if $x = 0$ then $y \ne b$

Question

I'm trying to model the following:

if $x=0$ then $y \ne b$

$y$ is a positive integer number( $y\le U$) and $x$ is binary and $b$ is a constant.

Answer 1

Introduce binary variables $z_1$, $z_2$, and $z_3$, and impose linear constraints \begin{align} z_1+z_2 +z_3&= 1 \tag1\\ 1z_1+bz_2+(b+1)z_3 \le y &\le (b-1)z_1+bz_2+Uz_3 \tag2\\ z_2&\le x\tag3 \end{align} Constraints $(1)$ and $(2)$ enforce the three disjoint cases $y<b$, $y=b$, and $y>b$. Constraint $(3)$ enforces $x=0\implies z_2=0$, and $z_2=0$ excludes the $y=b$ case.