f(x) + g(y), \quad s.t.  \>\> Ax = y.\end{aligned}$$ This technique is often called variable-splitting [@combettes2011proximal]. The IAL method approximately minimizes in each iteration the augmented Lagrangian of (\[eq:struct\_c\]) defined by $\mathcal{L}(x,y,\gamma) := f(x) + g(y) + \gamma^T(y-Ax) + \frac{\mu}{2}\|Ax-y\|_2^2$ , followed by an update to the Lagrange multiplier $\gamma \gets \gamma - \mu(Ax - y)$. The IAL method is guaranteed to converge to the optimal solution of (\[eq:struct\_unc\]), as long as the subproblem of approximately minimizing the augmented Lagrangian is solved with an increasing accuracy [@rockafellar1973multiplier]. ADMM can be viewed as a practical implementation of IAL, where the subproblem is solved approximately by minimizing $\mathcal{L}(x,y;\gamma)$ with respect to $x$ and $y$ [*alternatingly once*]{}. Eckstein and Bertsekas [@eckstein1992douglas] established the convergence of ADMM for the case of two-way splitting. Now applying variable-splitting and ADMM to problem (\[eq:en\_svm\]), [@yeefficient] introduced auxiliary variables $({\textbf{a}},{\textbf{c}})$ and linear constraints so that the non-smooth hinge loss and $L_1$-norm in the objective function are decoupled, making it easy to optimize over each of the variables. Specifically, problem (\[eq:en\_svm\]) is transformed into an equivalent constrained form $$\begin{aligned}
\label{eq:en_svm_constr}
    \min_{{\textbf{w}},b,{\textbf{a}},{\textbf{c}}} && \frac{1}{N}\sum_{i=1}^N (a_i)_+ + \lambda_1\|\textbf{c}\|_1 + \frac{\lambda_2}{2}\|{\textbf{w}}\|_2^2 \\
    \nonumber s.t. && \textbf{a} = {\textbf{e}}- Y(X{\textbf{w}}+ b{\textbf{e}}) \quad \mbox{and} 