Algorithm 2: Computing the superposition coding inner bound for $\theta \in (0,\frac{1}{2})$
Input: $p(y,z|x)$, maximum iterations K, N, thresholds $\eta$, $ϵ>0$, step size $\tau >0$;
Initialization: ${\alpha }_0\in (0,1)$, $q_0(u,x)>0$, ${\eta }_{\alpha }>\eta$, $k=0$;
while $k<K$ and ${\eta }_{\alpha }>\eta$ do
		initialize ${ϵ}_q>ϵ$, $n=0$;
		while $n<N$ and ${ϵ}_q>ϵ$ do
				$n\leftarrow n+1$;
				$Q_n=Q\left[q_{n-1}\right]$ using Equation (5) similarly;
				$q_n=q\left[Q_n\right]$ using Equation (7) similarly;
				$F({\alpha }_k,q_n,Q_n)=\overline{\theta }·ln{\sum }_{u,x}exp\left\{d\left[Q_n\right]\right\}$ using Equation (20) or (18);
				${ϵ}_q=\overline{\theta }·max\{d\left[Q_n\right]-ln{q}_{n-1}\}-F({\alpha }_k,q_n,Q_n)$;
		end
		$\phantom{(}$ $k\leftarrow k+1$;
		calculate ${\alpha }_k$ using Equations (21) and (22);
		${\alpha }_k\leftarrow min\{1,max\{0,{\alpha }_k\}\}$;
		${\eta }_{\alpha }=|{\alpha }_k-{\alpha }_{k-1}|$;
		$q_0\leftarrow q_n$;
end
Output: ${\alpha }_k$, $q_n(u,x)$, $Q_n$, $F({\alpha }_{k-1},q_n,Q_n)$