does not enforce sparsity in the solution, and $w$ is generally dense. The ENSVM adds an $L_1$ regularization on top of the ridge regularization term, giving $$\label{eq:en_svm}
    \min_{{\textbf{w}},b}\frac{1}{N}\sum_{i=1}^{N}(1-y_i({\textbf{x}_i}^T{\textbf{w}}+ b))_+ + \lambda_1\|{\textbf{w}}\|_1 + \frac{\lambda_2}{2}\|{\textbf{w}}\|_2^2.$$ Compared to the Lasso ($L_1$-regularized regression) [@tibshirani1996regression], the elastic-net has the advantage of selecting highly correlated features in groups (i.e. the grouping effect) while still enforcing sparsity in the solution. This is a particularly attractive feature for text document data, which is common in the hierarchical classification setting. Adopting the elastic-net regularization as in (\[eq:en\_svm\]) brings the same benefit to SVM for training classifiers.

To approximately solve problem (\[eq:en\_svm\]), we adopt the alternating direction method of multipliers (ADMM) for elastic-net SVM recently proposed in [@yeefficient]. ADMM has a long history dating back to the 1970s [@gabay1976dual]. Recently, it has been successfully applied to problems in machine learning [@boyd2010distributed]. ADMM is a special case of the inexact augmented Lagrangian (IAL) method [@rockafellar1973multiplier] for the structured unconstrained problem $$\label{eq:struct_unc}
    \min_x F(x) \equiv f(x) + g(Ax),$$ where both functions $f(\cdot)$ and $g(\cdot)$ are convex. We can decouple the two functions by introducing an auxiliary variable $y$ and convert problem (\[eq:struct\_unc\]) into an equivalent constrained optimization problem $$\begin{aligned}
\label{eq:struct_c}
  \min_{x,y} && 