non-differentiability of the square-root terms

non-differentiability of the square-root terms (and l1 norms) (see Xu et al., ICML 2013)

Given any g(x) > 0, the following holds:

\sqrt(g(x)) = \min_{z>0} \frac{1}{2}[\frac{g(x)}{z}+z].

Then add this auxiliary variable to your cost.