What is the difference between $min_x$ and $min$

Question

I have three functions $f(x)$, $g(x)$, and $h(x)$. I want to know the definition and result of the following minimizations:

1. $\min_x(f(x),g(x),h(x))$ - with $x$ as subscript

2. $\min(f(x),g(x),h(x))$ - without $x$

3. $\min_x\big[\min_x(f(x),g(x),h(x))\big]$

4. $\min\big[\min(f(x),g(x),h(x))\big]$

Update:

5. $\min_{\mathbf{x}}(f(x_1),g(x_2),h(x_3))$

Answer 1

As I understand the (quite unclear) notation :

$\min(f(x),g(x),h(x))$ means that we take the minimum between $f(x), g(x)$ and $h(x)$. Then it is implicitely a function of $x$. The function $x \mapsto \min (f(x),g(x),h(x))$.

$\min_x (f(x),g(x),h(x))$ means that we take the minimum of $\min(f(x),g(x),h(x)) $ for $x$ over a set. It is then a number. A better notation would be :

$$\min_x \min(f(x),g(x),h(x)).$$

To make it (almost) perfect, just specify where $x$ is allowed to vary. For example :

$$\min_{x \in A} \min(f(x),g(x),h(x)).$$

I hope this is enough for 1 and 2.

3 and 4 are not meaningless but the first minimum is irrelevant. Indeed, for 3, you are taking the minimum over a set of a function being constant, then the result is this constant. For 4, it is almost the same, you are taking the minimum of a finite set with only one element.

I may add that you should be carefull with the use of $\min$ in optimization. Indeed, we are talking about minimums only when it can be achieved. It is safer to use $\inf$ unless you know (by assumptions or arguments), that the minimum exists.