Algorithm 1: Augmented Lagrangian method.

Fixed:$k=0, \rho >0, {\lambda }_0, u_0$

Iterations: for $k=k+1 \mathrm{until} \mathrm{the} \mathrm{stopping} \mathrm{criterion} \mathrm{is}$

satisfied, repeat steps 1–2

1. Keep λ fixed and update u:

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{u}_{k+1}=\underset{u}{ \min }f\left(u\right)-{\lambda }_k^T\left(Hu-g\right)+\frac{\rho }{2}\Vert Hu-g{\Vert }_2^2$$

2. Keep u fixed and update λ:

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\lambda }_{k+1}={\lambda }_k+\rho \left(H{u}_{k+1}-g\right)$$

Output:u the optimal solution.