System of Linear Inequalities to create feasible solution $y=\min(x_1,x_2)$

Question

I have a question in optimization. The question is $L_{1}\leq x_{1}\leq U_{1}$,..., $L_{n}\leq x_{n}\leq U_{n}$. Can we introduce decision variables and define a system of mixed-integer linear inequalities whose feasible solution is $y=\min(x_{1},...,x_{n})$?

Additional requirement for the question is include the fewest possible number of constraints and decision variables.

Thank you.

Answer 1

I assume that there are other constraints on the $x$ variables that are not listed; otherwise, just add the constraints $x_i=L_i$ and $y=\min_i L_i$. With that assumption in place, you could add binary variables $z_1,\dots, z_n$, along with constraints $y_i \le x_i \,\forall i$, $y_i \ge x_i - U_i (1-z_i)\,\forall i$, and $\sum_i z_i = 1$.