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 a pair of constraints $x_1 \le x_2$, $x_1 \le x_3$ but it doesn't work if the coefficient of $x_1$ is positive in $\mathbf c$. And this is the case in my problem (actually all the entries of $\mathbf c$ a positive/nonnegative).