\appendix

\chapter{\acs{LP} Decoding using \acs{ADMM} as a Proximal Algorithm}%
\label{chapter:LD Decoding using ADMM as a Proximal Algorithm}

\todo{Find out how to properly title and section appendix}

%For problems of the form%
%\begin{align*}
%    \text{minimize}\hspace{2mm} & f\left( \boldsymbol{x} \right)
%        + g\left( \boldsymbol{A}\boldsymbol{x} \right) \\
%    \text{subject to}\hspace{2mm} & \boldsymbol{x} \in \mathbb{R}^n
%,\end{align*}%
%%
%a version of \ac{ADMM}, \textit{linearized \ac{ADMM}}, can be expressed
%as a proximal algorithm \cite[Sec. 4.4.2]{proximal_algorithms}:
%%
%\begin{align*}
%    \boldsymbol{x} &\leftarrow \textbf{prox}_{\mu f}\left( \boldsymbol{x}
%        - \frac{\mu}{\lambda}\boldsymbol{A}^\text{T}\left( \boldsymbol{A}\boldsymbol{x}
%        - \boldsymbol{z} + \boldsymbol{u} \right)  \right) \\
%    \boldsymbol{z} &\leftarrow \textbf{prox}_{\lambda g}\left( \boldsymbol{A}\boldsymbol{x}
%        + \boldsymbol{u} \right) \\
%    \boldsymbol{u} &\leftarrow \boldsymbol{u} + \boldsymbol{A} \boldsymbol{x} - \boldsymbol{z}
%.\end{align*}

In order to express \ac{LP} decoding using \ac{ADMM} through proximal operators,
it can be rewritten to fit the template for \textit{linearized \ac{ADMM}} given
in \cite[Sec. 4.4.2]{proximal_algorithms}.
We start with the general formulation of the \ac{LP} decoding problem:%
%
\begin{align*}
    \text{minimize}\hspace{2mm} & \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}} \\
    \text{subject to}\hspace{2mm} & \boldsymbol{T}_j \tilde{\boldsymbol{c}} \in \mathcal{P}_{d_j}
        \hspace{5mm} \forall j \in \mathcal{J}
.\end{align*}
%
The constraints can be moved into the objective function: %
%
\begin{align}
    \begin{aligned}
        \text{minimize}\hspace{2mm} & \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
            + \sum_{j\in \mathcal{J}} I_{P_{d_j}}\left(
                \boldsymbol{T}_j\tilde{\boldsymbol{c}} \right) \\
        \text{subject to}\hspace{2mm} & \tilde{\boldsymbol{c}} \in \mathbb{R}^n,
    \end{aligned}
    \label{eq:app:sum_reformulated}
\end{align}%
%
using the \textit{indicator functions}
$I_{\mathcal{P}_{d_j}} : \mathbb{R}^{d_j} \rightarrow \left\{ 0, +\infty \right\},
\hspace{3mm} j\in \mathcal{J}$, defined as%
%
\begin{align*}
    I_{\mathcal{P}_{d_j}}\left( \boldsymbol{t} \right) :=
    \begin{cases}
        0 & \boldsymbol{t} \in \mathcal{P}_{d_j} \\
        +\infty & \boldsymbol{t} \not\in \mathcal{P}_{d_j}
    \end{cases}
.\end{align*}%
%
Further defining
%
\begin{align*}
    \boldsymbol{T} := \begin{bmatrix}
        \boldsymbol{T}_1 \\
        \boldsymbol{T}_2 \\
        \vdots \\
        \boldsymbol{T}_m
    \end{bmatrix}
    \hspace{5mm}\text{and}\hspace{5mm}
    g\left( \boldsymbol{t} \right) = \sum_{j\in\mathcal{J}} I_{\mathcal{P}_{d_j}}\left(
        \boldsymbol{B}_j \boldsymbol{t} \right)
,\end{align*}%
%
\todo{Define $\boldsymbol{B}_j$}%
problem (\ref{eq:app:sum_reformulated}) becomes%
%
\begin{align}
    \begin{aligned}
        \text{minimize}\hspace{2mm} & \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
            + g\left( \boldsymbol{T}\tilde{\boldsymbol{c}} \right) \\
        \text{subject to}\hspace{2mm} & \tilde{\boldsymbol{c}} \in \mathbb{R}^n.
    \end{aligned}
    \label{eq:app:func_reformulated}
\end{align}
%
\todo{Fix $\mu f$ and $\lambda g$ in the steps below}%
In this form, it fits the template for linearized \ac{ADMM}.
The iterative algorithm can then be expressed as%
%
\begin{align}
    \begin{aligned}
        \tilde{\boldsymbol{c}} &\leftarrow \textbf{prox}_{\mu f}\left( \tilde{\boldsymbol{c}}
            - \frac{\mu}{\lambda}\boldsymbol{T}^\text{T}\left( \boldsymbol{T}\tilde{\boldsymbol{c}}
            - \boldsymbol{z} + \boldsymbol{u} \right)  \right) \\
        \boldsymbol{z} &\leftarrow \textbf{prox}_{\lambda g}\left(\boldsymbol{T}\tilde{\boldsymbol{c}}
            + \boldsymbol{u} \right) \\
        \boldsymbol{u} &\leftarrow \boldsymbol{u} + \boldsymbol{T} \tilde{\boldsymbol{c}}
            - \boldsymbol{z}.
    \end{aligned}
    \label{eq:app:admm_prox}
\end{align}
%

Using the definition of the proximal operator, the $\tilde{\boldsymbol{c}}$ update step
can be rewritten to match the definition given in section \ref{sec:lp:Decoding Algorithm}:%
%
\begin{align*}
    \tilde{\boldsymbol{c}} &\leftarrow \textbf{prox}_{\mu f}\left( \tilde{\boldsymbol{c}}
        - \frac{\mu}{\lambda}\boldsymbol{T}^\text{T}\left( \boldsymbol{T}\tilde{\boldsymbol{c}}
        - \boldsymbol{z} + \boldsymbol{u} \right)  \right) \\
    &= \argmin_{\tilde{\boldsymbol{c}}}\left( \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
        - \frac{\mu}{2} \left\Vert \boldsymbol{T}^\text{T}
            \left( \boldsymbol{T}\tilde{\boldsymbol{c}}
        - \boldsymbol{z} + \boldsymbol{u} \right)  \right\Vert_2^2 \right) \\
    &\overset{\text{(a)}}{=} \argmin_{\tilde{\boldsymbol{c}}}\left( \boldsymbol{\gamma}^\text{T}
        \tilde{\boldsymbol{c}}
        - \frac{\mu}{2} \left\Vert \boldsymbol{T}\tilde{\boldsymbol{c}}
        - \boldsymbol{z} + \boldsymbol{u} \right\Vert_2^2 \right) \\
    &= \argmin_{\tilde{\boldsymbol{c}}}\left( \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
    - \frac{\mu}{2} \left\Vert \begin{bmatrix}
        \boldsymbol{T}_1 \\
        \boldsymbol{T}_2 \\
            \vdots \\
            \boldsymbol{T}_m
        \end{bmatrix}
        \tilde{\boldsymbol{c}}
        - \begin{bmatrix}
            \boldsymbol{z}_1 \\
            \boldsymbol{z}_2 \\
            \vdots \\
            \boldsymbol{z}_m
        \end{bmatrix}
        + \begin{bmatrix}
            \boldsymbol{u}_1 \\
            \boldsymbol{u}_2 \\
            \vdots \\
            \boldsymbol{u}_m
    \end{bmatrix} \right\Vert_2^2 \right),
        \hspace{5mm}\boldsymbol{z}_j,\boldsymbol{u}_j \in \mathbb{F}_2^{d_j},
            \hspace{2mm} j\in\mathcal{J}\\
    &= \argmin_{\tilde{\boldsymbol{c}}} \left( \boldsymbol{\gamma}^\text{T} \tilde{\boldsymbol{c}}
        - \frac{\mu}{2} \sum_{j \in J} \left\Vert \boldsymbol{T}_j \tilde{\boldsymbol{c}}
        - \boldsymbol{z}_j + \boldsymbol{u}_j \right\Vert_2^2 \right)
.\end{align*}
%
Step (a) can be justified by observing that multiplication with $\boldsymbol{T}^\text{T}$
only reorders components, leaving their values unchanged.
Similarly to the $\boldsymbol{c}$ update, the $\boldsymbol{z}$ update step can be rewritten.
Since $g\left( \cdot \right)$ is separable, so is its proximal operator
\cite[Sec. 2.1]{proximal_algorithms}. The $\boldsymbol{z}$ update step can then
be expressed as a number of smaller steps:%
%
\begin{gather*}
    \boldsymbol{z} \leftarrow \textbf{prox}_{\lambda g} \left(\boldsymbol{T}\tilde{\boldsymbol{c}}
    + \boldsymbol{u} \right) \\[0.5em]
    \iff \\[0.5em]
    \begin{alignedat}{3}
        \boldsymbol{z}_j &\leftarrow \textbf{prox}_{\lambda I_{\mathcal{P}_{d_j}}}\left(
            \boldsymbol{T}_j \tilde{\boldsymbol{c}} + \boldsymbol{u}_j \right),
            \hspace{5mm} && \forall j\in\mathcal{J} \\
        & \overset{\text{(b)}}{=} \Pi_{\mathcal{P}_{d_j}}\left( \boldsymbol{T}_j
            \tilde{\boldsymbol{c}}
            + \boldsymbol{u}_j \right), \hspace{5mm} && \forall j\in\mathcal{J}
    ,\end{alignedat}
\end{gather*}
%
where (b) results from the fact that appying the proximal operator on the
indicator function of a convex set amounts to a projection onto the set
\cite[Sec. 1.2]{proximal_algorithms}.