Things are clearer to me now. It’s merely about the KKT system of equations.
At a local optimum, which Ipopt ensures, there are 2 possibilities for an inequality constraint
- it is binding at this local optimum
- it is nonbinding, i.e. it can be safely removed, concerning only this specific local optimum
If it is the situation 2, then the multiplier returned is simply 0.
If it is the situation 1, then that inequality constraint can be deemed a == constraint, which owns a normal lagrange multiplier (typically will be nonzero).