$$f(x)=\begin{cases} -\frac{5}{2} + 2 |x| & |x|>0\\ -|\frac{x}{2}| & |x|\le0 \end{cases}.$$
How to calculate the conjugate gradient?
I found out that I can rewrite it to $max(-\frac{5}{2} + 2 |x|, -|\frac{x}{2}|)$...
I have found related examples: http://www.onmyphd.com/?p=legendre.fenchel.transform