Formulating a constraint to exclude a single point from the feasible region of an IP?

Consider a basic integer program such as:

$$\begin{align}
\min_x & \quad c^Tx \\
\text{s.t.} & \quad Ax \leq b \\
&\quad x_i \in \{-100,\ldots,100\}
\end{align}
$$

where $x \in \mathbb{Z}^n, A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R^m}$.

Say that I have a point $y \in \mathbb{Z}^n \cap [-100,100]^n$ that I would like to exclude from the feasible region.

I am wondering if there is an elegant way to do this without introducing new variables to the formulation? If there is no way to exclude $y$ from the feasible region without adding new variables to the formulation, then is an approach that allows me to do this by adding the fewest possible variables?

I should add that this question is loosely related to a different question that I posted here a few months ago (though there was no concrete answer given to that one either).

You need to add an integer cut of the form:
$$
\sum_i |x_i - y_i| \ge 1
$$
where $y_i$ is your point your want to exclude.

We can rewrite this as:
$$
\begin{align}
&\sum_i z_i \ge 1 \\
&z_i \le |x_i - y_i|
\end{align}
$$
The last inequality is identical to $z_i \le \max\{x_i-y_i, -(x_i-y_i)\}$. This again can be implemented using additional binary variables:
$$
\begin{align}
&z_i\le x_i-y_i+M_i\delta_i \\
&z_i\le -(x_i-y_i)+M_i(1-\delta_i) \\
&\delta_i \in \{0,1\} \\
&M_i = U_i - L_i = 200
\end{align}
$$
There are two special cases we can handle more efficiently. If $y_i=L_i=-100$ then we can write $z_i = x_i - y_i$. If $y_i=U_i=100$ then we can write $z_i=-(x_i-y_i)$.

StackExchange Computer Science Q#51645, answer score: 3