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I wonder how systems like CPLEX, GUROBI, etc implement indicator constraints. Do they just implement Big M equivalents? If yes, what is then the justification for using them?
Edit
The question does not concern when to prefer using BigM constraints vs Indicator constraints. The question is if the developers of the solver coded the idea of an indicator constraint by coding its equivalent BigM constraint.
You can have a look at SCIP's implementation in cons_indicator. They say that:
An indicator constraint is given by a binary variable $z$ and an inequality $ax \le b$. It states that if $z=1$ then $ax \le b$ holds.
This constraint is handled by adding a slack variable $s: ax−s \le b$ with $s \le 0$. The constraint is enforced by fixing $s$ to $0$ if $z=1$.
This constraint is equivalent to a linear constraint $ax−s \le b$ and an SOS1 constraint on $z$ and $s$ (at most one should be nonzero). In the indicator context we can, however, separate more inequalities.
