Smooth Reformulation of NonSmooth Constraints If I have something like :
\begin{align}
\min_x \max_i f_i(x)
\end{align}
I can reformulate this nonsmooth formulation as:
$$\min_x z$$
$$z\geq f_i(x)$$
and I have a smooth formulation of the problem.

Now, Suppose I have :
$$\min_x \min_i f_i(x)$$
Can I reformulate it as a smooth version?

I tried:
$$\min_x z$$
$$z\leq f_i(x)$$
but then, $z$ can always take the value $-\infty$. 

My question is: Is a smooth formulation of min of mins even possible? My gut feeling is no but I wanted to be sure.
In the first case you put a lower bound for a minimization problem. In the second case you put an upper bound for a minimization problem which is practically non-sense.
In the first case you put a lower bound for a minimization problem. In the second case you put an upper bound for a minimization problem which is practically non-sense.