How to model $C_1=C_2$ implies $b_1 = b_2$

Question

Suppose $C_1 \ge 0$, $C_2 \ge 0$ are continuous variables and $b_1$, $b_2$ are binary variables.

How could I model the following?

$C_1 = C_2 \implies b_1 = b_2$, the opposite does not hold.

Answer 1

Equivalently, you want to enforce the contrapositive $$b_1 \not= b_2 \implies C_1 \not= C_2$$ Because the $b_i$ are binary, this is the same as $$b_1 + b_2 = 1 \implies C_1 \not= C_2$$ Let $\epsilon>0$ be a small constant tolerance, and suppose that $C_i \le M_i$. Introduce binary variables $y_i$ and $z_i$, and impose linear constraints \begin{align} b_1 + b_2 &= 2y_1 + y_2 \tag1\label1 \\ y_2 &\le z_1 + z_2 \tag2\label2 \\ C_1 - C_2 + \epsilon &\le (M_1 + \epsilon) (1-z_1) \tag3\label3 \\ C_2 - C_1 + \epsilon &\le (M_2 + \epsilon) (1-z_2) \tag4\label4 \end{align} Constraint \eqref{1} enforces $b_1 + b_2 = 1 \implies y_2$. Constraint \eqref{2} enforces $y_2 \implies (z_1 \lor z_2)$. Constraint \eqref{3} enforces $z_1 \implies C_1 + \epsilon \le C_2$. Constraint \eqref{4} enforces $z_2 \implies C_2 + \epsilon \le C_1$.

Answer 2

You can try the following
Lets define 3 binary variables $y_1,y_2,z$
Additional constraints with $\delta$ as small positive number
$c_1-c_2 + \delta \le My_1$
$c_2-c_1-\delta \le M(1-y_1)$
$c_2-c_1+\delta \le My_2$
$c_1-c_2-\delta \le M(1-y_2)$
$y_1+y_2 \le 1+z$
$z\le y_1,y_2$
Then
$ b_2+z-1 \le b_1 \le b_2 + 1-z$