What are the objective values of min (min($x_1,x_2$)) such that $x_1,x_2 \geq 0$

Question

1. min (min($x_1,x_2$)) such that $x_1,x_2 \geq 0$
2. min $t$ such that $t \leq x_1, t \leq x_2, x_1,x_2 \geq 0$

What are the objective values of 1 and 2.

For part 1, it is a one dimensional question? So i imagine a real line where variables $x_1,x_2$ are on it, and so to minimize the minimum, without loss of generality, it is $0$. Hope i am right.

For question 2, i do not quite know how to do, is not part 2 the same formulation as question 1? The answer says it is unbounded...

Answer 1

Unclear what the question is.

The minimum to the first problem is $0$ (just let any or both of $x_1$ and $x_2$ be zero), while in the second model the minimum (well, infimum) is $-\infty$ ($t$ can tend to $-\infty$ while $x_1\geq 0$ and $x_2\geq 0$ are arbitrary).

It vaguely looks like a question/statement trying to illustrate that you cannot use an epigraph reformulation to model the concave min-operator when minimizing it.