How many methods for smoothing an unsmoothed function? Which is the simplest one? For example, we smooth $f(x)=|x|$ to $$f(x)=\begin{cases} \frac{x^2}{\epsilon}+\frac{\epsilon}{2} & |x| \le \epsilon\\\ |x| & |x|\ge epsilon \end{cases}$$

In convex analysis, maybe the most natural or best way to smooth a convex function $f$ is to use the Moreau-Yosida regularization of $f$:

\begin{equation} f^{(\mu)}(x) = \inf_u f(u) + \frac{1}{2\mu} \|u - x\|_2^2. \end{equation}

This is discussed in lecture 15 ("Multiplier methods") of Vandenberghe's 236c notes.

If $f(x) = |x|$ for all $x \in \mathbb R$, and $\mu > 0$, then \begin{equation} f^{(\mu)}(x) = \begin{cases} |x| - \frac{\mu}{2} \quad \text{if } |x| > \mu \\\ \frac{x^2}{2\mu} \quad \text{otherwise}. \end{cases} \end{equation}