I have some trouble understanding this apparently simple concept. Let's say we have to do a minimization of an integral and I need to find L* $$ L^{*} = \min_{z(x)} \int f(x,z(x)) dx$$
Can I bring the min inside and thus write:
$$ L^{*} = \int \min_{z(x)} f(x,z(x)) dx$$
If so, why? Intuitively make sense, but I can't really see the mathematical reason behind.
On
First, the notation $ \min\limits_{z(x)}$ is very confusing and should not be used.
The minimum is defined for an object belonging to a set. You should write $\min\limits_{z \in A}$ and define what the set $A$ is.
Second in the case you describe, I imagine that the set $A$ is a set of functions. But which one? Continuous? Integrable?
Finally when everything will be properly defined, but only then, it is unlikely that the equality you wrote holds.
No, you cannot do this. Let's look at a simpler example, where you're minimizing over a real number rather than a function:
$$\min_a \int_0^1 (x-a)^2\,dx.$$
Here you need to pick a single value of a for which the integral is minimized; the best you can do is $a=\frac{1}{2}$, which gives a value of $\frac{1}{12}$.
Now if the minimimum is on the inside,
$$\int_0^1 \min_a\ (x-a)^2\,dx,$$
you are allowed to pick a different $a$ for every $x$ and can do much better than the single-$a$ case. In particular, you can just pick $a=x$ for every $x$, giving the integral a value of zero.