It is widely known that the following optimization problem \begin{equation} \text{min}_{x\in\mathbb{R}^n}\lVert{x}\rVert_2 \end{equation} is equivalent to the problem \begin{equation} \text{min}_{x,q} \hspace{3pt}q\\ \hspace{25pt} s.t. \hspace{15pt} q\geq\lVert{x}\rVert_2. \end{equation}
I don't quite understand the constraint here: in the original problem, we try to minimize $\lVert x \rVert_2$, but in the "equivalent" problem, we try to minimize a quantity that may be strictly greater than $\lVert x \rVert_2$. In this case, why are the two problems equivalent?
Besides this specific problem, when we call two optimization problems equivalent, what do we actually mean?
At optimal solution, we must have $q = \|x\|_2$.
If you have $(q_1, x)$ that satifies $q_1 > \|x\|_2$ and you can claim that it is optimal, I will choose $(\|x\|_2, x)$ and we can check that it is a better solution and contradicts your optimality claim.
We say two problems are equivalent if by solving one, we can recover the solution for the other one.