Model disjunction in a $\{0,1\}$ integer linear program
Let's assume that all variables are Boolean (constrained to $\{0,1\}$). Here is how to encode the constraint "$x+y=1$ or $z=1$". Presumably you can extend this.

$$
\begin{align*}
&amp;x+y+u-v = 1 \\
&amp;z+w = 1 \\
&amp;w+u+v \leq 1
\end{align*}
$$

If $x+y = 1$ then we take $u=v=0$ and $w=1-z$.

If $z = 1$ and $x+y = 0$ then we take $u=1$, $v=0$ and $w=0$.

If $z = 1$ and $x+y = 2$ then we take $u=0$, $v=1$ and $w=0$.

This shows that these equations are solvable when $x+y=1$ and when $z=1$.

Conversely, suppose that these equations are solvable, and so at most one of $w,u,v$ is non-zero. If $w = 0$ then $z = 1$. Otherwise, $u=v=0$ and so $x+y = 1$.