Algorithm 6.

Multi-block ADMM to solve iGecco+

while not converged do

for all $k=1,\cdots ,K$ do

$U^{\left(k\right)}=\underset{U}{\mathrm{argmin}}{\pi }_k{\ell }_k(X^{\left(k\right)},U)+\frac{\rho }{2}{\Vert DU-V^{\left(k\right)}+{\mathbf{\Lambda }}^{\left(k\right)}\Vert }_F^2+\alpha {\sum }_{j=1}^{p_k}{\zeta }_j^{\left(k\right)}{\Vert U_{.j}-{\tilde{x}}_j^{\left(k\right)}\cdot {\mathbf{1}}_n\Vert }_2$

$U^{(k^{\prime })}=\underset{U}{\mathrm{argmin}}\frac{\rho }{2}{\Vert X^{(k^{\prime })}-U+Z^{(k^{\prime })}+{\mathbf{\Psi }}^{(k^{\prime })}\Vert }_F^2+\frac{\rho }{2}{\Vert U-{\tilde{X}}^{(k^{\prime })}-R^{(k^{\prime })}+N^{(k^{\prime })}\Vert }_F^2+\frac{\rho }{2}{\Vert DU-V^{(k^{\prime })}+{\mathbf{\Lambda }}^{(k^{\prime })}\Vert }_F^2$

$Z^{(k^{\prime })}=\underset{Z}{\mathrm{argmin}}{\pi }_{k^{\prime }}{f}_{k^{\prime }}\left(Z\right)+\frac{\rho }{2}{\Vert X^{(k^{\prime })}-U^{(k^{\prime })}-Z+{\mathbf{\Psi }}^{(k^{\prime })}\Vert }_F^2$

$R^{(k^{\prime })}=\underset{R}{\mathrm{argmin}}\alpha {\sum }_{j=1}^{p_{k^{\prime }}}{\zeta }_j^{(k^{\prime })}{\Vert U_{.j}^{(k^{\prime })}-{\tilde{x}}_j^{(k^{\prime })}\cdot {\mathbf{1}}_n\Vert }_2+\frac{\rho }{2}{\Vert U^{(k^{\prime })}-{\tilde{X}}^{(k^{\prime })}-R+N^{(k^{\prime })}\Vert }_F^2$

${\mathbf{\Psi }}^{(k^{\prime })}={\mathbf{\Psi }}^{(k^{\prime })}+(X^{(k^{\prime })}-U^{(k^{\prime })}-Z^{(k^{\prime })})$

$N^{(k^{\prime })}=N^{(k^{\prime })}+(U^{(k^{\prime })}-{\tilde{X}}^{(k^{\prime })}-R^{(k^{\prime })})$

end for

$V=\underset{V}{\mathrm{argmin}}\frac{\rho }{2}{\Vert D{U}^{\left(k\right)}-V^{\left(k\right)}+{\mathbf{\Lambda }}^{\left(k\right)}\Vert }_F^2+\frac{\rho }{2}{\Vert D{U}^{(k^{\prime })}-V^{(k^{\prime })}+{\mathbf{\Lambda }}^{(k^{\prime })}\Vert }_F^2+\gamma \left({\sum }_{l\in \epsilon }{w}_l{\Vert V_{l.}\Vert }_2\right)$

${\mathbf{\Lambda }}^{\left(k\right)}={\mathbf{\Lambda }}^{\left(k\right)}+(D{U}^{\left(k\right)}-V^{\left(k\right)})$ for all k and k′

end while