Answers
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The following constraints should be work:
$z \leq M(1-w)$
$z \geq -Mw$
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Generally, in LP, we do not use strict inequalities. When we consider non-strict inequalities, Ytsao's answer works using $M=1$, $-w \leq z \leq 1-w$.
