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Consider the following problem. I have two regions $X1=[a,b]$ and $X2=[b,c]$. Notice that they are disjoint except for point $b$.

I have a continuous variable $x$ and two binary variables, $y_1,y_2$ which indicate membership of the two sets. I want to model $$x\in X1\implies y_1=1$$ $$x\in X2\implies y_2=1$$ So far so good.

However, the two regions intersect at one end (point $b$). So, I want to enforce that $$x=b\implies y_1=1,y_2=0$$ That is, if $x$ is at level $b$ we consider region $X1$ active instead of $X2$.

I am finding this surprisingly difficult to model explicitly, without adding a tolerance to $b$ in set $X2$. Any hints?

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  • $\begingroup$ $Pr(x=b)\approx 0$ for continuous variables and there are also all kind of tolerances in a MIP solver. So, the question is somewhat moot. I would allow the model to choose $y_1$, $y_2$ when $x\approx b$. It can pick the best one (and make you some profit). $\endgroup$ Jan 3 at 15:49
  • $\begingroup$ I don't think there is a way to accomplish what you want without putting some space between $X1$ and $X2$ (i.e., $X2=[b + \epsilon, c]$). $\endgroup$
    – prubin
    Jan 3 at 16:49

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