Modeling a continuous variable which can't take values between a and b

Question

Consider two binary variables $p_1$ and $p_2$. Suppose, $x$ is a continuous variable that should not take values between $a$ and $b$.

Here is my try:

$$ p_1 =1 \mbox{ if } x \le a \\ p_1 =0 \mbox{ if } x >a \\ p_2 =1 \mbox{ if } x \ge b \\ p_2 =0 \mbox{ if } x < b $$ Now what will be the final equations?? $$ p_1+p_2=1 $$

Answer 1

You want to model $$ x \le a \mbox{ or } x \ge b $$ which you can enforce with a binary variable $p\in \{0,1\}$ and big M constraints: \begin{align} b - M_1p \le x&\le a +M_2(1-p) \\ \end{align}

The LHS enforces $p=0 \implies x\ge b$, while the RHS enforces $p=1 \implies x \le a$:

• if $p=0$, the constraint becomes $$ b \le x\le a +M_2 $$
• if $p=1$, the constraint becomes $$ b - M_1 \le x \le a $$

If $x \ge 0$, you could set $M_1$ to $b$. $M_2$ is either an upper bound on $x-a$ (if available), or a large constant.

Answer 2

Assuming $ b \lt a$ you can try

$ x + \epsilon \le b + Mp$
if $ p=0$ then $ x \lt b$, the 2nd constr becomes meaningless.

$ a+\epsilon \le x + M(1-p)$
if $p=1$ $x \gt a$, 1st constr turns meaningless & since $ b \lt a$ so $x$ cant be $[b,a]$

$M$ is a reasonably big number, maybe upper bound of $x$ and $\epsilon$ is a very small number to take care $\lt$.