Linearizing a disjunctive expression into MILP

Question

I want to linearize the following disjunctive form.

$$\left[\begin{gathered}w_{1}\\x \geq a\end{gathered}\right] \vee \left[\begin{gathered}w_{2}\\x \geq b\end{gathered}\right]$$

where $w_1$ and $w_2$ are binary and $x$ is a free variable $\in \{ -2, \cdots, 10\}$ and $\{ -2 \lt a \lt b \lt 10 \}$. I think one possible way is as follows:

$$\left[\begin{gathered}(w_{1}=1) \land (x \geq a) \end{gathered}\right] \vee \left[\begin{gathered}(w_{2}=1) \land (x \geq b)\end{gathered}\right]$$

Then:

$$(w_{1} \lor w_{2}) \bigwedge (w_{1} \implies x \geq a) \bigwedge (w_{2} \implies x \geq b)$$

I would like to know if, is there a way to linearize such a disjunctive form by introducing the new auxiliary variable $z$ for each disjunct? (Also, the answer by Rob is a perfect one).

Answer 1

Because $a<b$, the lower bound $x \ge a$ is valid for both parts. So linearize the following: $$ a\le x \le 10\\ z_1 \lor z_2 \\ z_1 \implies w_1 \\ z_2 \implies w_2 \\ z_2 \implies x\ge b $$ The resulting linear constraints are: \begin{align} z_1 + z_2 &\ge 1 \\ z_1 &\le w_1 \\ z_2 &\le w_2 \\ a + (b-a)z_2 \le x &\le 10 \end{align}

---

If the disjunction is instead intended to be exclusive, you can omit the $z_i$ variables and impose: \begin{align} w_1 + w_2 &= 1 \\ a + (b-a)w_2 \le x &\le 10 \end{align}

Answer 2

From the modeling language point of view, the above disjunctive form can be written by CPLEX in the following form:

  c1: (y1==1 && x>=2) || (y2==1 && x>=3);
  c2: (y3==1 && x<=8) || (y4==1 && x==2.5);

or

  c1: (y1==1 || y2==1);
  c2: (y3==1 || y4==1);

and adding appropriate linearization constraints.