Used scaled version of admm instead of unscaled

latex/thesis/chapters/decoding_techniques.tex

@@ -800,6 +800,25 @@ The same is true for $\left( \boldsymbol{\lambda}_j \right)_i$.}
 It should be noted that all of the $\boldsymbol{z}_j$-updates can be computed simultaneously,
 as they are independent of one another.
 The same is true for the updates of the individual components of $\tilde{\boldsymbol{c}}$.
+This representation can be slightly simplified by substituting
+$\boldsymbol{\lambda}_j = \mu \cdot \boldsymbol{u}_j \,\forall\,j\in\mathcal{J}$:%
+%
+\begin{alignat*}{3}
+    \tilde{c}_i &\leftarrow \frac{1}{\left| N_v\left( i \right) \right|} \left( 
+        \sum_{j\in N_v\left( i \right) } \Big( \left( \boldsymbol{z}_j \right)_i
+            - \left( \boldsymbol{u}_j \right)_i \Big)
+        - \gamma_i \right)
+    \hspace{3mm} && \forall i\in\mathcal{I} \\
+    \boldsymbol{z}_j &\leftarrow \Pi_{\mathcal{P}_{d_j}}\left(
+        \boldsymbol{T}_j\tilde{\boldsymbol{c}} + \boldsymbol{u}_j \right)
+    \hspace{3mm} && \forall j\in\mathcal{J} \\
+    \boldsymbol{u}_j &\leftarrow \boldsymbol{u}_j
+        + \boldsymbol{T}_j\tilde{\boldsymbol{c}}
+        - \boldsymbol{z}_j
+    \hspace{3mm} && \forall j\in\mathcal{J}
+.\end{alignat*}
+%
+
 
 The reason \ac{ADMM} is able to perform so well is due to the relocation of the constraints
 $\boldsymbol{T}_j\tilde{\boldsymbol{c}}_j\in\mathcal{P}_{d_j}\,\forall\, j\in\mathcal{J}$
@@ -819,19 +838,19 @@ as is shown in figure \ref{fig:lp:message_passing}.%
     \begin{genericAlgorithm}[caption={}, label={},
         basicstyle=\fontsize{11}{16}\selectfont
         ]
-Initialize $\tilde{\boldsymbol{c}}, \boldsymbol{z}_{[1:m]}$ and $\boldsymbol{\lambda}_{[1:m]}$
+Initialize $\tilde{\boldsymbol{c}}, \boldsymbol{z}_{[1:m]}$ and $\boldsymbol{u}_{[1:m]}$
 while $\sum_{j\in\mathcal{J}} \lVert \boldsymbol{T}_j\tilde{\boldsymbol{c}} - \boldsymbol{z}_j \rVert_2 \ge \epsilon_{\text{pri}}$ or $\sum_{j\in\mathcal{J}} \lVert \boldsymbol{z}^\prime_j - \boldsymbol{z}_j \rVert_2 \ge \epsilon_{\text{dual}}$ do
-    for all $j$ in $\mathcal{J}$ do
+    for $j$ in $\mathcal{J}$ do
         $\boldsymbol{z}_j \leftarrow \Pi_{\mathcal{P}_{d_j}}\left(
-            \boldsymbol{T}_j\tilde{\boldsymbol{c}} + \frac{\boldsymbol{\lambda}_j}{\mu} \right)$
-        $\boldsymbol{\lambda}_j \leftarrow \boldsymbol{\lambda}_j
-            + \mu\left( \boldsymbol{T}_j\tilde{\boldsymbol{c}}
-            - \boldsymbol{z}_j \right)$
+            \boldsymbol{T}_j\tilde{\boldsymbol{c}} + \boldsymbol{u}_j \right)$
+        $\boldsymbol{u}_j \leftarrow \boldsymbol{u}_j
+            + \boldsymbol{T}_j\tilde{\boldsymbol{c}}
+            - \boldsymbol{z}_j$
     end for
-    for all $i$ in $\mathcal{I}$ do
+    for $i$ in $\mathcal{I}$ do
         $\tilde{c}_i \leftarrow \frac{1}{\left| N_v\left( i \right) \right|} \left( 
             \sum_{j\in N_v\left( i \right) } \Big(
-                \left( \boldsymbol{z}_j \right)_i - \frac{1}{\mu} \left( \boldsymbol{\lambda}_j
+                \left( \boldsymbol{z}_j \right)_i - \left( \boldsymbol{u}_j
                 \right)_i
             \Big) - \frac{\gamma_i}{\mu} \right)$
     end for