From a0a13dbb2d9f2edb6f4592524c9ec9a2d55f4097 Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Fri, 7 Apr 2023 11:17:50 +0200 Subject: [PATCH] Minor changes in appendix --- latex/thesis/chapters/appendix.tex | 27 +++++++++++++++------------ 1 file changed, 15 insertions(+), 12 deletions(-) diff --git a/latex/thesis/chapters/appendix.tex b/latex/thesis/chapters/appendix.tex index 1bfb7de..f655e4b 100644 --- a/latex/thesis/chapters/appendix.tex +++ b/latex/thesis/chapters/appendix.tex @@ -89,15 +89,18 @@ problem (\ref{eq:app:sum_reformulated}) becomes% In this form, it fits the template for linearized \ac{ADMM}. The iterative algorithm can then be expressed as% % -\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) \\ - \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{align*} +\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 @@ -134,12 +137,12 @@ can be rewritten to match the definition given in section \ref{sec:dec:LP Decodi \boldsymbol{u}_2 \\ \vdots \\ \boldsymbol{u}_m - \end{bmatrix} \right\Vert_2^2 \right) + \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) + - \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 $\boldsymbol{T}^\text{T}$