Algorithm 1: (Newton’s method)

Input: $b_y^0={\nabla }_y{l}_{}^{k+1}$  If ${\nabla }_y{l}_{}^{k+1}+t_y^k=0$   Output $b_y^{k+1}=0$  Else   while not converged $$b_y^{j+1}=\text{sign}({\nabla }_y{l}_{}^{k+1}+t_y^k)\text{max}\{b_y^j-\frac{T{(b_x^j)}^{p(x)-1}+2\gamma (b_y^j-|{\nabla }_y{l}_{}^{k+1}+t_y^k|)}{(p(x)-1)T{(b_x^j)}^{p(x)-2}+2\gamma b_y^j},0\}$$   End  Output $b_y^{k+1}$  If ${\nabla }_x{l}_{}^{k+1}+t_x^k=0$   Output $b_x^{k+1}=0$  Else $b_x^{k+1}=\frac{{\nabla }_x{l}_{}^{k+1}+t_x^k}{{\nabla }_y{l}_{}^{k+1}+t_y^k}{b}_y^{k+1}$  End End