Equivalence of two optimization problem when introducing an auxiliary variable

Question

Consider two optimization problems:

$$ (P1:) \ \ \min\limits_{x \in X} x $$

$$ (P2:) \ \ \min\limits_{x \in X, x \le y} y $$

Many authors said that P1 and P2 are equivalent, but without any further explanation. In my opinion, P1 can be seen as $\inf\nolimits_{x \in X}x$, which is the maximal lower bound of $x$. According to this idea, P1 should be equivalent to the following P3:

$$ (P3:) \ \ \max\limits_{x \in X, x \ge y} y $$

But the fact is that P1 is indeed not equivalent to P3. May I ask where my thinking is wrong? Why are P1 and P2 equivalent?

Answer 1

You just seem to read the notation the wrong way.

P3 means to maximize $y$ on the set

$$\{(x,y)~:~x\in X,\ x\geq y\}.$$

Whereas you seem to think that P3 means to maximize $y$ on the set

$$\{y~:~\forall_{x\in X} x\geq y\}.$$

Similar considerations apply to your reading of P2.