Indicator function constraints with three conditions
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I have an optimization problem using gurobi c++:
objective: minimize L(u),
where L(u) = {1/2 - u, if u<=0; 1/2(1-u)^2, if 0<u<1; 0, otherwise. }
How can I create constraints with this indicator function with binary variables and addGenConstrIndicator()?
Please help, thanks.
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In your case you could use the following formulation
\[\begin{align*}
b_0 = 1 &\rightarrow u \leq 0\\
b_0 = 1 &\rightarrow w = 0.5 - u\\
b_1 = 1 &\rightarrow u_{aux} = u\\
b_1 = 1 &\rightarrow w = 0.5 - u_{aux} + 0.5v\\
b_1 = 0 &\rightarrow u_{aux} = 0\\
b_2 = 1 &\rightarrow u \geq 1\\
b_2 = 1 &\rightarrow w = 0\\
b_0 + b_1 + b_2 &= 1\\
v &= u_{aux}^2\\
b_i &\in \{0,1\}, u_{aux} \in [0,1], v \in [0,1]
\end{align*}\]The constraints with \(\rightarrow\) are indicator constraints. Note that your model is nonconvex, thus you will have to set the NonConvex parameter. Note that the auxiliary variable \{v\} is needed to avoid the product \(u_{aux}^2 \cdot b_1\). Variable \(w\) holds the value of your function \(L(u)\).
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