Are the following notations equivalent for $\vec{x}\in\mathbb{R}^n$ and $f\colon \mathbb{R}^n \to \mathbb{R}$ ?
$$\underset{\vec{x}}{\text{min}}\ f(\vec{x}) \overset{?}{=} \underset{x_1,\dots,x_n}{\text{min}}\ f(x_1,\dots,x_n)\overset{?}{=}\underset{x_1}{\text{min}}\left\{\ \underset{x_2}{\text{min}}\left\{\dots\underset{x_n}{\text{min}}\ f(x_1,\dots,x_n)\right\}\right\} $$
I think the notations would be equivalent if you replaced $\min$ by $\inf$. However, with $\min$, you run into the problem that the inner minima may not exist. For instance, if you minimize $x^2\mathrm e^y$, the overall minimum is $0$ (attained on the line $x=0$), but in the componentwise minimization, the minimum over $y$ doesn't exist for $x\ne0$. The infimum, on the other hand, always exists; in this case the infimum over $y$ is $0$ for all $x$.