Let's consider the following minimization problem:
\begin{align} \min_{x,a,b}&\quad X\tag1\\ \text{s.t.}&\quad X = \min(A,B)\tag2\end{align}
with $A,B$ functions that depend on $X$.
Is there a way to represent $(2)$ as a continuous constraint? i.e., I don't want to use binary variables.
I can't do: \begin{align} X \leq A \\ X \leq B \end{align} Because the minimization of $X$ would give me $X = 0$, and \begin{align} X \geq A \\ X \geq B \end{align} would give me the max.