Linear programming: reduce a contstraint that includes minimun

I have an almost linear programme. However one of the constraints has a form $z = min(x,y)$ (all the other things are linear in the model). Is there a way to substitute this with something (or introduce additional variables) to turn this into a linear programme?

In other words, I have the problem that looks like the following:
$$
\mathbf c' \mathbf x \to \min,
$$
s.t.
$$
A \mathbf x = \mathbf b,\quad x_1 = \min(x_2, x_3).
$$

Update: I thought about substituting the constraint with $\min$ to the following pair: $x_1 \le x_2$ and $x_1 \le x_3$. However, this doesn't work if $x_1$ has a positive coefficient in $\mathbf c$, which is exactly the case for me. In fact, all the entries are positive in $\mathbf c$.

Solve two linear programs. The first has the constraints
$$
A \mathbf x = \mathbf b,\quad x_1 = x_2 \le x_3.
$$
The second has the constraints
$$
A \mathbf x = \mathbf b,\quad x_1 = x_3 \le x_2.
$$
Choose whichever solution gives you a smaller value for $\mathbf c' \mathbf x$.

This works for your particular situation. It doesn't scale to a case where you have a large number of $\min$ constraints.

StackExchange Computer Science Q#118981, answer score: 3