Is it possible to convert an equality constraint involving the minimum, to a linear inequality constraint?
Suppose I have an optimization problem which involves the variables $x_1,\,x_2,\,x_3$, with the constraint \begin{equation} x_1 =\min(x_2,\,x_3)\;. \end{equation} I would like to convert it to a standard linear inequality constraint. My naive attempt was \begin{equation} x_1 =\min(x_2,\,x_3) \to x_1 \leq \min(x_2,\,x_3)\, \land\, x_1 \geq \min(x_2,\,x_3) \end{equation} Then, \begin{equation} x_1 \leq \min(x_2,\,x_3) \to x_1 \leq x_2 \land x_1 \leq x_3\;. \end{equation} But I am stuck with \begin{equation} x_1 \geq \min(x_2,\,x_3)\;, \end{equation} which I don't know how to linearize.