# Expressing a logical constraint in integer programming

Let $x$ be an integer such that $x \in [ - 10,10]$ and $b$ a binary variable. Apply integer programming to express $b = 1 \leftrightarrow x \ge3$

My work:

$$\begin{cases} \text{if}\,\,\,\,\ b = 1, & \text{then}\,\,\,\,\,\,\ x \ge3 \\ \text{if}\,\,\,\,\ x \ge3, & \text{then}\,\,\,\,\,\,\ b = 1 \end{cases}$$

I got the following encoding $$3 + {M_1}(1 - b) \le x \le 2 + {M_2}b$$where $${M_1} \le- 13{\text{ and }}{M_2} \ge8$$

Checking:

Case: $b = 0$ $$3 + {M_1} \le x \le 2$$For ${M_1} \le - 13$, we have $x = - 10, - 9, \ldots ,2$

Case: $b = 1$ $$3 \le x \le 2 + {M_2}$$ For ${M_2} \ge 8$, we have $x = 3, 4, \ldots ,10$

Thus, by choosing $b = \{ 0,1\}$ we can force $x$ to take any value from its domain.

Let $x \in [-10,10] \cap \mathbb Z$ and $y \in \{0,1\}$. Suppose that

$$y = \begin{cases} 1 & \text{if } x \geq 3\\ 0 & \text{if } x \leq 2\end{cases}$$

To ensure that $y \neq 0$ when $x \geq 3$, use

$$y \geq \frac{x-2}{8}$$

To ensure that $y \neq 1$ when $x \leq 2$, use

$$y \leq \frac{x+10}{13}$$

Pictorially,

The trivial inequality constraints are $-10 \leq x \leq 10$ and $0 \leq y \leq 1$. Lastly, the integrality constraints $x, y \in \mathbb Z$ should not be forgotten.