  We deal with the problem of numerically computing the dual norm, which is
important to study sparsity-inducing regularizations (Jenatton et al. 2011,Bach
et al. 2012). The dual norms find application in optimization and statistical
learning, for example, in the design of working-set strategies, for
characterizing dual gradient methods, for dual decompositions and in the
definition of augmented Lagrangian functions. Nevertheless, the dual norm of
some well-known sparsity-inducing regolarization methods are not analytically
available. Examples are the overlap group $\ell_2$-norm of (Jenatton et al.
2011) and the elastic net norm of Zhou and Hastie (2005). Therefore we resort
to the Majorization-Minimization principle of Lange (2016) to provide an
efficient algorithm that leverages a reparametrization of the dual constrained
optimization problem as unconstrained optimization with barrier. Extensive
simulation experiments have been performed in order to verify the correctness
of operation, and evaluate the performance of the proposed method. Our results
demonstrate the effectiveness of the algorithm in retrieving the dual norm even
for large dimensions.
