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Assume a mathematical optimization problem with two positive continuous variables:

0 <= x <= 1
0 <= y <= 1000

I am seeking an efficient way to express the following nonlinear relationship in form of linear constraints (possibly with the use of binary/integer variables and big M), so the problem can be solved with MILP solvers:

when   0 <= y < 200      then   x = 0
when   y = 200           then   0 <= x <= 1
when   200 < y <= 1000   then   x = 1

The numbers 200 and 1000 are indicatively big.

Are there any direct suggestions or papers/books addressing similar problems?

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    $\begingroup$ Welcome to OR.SE. These constraints can be directly handled by the indicator constraints and, almost, all of the modern solvers have had such facilities. Also, what you mentioned is something like logical constraints than being nonlinear constraints. Would you say please, why do you think those should be nonlinear? $\endgroup$
    – A.Omidi
    Commented Jan 29, 2022 at 13:13

1 Answer 1

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Introduce three binary decision variables $z_1,z_2,z_3 $ and impose linear constraints $$ z_1+z_2+z_3=1\\ 0z_1+0z_2+1z_3\le x\le 0z_1+1z_2+1z_3 \\ 0z_1+200z_2+200z_3\le y\le 200z_1+200z_2+1000z_3 $$

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