Algorithm 1. Solving BLSR by Inexact ALM

Input: Training data $X$. Parameters $\alpha$, $\beta$ and $\lambda$.

Initialization: $Z^0=J^0=E^0=0$, $Y_1^0=Y_2^0=Y_3^0=0$,

${\mathsf{\mu }}^0={10}^{-5}$, ${\mathsf{\mu }}_{max}={10}^8$, $\mathsf{\epsilon }={10}^{-6}$, $\mathsf{\rho }=1.1$.

1: While not converged do

2: Fix other variables and optimize $Z^{k+1}$ via (14).

3: Fix other variables and optimize $E^{k+1}$ via (15).

4: Fix other variables and optimize $J^{k+1}$ via (16).

5: Fix other variables and optimize $L^{k+1}$ via (18).

6: Update the multipliers and $\mathsf{\mu }$ via (19).

7: Check the convergence conditions:

$\Vert X-X{Z}^{k+1}-E^{k+1}{\Vert }_{\infty }<\epsilon$, $\Vert Z^{k+1}-L^{k+1}{\Vert }_{\infty }<\epsilon$

$\Vert Z^{k+1}-J^{k+1}{\Vert }_{\infty }<\epsilon$

8: End while

Output: $Z$