linear constraint of semi-continuous variable

Answered

Bahar Nasirian

May 18, 2022 00:40

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3 comments

Official comment
Simranjit Kaur

Gurobi Staff

November 16, 2025 23:37
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Jaromił Najman

Gurobi Staff

May 18, 2022 09:44
Hi Bahar,

You can model \(x\) directly as a semicontinuous variable in Gurobi with lower bound \(u_1\) and upper bound \(u_2\). You can then model a conditional statement if \(x \geq \frac{u_1}{2}\) then \(z=1\) and \(z=0\) else. With this, you can define \(y\) as \(y=b_0 z + b_1 x\). To avoid numerical issues, \(u_1\) should have a value strictly above the FeasibilityTol value. A pseudo Python code could look something like
```
u_1 = 2
u_2 = 4
b0  = 1
b1  = 3

x = model.addVar(lb=u_1, ub=u_2, vtype=GRB.SEMICONT, name="x")
z = model.addVar(vtype=GRB.BINARY, name="z")
y = model.addVar(name="y")

model.addConstr(x >= 0.5*u_1 - u_2*(1-z), name="cond_constr1")
model.addConstr(x <= 0.5*u_1 + u_2*z, name="cond_constr2")
model.addConstr(y == b0*z + b1*x)
```
With the above, if \(x=0\), then \(z=0\) and \(z=1\) else. Thus, \(y=0\) if \(x=0\) and \(y = b_0\cdot 1 + b_1\cdot x\) else.

Best regards,
Jaromił
0
Bahar Nasirian

May 19, 2022 00:08
Thank you for your guidance.

Regards,

Bahar
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