I'm trying to minimize a convex (not necessarily strictly convex) function involving an L1 norm (similar to lasso), which makes it non-differentiable at some points. So I'd like to smooth it and treat it as an L2 norm problem.

The two approaches I've seen (http://www.ee.ucla.edu/~vandenbe/236C/lectures/smoothing.pdf) are directly smoothing the L1 norm using the Huber function, and smoothing the conjugate (i.e, derive the dual norm, here it's L-infinity, which is still non-differentiable, then smooth that).

The Huber approach is much simpler, is there any advantage in the conjugate method over Huber? I can't see the point of smoothing the dual instead of just smoothing the primal.

I'm trying to minimize a convex (not necessarily strictly convex) function involving an L1 norm (similar to lasso), which makes it non-differentiable at some points. So I'd like to smooth it and treat it as an L2 norm problem.

The two approaches I've seen (  http://www.ee.ucla.edu/~vandenbe/236C/lectures/smoothing.pdf  ) are directly smoothing the L1 norm using the Huber function, and smoothing the conjugate (i.e, derive the dual norm, here it's L-infinity, which is still non-differentiable, then smooth that).

The Huber approach is much simpler, is there any advantage in the conjugate method over Huber? I can't see the point of smoothing the dual instead of just smoothing the primal.