\frac{1}{N}\sum_{i=1}^{N}(1-y_i(x_i^Tw + b))_+
    + \frac{\rho_1}{2}\|B^T{\textbf{u}}+{\textbf{w}}\|_2^2 + \rho_2({\textbf{d}}^T{\textbf{u}}- b + 1)_+ + \frac{\rho_3}{2}\|D^T{\textbf{v}}-{\textbf{w}}\|_2^2
    + \rho_4({\textbf{g}}^T{\textbf{v}}+ b + 1)_+$. We then apply variable-splitting to decouple the $L_1$-norms and hinge losses and obtain the following equivalent constrained optimization problem: $$\begin{aligned}
\label{eq:ensvm_ksvm_c}
  \min_{{\textbf{w}},b,{\textbf{u}},{\textbf{v}},{\textbf{a}},{\textbf{c}},p,q} &&\>\>\>  F({\textbf{w}},b,{\textbf{u}},{\textbf{v}},{\textbf{a}},{\textbf{c}},p,q) \\
  \nonumber s.t. && \>\>\> {\textbf{a}}= {\textbf{e}}- (\bar{X}{\textbf{w}}+ {\textbf{y}}b), \>\>\>  {\textbf{c}}= {\textbf{w}}, \\
  \nonumber && \>\>\> q = {\textbf{d}}^T{\textbf{u}}- b + 1,   \>\>\> p = {\textbf{g}}^T{\textbf{v}}+ b + 1, \\
  \nonumber && \>\>\> {\textbf{u}}\geq {\textbf{0}}, \>\>\> {\textbf{v}}\geq {\textbf{0}}.\end{aligned}$$ with $F({\textbf{w}},b,{\textbf{u}},{\textbf{v}},{\textbf{a}},{\textbf{c}},p,q)
  \equiv \frac{\lambda_2}{2}\|{\textbf{w}}\|_2^2 + \lambda_1\|{\textbf{c}}\|_1 + \frac{1}{N}{\textbf{e}}^T({\textbf{a}})_+
    + \frac{\rho_1}{2}\|B^T{\textbf{u}}+{\textbf{w}}\|_2^2 + \rho_2(q)_+
+ \frac{\rho_3}{2}\|D^T{\textbf{v}}-{\textbf{w}}\|_2^2 + \rho_4(p)_+$. As usual, we form the augmented Lagrangian $\mathcal{L}$ of problem (\[eq:ensvm\_ksvm\_c\]), $$\begin{gathered}
    \mathcal{L} := F({\textbf{w}},b,{\textbf{u}},{\textbf{v}},{\textbf{a}},{\textbf{c}},p,q)
    + \gamma_1^T({\textbf{e}}-({\bar{X}}{\textbf{w}}+{\textbf{y}}b)-{\textbf{a}})
    + \frac{\mu_1}{2}\|{\textbf{e}}-({\bar{X}}+{\textbf{y}}b)-{\textbf{a}}\|_2^2\\
    + \gamma_2^T({\textbf{w}}-{\textbf{c}}) + \frac{\mu_2}{2}\|{\textbf{w}}-{\textbf{c}}\|_2^2
    + \gamma_3({\textbf{d}}^T{\textbf{u}}-b+1-q) + \frac{\mu_3}{2}\|{\textbf{d}}^T{\textbf{u}}-b+1-q\|_2^2 \\
    + \gamma_4({\textbf{g}}^T{\textbf{v}}+b+1-p) + \frac{\mu_4}{2}\|{\textbf{g}}^T{\textbf{v}}+b+1-p\|_2^2\end{gathered}$$ and minimize $\mathcal{L}$ with respect to ${\textbf{w}},b,{\textbf{c}},{\textbf{a}},p,q,{\textbf{u}},{\textbf{v}}$ individually and in order. For the sake of readability, we do not penalize the non-negative constraints for ${\textbf{u}}$ and ${\textbf{v}}$ in the augmented Lagrangian.

Given $({\textbf{a}}^k, {\textbf{c}}^k, p^k, q^k)$, solving for $({\textbf{w}},b)$ again involves solving a linear system $$\label{eq:ksvm_admm_linsys_wb}
        \left(
          \begin{array}{cc}
            \kappa_1 I + \mu_1 X^TX & \mu_1 X^T{\textbf{e}}\\
            \mu_1{\textbf{e}}^TX & \mu_1 N + \kappa_2 \\
          \end{array}
        \right)
         \left(
          \begin{array}{c}
            {\textbf{w}}^{k+1} \\
            b^{k+1} \\
          \end{array}
        \right) 