Algorithm 1. Optimization of CDNSC via ADMM

Input:$\mathit{A},\mathit{y},\alpha ,\beta ,\mu ,{\rho }_{\max },k_{\max }$, and $\epsilon$.

Output: The optimal ${\tilde{\mathit{y}}}^{\ast }$, ${\tilde{\mathit{A}}}^{\ast }$, ${\tilde{\mathit{e}}}^{\ast }$, and ${\mathit{x}}^{\ast }$.

Initialization:$k=0$, $x^k=1/L$.

Repeat

1: $k=k+1$;

2: Estimate weight matrix ${\mathit{W}}^k$ by (17) and (20);

Update:$\tilde{\mathit{A}}=\sqrt{{\mathit{W}}^k}\mathit{A}$ and $\tilde{\mathit{y}}=\sqrt{{\mathit{W}}^k}\mathit{y}$.

Initialization:$l=0$, ${\mathit{x}}^l={\mathit{x}}^k$, ${\mathit{\varphi }}_1^l$, ${\mathit{\varphi }}_2^l$, ${\mathit{\varphi }}_3^l$, and ${\rho }^l$.

Repeat

3: $l=l+1$;

4: Estimate ${\tilde{\mathit{e}}}^l$ by (43);

5: Update ${\mathit{B}}^l$ by (32);

6: Estimate ${\mathit{u}}^l$ by (45);

7: Estimate ${\tilde{\mathit{x}}}^l$ by (48);

8: Update ${\mathit{\varphi }}_1^l$, ${\mathit{\varphi }}_2^l$, ${\mathit{\varphi }}_3^l$, and ${\rho }^l$ by (38), (39), (40), and (41);

9: Check the termination criterion by (49);

Until convergence

11: ${\mathit{x}}^k={\mathit{x}}^l$

Until$k>k_{\max }$