From 81837b34f3da7cb452fa3aee4f6bc0fa42b5fe94 Mon Sep 17 00:00:00 2001
From: Andreas Tsouchlos <an.tsouchlos@gmail.com>
Date: Fri, 17 Mar 2023 00:23:50 +0100
Subject: [PATCH] Added dual ascent; Minor changes

---
 .../chapters/theoretical_background.tex       | 32 +++++++++++++++----
 1 file changed, 26 insertions(+), 6 deletions(-)

diff --git a/latex/thesis/chapters/theoretical_background.tex b/latex/thesis/chapters/theoretical_background.tex
index 2a3a2a6..e1f0c3f 100644
--- a/latex/thesis/chapters/theoretical_background.tex
+++ b/latex/thesis/chapters/theoretical_background.tex
@@ -101,11 +101,15 @@ Lastly, the optimization methods utilized are described.
 \section{Optimization Methods}
 \label{sec:theo:Optimization Methods}
 
+\begin{itemize}
+    \item \ac{ADMM}
+    \item proximal decoding
+\end{itemize}
+
 Generally, any linear program \todo{Acronym} can be expressed in \textit{standard form}%
-\todo{Citation needed}%
 \footnote{The inequality $\boldsymbol{x} \ge \boldsymbol{0}$ is to be
-interpreted componentwise.}%
-:%
+interpreted componentwise.}
+\cite[Sec. 1.1]{intro_to_lin_opt_book}:%
 %
 \begin{alignat}{3}
     \begin{alignedat}{3}
@@ -116,7 +120,9 @@ interpreted componentwise.}%
     \label{eq:theo:admm_standard}
 \end{alignat}%
 %
-A technique called \textit{lagrangian relaxation} can then be applied - some of the
+A technique called \textit{lagrangian relaxation}%
+\todo{Citation needed}%
+can then be applied - some of the
 constraints are moved into the objective function itself and the weights
 $\boldsymbol{\lambda}$ are introduced. A new, relaxed problem is formulated:
 %
@@ -172,8 +178,9 @@ bound actually reaches the value itself:
 .\end{align*}
 %
 In other words, with the optimal choice of $\boldsymbol{\lambda}$,
-the optimal objectives of the problems (\ref{eq:theo:admm_standard})
-and (\ref{eq:theo:admm_relaxed}) have the same value.
+the optimal objectives of the problems (\ref{eq:theo:admm_relaxed})
+and (\ref{eq:theo:admm_standard}) have the same value.
+
 Thus, we can define the \textit{dual problem} as the search for the tightest lower bound:%
 %
 \begin{align}
@@ -194,4 +201,17 @@ by computing \cite[Sec. 2.1]{admm_distr_stats}%
             \boldsymbol{\lambda}_{\text{opt}} \right)
     \label{eq:theo:admm_obtain_primal}
 .\end{align}
+%
+The dual problem can then be solved using \textit{dual ascent}: starting with an
+initial estimate of $\boldsymbol{\lambda}$, calculate an estimate for $\boldsymbol{x}$
+using equation (\ref{eq:theo:admm_obtain_primal}); then, update $\boldsymbol{\lambda}$
+using gradient descent \cite[Sec. 2.1]{admm_distr_stats}:%
+%
+\begin{align*}
+    \boldsymbol{x} &\leftarrow \argmin_{\boldsymbol{x}} \mathcal{L}\left(
+        \boldsymbol{x}, \boldsymbol{b}, \boldsymbol{\lambda} \right) \\
+    \boldsymbol{\lambda} &\leftarrow \boldsymbol{\lambda}
+        + \alpha\left( \boldsymbol{A}\boldsymbol{x} - \boldsymbol{b} \right),
+    \hspace{5mm} \alpha > 0
+.\end{align*}