Algorithm 4: Computing the UV outer bound
Input: $p(y,z|x)$, maximum iterations K, N, thresholds $\eta$, $ϵ>0$, step size $\tau >0$;
Initialization: ${\alpha }_0,{\beta }_0\in (0,1)$, $q_0(u,v,x)>0$, $\overline{\theta }{\alpha }_0+\theta {\beta }_0\ge max\{\theta ,\overline{\theta }\}$, ${\eta }_{\alpha },{\eta }_{\beta }>\eta$,
$k=0$;
while $k<K$ and $max\{{\eta }_{\alpha },{\eta }_{\beta }\}>\eta$ 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 Equations (33), (34) and (37);
				$G({\alpha }_k,{\beta }_k,q_n,Q_n)=(\overline{\theta }\alpha +\theta \beta )·ln{\sum }_{x}exp\left\{d\left[Q_n\right]\right\}$ using Equation (36);
				${ϵ}_q=(\overline{\theta }\alpha +\theta \beta )·max\{d\left[Q_n\right]\left(x\right)-ln{q}_{n-1}\left(x\right)\}-G({\alpha }_k,{\beta }_k,q_n,Q_n)$;
		end
		$k\leftarrow k+1$;
		calculate ${\alpha }_k$ and ${\beta }_k$ using Equations (39) and (40);
		${\alpha }_k\leftarrow min\{1,max\{0,{\alpha }_k\}\}$;
		${\beta }_k\leftarrow min\{1,max\{0,{\beta }_k\}\}$;
		if $\overline{\theta }{\alpha }_k+\theta {\beta }_k<max\{\theta ,\overline{\theta }\}$, scale up to equality;
		${\eta }_{\alpha }=|{\alpha }_k-{\alpha }_{k-1}|$
		${\eta }_{\beta }=|{\beta }_k-{\beta }_{k-1}|$;
		$q_0\leftarrow q_n$;
end
Output: ${\alpha }_k$, ${\beta }_k$, $q_n(u,v,x)$, $Q_n$, $G({\alpha }_{k-1},{\beta }_{k-1},q_n,Q_n)$