${\textbf{t}}$ and solve the linear system $$\label{eq:ksvm_admm_linsys_v}
  (\rho_3DD^T + \mu_4 {\textbf{g}}{\textbf{g}}^T + \mu_6 I){\textbf{v}}^{k+1} = {\textbf{r}}_{\textbf{v}},$$ where ${\textbf{r}}_{\textbf{v}}= \rho_3D{\textbf{w}}^{k+1} - \mu_4{\textbf{g}}b^{k+1} - {\textbf{g}}\gamma_4^k - \mu_4{\textbf{g}}(1-p^k) + \gamma_6 + \mu_6{\textbf{t}}^k$, and ${\textbf{t}}^{k+1} = \max(0,{\textbf{v}}^{k+1} - \frac{\gamma_6^k}{\mu_6})$.

Now given $({\textbf{w}}^{k+1},b^{k+1},{\textbf{u}}^{k+1},{\textbf{v}}^{k+1})$, the solutions for ${\textbf{a}}$ and ${\textbf{c}}$ are exactly the same as in Lines \[line:a\_sol\] and \[line:c\_sol\] of Algorithm \[alg:hipad\], i.e. $$\begin{aligned}
  {\textbf{a}}^{k+1} &=& \mathcal{S}_{\frac{1}{N\mu_1}}\left( {\textbf{e}}+ \frac{\gamma_1^k}{\mu_1} - Y(X{\textbf{w}}^{k+1} + b^{k+1}{\textbf{e}}) \right), \\
  {\textbf{c}}^{k+1} &=& \mathcal{T}_{\frac{\lambda_1}{\mu_2}}\left( \frac{\gamma_2^k}{\mu_2}+{\textbf{w}}^{k+1} \right).\end{aligned}$$ The subproblem with respect to $q$ is $$\begin{aligned}
\label{eq:ksvm_admm_qsub}
    \nonumber \min_q && \quad \rho_2(q)_+ - \gamma_3^k q  + \frac{\mu_3}{2}\|{\textbf{d}}^T{\textbf{u}}^k - b^{k+1} + 1 - q\|_2^2 \equiv \\
     && \rho_2(q)_+
     + \frac{\mu_3}{2}\|q - ({\textbf{d}}^T{\textbf{u}}^k - b^{k+1} + 1 + \frac{\gamma_3^k}{\mu_3})\|_2^2.\end{aligned}$$ The solution is given by a (one-dimensional) proximal operator associated with the hinge loss $$\label{eq:ksvm_admm_qsol}
    q^{k+1} = \mathcal{S}_{\frac{\rho_2}{\mu_3}}\left( {\textbf{d}}^T{\textbf{u}}^k - b^{k+1} + 1 + \frac{\gamma_3^k}{\mu_3} \right).$$ Similarly, the subproblem with respect to $p$ is $$\min_p \quad\quad \rho_4(p)_+ - \gamma_4^k p + \frac{\mu_4}{2}\|{\textbf{g}}^T{\textbf{v}}^k + b^{k+1} + 1 - p\|_2^2,$$ and the solution is given by $$\label{eq:ksvm_admm_psol}
    p^{k+1} = \mathcal{S}_{\frac{\rho_4}{\mu_4}}\left( {\textbf{g}}^T{\textbf{v}}^k + b^{k+1} + 1 + \frac{\gamma_4^k}{\mu_4} \right).$$ Due to lack of space in the paper, we summarize the detailed solution steps in the Appendix (see Algorithm \[alg:admm\_enk\] 