\quad {\textbf{c}}= {\textbf{w}}$$ where ${\textbf{x}_i}^T$ is the $i$-th row of $X$, and $Y = \textrm{diag}({\textbf{y}})$. The augmented Lagrangian $$\begin{gathered}
    \mathcal{L}({\textbf{w}},b,{\textbf{a}},{\textbf{c}},\gamma_1,\gamma_2) := \frac{1}{N}\sum_{i=1}^N a_i + \lambda_1\|\textbf{c}\|_1 + \frac{\lambda_2}{2}\|{\textbf{w}}\|_2^2  + \gamma_1^T({\textbf{e}}- Y(X{\textbf{w}}+ b{\textbf{e}})-{\textbf{a}})\\
    +\gamma_2^T({\textbf{w}}-{\textbf{c}})  + \frac{\mu_1}{2}\|{\textbf{e}}- Y(X{\textbf{w}}+ b{\textbf{e}})-{\textbf{a}}\|_2^2 + \frac{\mu_2}{2}\|{\textbf{w}}-{\textbf{c}}\|_2^2\end{gathered}$$ is then minimized with respect to $({\textbf{w}},b), {\textbf{a}},$ and ${\textbf{c}}$ sequentially in each iteration, followed by an update to the Lagrange multipliers $\gamma_1$ and $\gamma_2$. The original problem is thus decomposed into three subproblems consisting of computing the proximal operator of the hinge loss function (with respect to ${\textbf{a}}$), solving a special linear system (with respect to $({\textbf{w}},b)$ ), and performing a soft-thresholding operation (with respect to ${\textbf{c}}$), which can all be done in an efficient manner. Due to lack of space in the paper, we have included the detailed solution steps in the Appendix (see Algorithm \[alg:admm-ensvm\] ADMM-ENSVM), where we define by $\mathcal{S}_\lambda(\cdot)$ the proximal operator associated with the hinge loss $$\begin{aligned}
  \mathcal{S}_\lambda(\omega) &=& \left\{
                                    \begin{array}{ll}
                                      \omega - \lambda, & \hbox{$\omega > \lambda$;} \\
                                      0, & \hbox{$0 \leq \omega \leq \lambda$;} \\
                                      \omega, & \hbox{$\omega < 0$.}
                                    \end{array}
                                  \right.\end{aligned}$$ and $\mathcal{T}_\lambda(\omega) = sgn(\omega)\max\{0,|\omega|-\lambda \}$ is the shrinkage operator.

SVM via Interior-Point Method {#sec:ipm}
-----------------------------

Interior Point Methods enjoy fast convergence rates 