From 5b7c0454bb165d7aa3b49a0c3a6efcf565c7557f Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Wed, 19 Apr 2023 11:08:12 +0200 Subject: [PATCH] Replaced splicing with tuple --- latex/thesis/chapters/lp_dec_using_admm.tex | 23 +++++++++++-------- .../chapters/theoretical_background.tex | 16 ++++++------- 2 files changed, 20 insertions(+), 19 deletions(-) diff --git a/latex/thesis/chapters/lp_dec_using_admm.tex b/latex/thesis/chapters/lp_dec_using_admm.tex index 9cc72a4..e232220 100644 --- a/latex/thesis/chapters/lp_dec_using_admm.tex +++ b/latex/thesis/chapters/lp_dec_using_admm.tex @@ -562,7 +562,7 @@ complexity has been demonstrated to compare favorably to \ac{BP} \cite{original_ The \ac{LP} decoding problem in (\ref{eq:lp:relaxed_formulation}) can be slightly rewritten using the auxiliary variables -$\boldsymbol{z}_{[1:m]}$:% +$\boldsymbol{z}_1, \ldots, \boldsymbol{z}_m$:% % \begin{align} \begin{aligned} @@ -592,8 +592,8 @@ The multiple constraints can be addressed by introducing additional terms in the augmented lagrangian:% % \begin{align*} - \mathcal{L}_{\mu}\left( \tilde{\boldsymbol{c}}, \boldsymbol{z}_{[1:m]}, - \boldsymbol{\lambda}_{[1:m]} \right) + \mathcal{L}_{\mu}\left( \tilde{\boldsymbol{c}}, \left( \boldsymbol{z} \right)_{j=1}^m, + \left( \boldsymbol{\lambda} \right)_{j=1}^m \right) = \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}} + \sum_{j\in\mathcal{J}} \boldsymbol{\lambda}^\text{T}_j \left( \boldsymbol{T}_j\tilde{\boldsymbol{c}} - \boldsymbol{z}_j \right) @@ -606,18 +606,21 @@ The additional constraints remain in the dual optimization problem:% \begin{align*} \text{maximize } \min_{\substack{\tilde{\boldsymbol{c}} \\ \boldsymbol{z}_j \in \mathcal{P}_{d_j}\,\forall\,j\in\mathcal{J}}} - \mathcal{L}_{\mu}\left( \tilde{\boldsymbol{c}}, \boldsymbol{z}_{[1:m]}, - \boldsymbol{\lambda}_{[1:m]} \right) + \mathcal{L}_{\mu}\left( \tilde{\boldsymbol{c}}, \left( \boldsymbol{z} \right)_{j=1}^m, + \left( \boldsymbol{\lambda} \right)_{j=1}^m \right) .\end{align*}% % The steps to solve the dual problem then become: % \begin{alignat*}{3} - \tilde{\boldsymbol{c}} &\leftarrow \argmin_{\tilde{\boldsymbol{c}}} \mathcal{L}_{\mu} \left( - \tilde{\boldsymbol{c}}, \boldsymbol{z}_{[1:m]}, \boldsymbol{\lambda}_{[1:m]} \right) \\ + \tilde{\boldsymbol{c}} &\leftarrow \argmin_{\tilde{\boldsymbol{c}}} + \mathcal{L}_{\mu} \left( + \tilde{\boldsymbol{c}}, \left( \boldsymbol{z} \right)_{j=1}^m, + \left( \boldsymbol{\lambda}\right)_{j=1}^m \right) \\ \boldsymbol{z}_j &\leftarrow \argmin_{\boldsymbol{z}_j \in \mathcal{P}_{d_j}} \mathcal{L}_{\mu} \left( - \tilde{\boldsymbol{c}}, \boldsymbol{z}_{[1:m]}, \boldsymbol{\lambda}_{[1:m]} \right) + \tilde{\boldsymbol{c}}, \left( \boldsymbol{z} \right)_{j=1}^m, + \left( \boldsymbol{\lambda} \right)_{j=1}^m \right) \hspace{3mm} &&\forall j\in\mathcal{J} \\ \boldsymbol{\lambda}_j &\leftarrow \boldsymbol{\lambda}_j + \mu\left( \boldsymbol{T}_j\tilde{\boldsymbol{c}} @@ -794,7 +797,7 @@ Defining% the $\tilde{\boldsymbol{c}}$ update can then be rewritten as% % \begin{align*} - \tilde{\boldsymbol{c}} \leftarrow \boldsymbol{D}^{\circ -1} \circ + \tilde{\boldsymbol{c}} \leftarrow \boldsymbol{D}^{\circ \left(-1\right)} \circ \left( \boldsymbol{s} - \frac{1}{\mu}\boldsymbol{\gamma} \right) .\end{align*} % @@ -820,7 +823,7 @@ while $\sum_{j\in\mathcal{J}} \lVert \boldsymbol{T}_j\tilde{\boldsymbol{c}} \left( \boldsymbol{z}_j - \boldsymbol{u}_j \right) $ end for for $i$ in $\mathcal{I}$ do - $\tilde{\boldsymbol{c}} \leftarrow \boldsymbol{D}^{\circ -1} \circ + $\tilde{\boldsymbol{c}} \leftarrow \boldsymbol{D}^{\circ \left( -1\right)} \circ \left( \boldsymbol{s} - \frac{1}{\mu}\boldsymbol{\gamma} \right) $ end for end while diff --git a/latex/thesis/chapters/theoretical_background.tex b/latex/thesis/chapters/theoretical_background.tex index 4ba7197..80bf172 100644 --- a/latex/thesis/chapters/theoretical_background.tex +++ b/latex/thesis/chapters/theoretical_background.tex @@ -24,13 +24,11 @@ For example:% c \in \mathbb{F}_2 &\to \tilde{c} \in \left[ 0, 1 \right] \subseteq \mathbb{R} .\end{align*} % -Additionally, a shorthand notation will be used to denote series of indices and series -of indexed variables:% +Additionally, a shorthand notation will be used to denote a set of indices:% % \begin{align*} \left[ m:n \right] &:= \left\{ m, m+1, \ldots, n-1, n \right\}, - \hspace{5mm} m < n, \hspace{2mm} m,n\in\mathbb{Z}\\ - x_{\left[ m:n \right] } &:= \left\{ x_m, x_{m+1}, \ldots, x_{n-1}, x_n \right\} + \hspace{5mm} m < n, \hspace{2mm} m,n\in\mathbb{Z} .\end{align*} \todo{Not really slicing. How should it be denoted?} % @@ -619,7 +617,7 @@ $\boldsymbol{x}$, the Lagrangian is as well: = \boldsymbol{b} \end{align*} \begin{align*} - \mathcal{L}\left( \boldsymbol{x}_{[1:N]}, \boldsymbol{\lambda} \right) + \mathcal{L}\left( \left( \boldsymbol{x}_i \right)_{i=1}^N, \boldsymbol{\lambda} \right) = \sum_{i=1}^{N} g_i\left( \boldsymbol{x}_i \right) + \boldsymbol{\lambda}^\text{T} \left( \boldsymbol{b} - \sum_{i=1}^{N} \boldsymbol{A}_i\boldsymbol{x_i} \right) @@ -641,7 +639,7 @@ This modified version of dual ascent is called \textit{dual decomposition}: % \begin{align*} \boldsymbol{x}_i &\leftarrow \argmin_{\boldsymbol{x}_i}\mathcal{L}\left( - \boldsymbol{x}_{[1:N]}, \boldsymbol{\lambda}\right) + \left( \boldsymbol{x}_i \right)_{i=1}^N, \boldsymbol{\lambda}\right) \hspace{5mm} \forall i \in [1:N]\\ \boldsymbol{\lambda} &\leftarrow \boldsymbol{\lambda} + \alpha\left( \sum_{i=1}^{N} \boldsymbol{A}_i\boldsymbol{x}_i @@ -652,13 +650,13 @@ This modified version of dual ascent is called \textit{dual decomposition}: \ac{ADMM} works the same way as dual decomposition. It only differs in the use of an \textit{augmented Lagrangian} -$\mathcal{L}_\mu\left( \boldsymbol{x}_{[1:N]}, \boldsymbol{\lambda} \right)$ +$\mathcal{L}_\mu\left( \left( \boldsymbol{x} \right)_{i=1}^N, \boldsymbol{\lambda} \right)$ in order to strengthen the convergence properties. The augmented Lagrangian extends the ordinary one with an additional penalty term with the penaly parameter $\mu$: % \begin{align*} - \mathcal{L}_\mu \left( \boldsymbol{x}_{[1:N]}, \boldsymbol{\lambda} \right) + \mathcal{L}_\mu \left( \left( \boldsymbol{x} \right)_{i=1}^N, \boldsymbol{\lambda} \right) = \underbrace{\sum_{i=1}^{N} g_i\left( \boldsymbol{x_i} \right) + \boldsymbol{\lambda}^\text{T}\left( \boldsymbol{b} - \sum_{i=1}^{N} \boldsymbol{A}_i\boldsymbol{x}_i \right)}_{\text{Ordinary Lagrangian}} @@ -672,7 +670,7 @@ condition that the step size be $\mu$:% % \begin{align*} \boldsymbol{x}_i &\leftarrow \argmin_{\boldsymbol{x}_i}\mathcal{L}_\mu\left( - \boldsymbol{x}_{[1:N]}, \boldsymbol{\lambda}\right) + \left( \boldsymbol{x} \right)_{i=1}^N, \boldsymbol{\lambda}\right) \hspace{5mm} \forall i \in [1:N]\\ \boldsymbol{\lambda} &\leftarrow \boldsymbol{\lambda} + \mu\left( \sum_{i=1}^{N} \boldsymbol{A}_i\boldsymbol{x}_i