# Smoothing L1 norm, Huber vs Conjugate

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.

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Smoothing the dual will not give you a smooth primal. However, you get a strongly convex primal by dual smoothing (as opposed to merely a strictly convex primal by Huber smoothing). Hence, it depends on what kind of regularity you are aiming at: A smoother primal or a "more convex" primal - both can be helpful algorithmically. Moreover, note that there are numerous methods to treat nonsmooth convex minimization problems efficiently. –  Dirk Jan 8 at 7:15
@Dirk: why don't you write an answer about these? What you write here might interest more people. –  András Bátkai Jan 8 at 7:18