148 lines
5.0 KiB
TeX
Executable File
148 lines
5.0 KiB
TeX
Executable File
\documentclass{scrartcl}
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\usepackage{pdfpages}
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\usepackage[utf8]{inputenc}
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\usepackage{float}
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\usepackage{graphicx}
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\usepackage[ruled,vlined]{algorithm2e}
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\usepackage{subcaption}
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\usepackage{hyperref}
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\usepackage{amsmath}
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\usepackage{pgfplots}
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\pgfplotsset{compat=newest}
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\usetikzlibrary{plotmarks}
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\usetikzlibrary{arrows.meta}
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\usepgfplotslibrary{patchplots}
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\usepackage{grffile}
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\usepackage{amsmath}
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\usepackage{subcaption}
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\usepgfplotslibrary{external}
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\tikzexternalize
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\usepackage[margin=2.5cm]{geometry}
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% To compile:
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% sed -i 's#title style={font=\\bfseries#title style={yshift=1ex, font=\\tiny\\bfseries#' *.tex
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% luatex -enable-write18 -shellescape main.tex
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\pgfplotsset{every x tick label/.append style={font=\tiny, yshift=0.5ex}}
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\pgfplotsset{every title/.append style={font=\tiny, align=center}}
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\pgfplotsset{every y tick label/.append style={font=\tiny, xshift=0.5ex}}
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\pgfplotsset{every z tick label/.append style={font=\tiny, xshift=0.5ex}}
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\setlength{\parindent}{0cm}
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\setlength{\parskip}{0.5\baselineskip}
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\title{Optimization methods -- Homework 3}
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\author{Claudio Maggioni}
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\begin{document}
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\maketitle
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\section{Exercise 1}
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\subsection{Exercise 1.1}
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Please consult the MATLAB implementation in the files \texttt{Newton.m}, \texttt{GD.m}, and \texttt{backtracking.m}.
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Please note that, for this and subsequent exercises, the gradient descent method without backtracking activated uses a
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fixed $\alpha=1$ despite the indications on the assignment sheet. This was done in order to comply with the forum post
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on iCorsi found here: \url{https://www.icorsi.ch/mod/forum/discuss.php?d=81144}.
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\subsection{Exercise 1.2}
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Please consult the MATLAB implementation in the file \texttt{main.m} in section 1.2.
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\subsection{Exercise 1.3}
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Please find the requested plots in figure \ref{fig:1}. The code used to generate these plots can be found in section 1.3 of \texttt{main.m}.
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\begin{figure}[h]
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\begin{subfigure}{0.5\textwidth}
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\resizebox{\textwidth}{\textwidth}{\includegraphics{ex1-3.jpg}}
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\caption{Zoomed plot on $x = (-1,1)$ and $y = (-1,1)$}
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\end{subfigure}
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\begin{subfigure}{0.5\textwidth}
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\resizebox{\textwidth}{\textwidth}{\input{ex1-3-gd}}
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\caption{Complete plot}
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\end{subfigure}
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\caption{Steps in the energy landscape for Newton and GD methods}\label{fig:1}
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\end{figure}
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\subsection{Exercise 1.4}
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Please find the requested plots in figure \ref{fig:gsppn}. The code used to generate these plots can be found in section 1.4 of \texttt{main.m}.
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\begin{figure}[h]
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\begin{subfigure}{0.45\textwidth}
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\resizebox{\textwidth}{\textwidth}{\includegraphics{1-4-grad-nonlog.jpg}}
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\caption{Gradient norms \\(zoomed, y axis is linear for this plot)}
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\end{subfigure}
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\begin{subfigure}{0.45\textwidth}
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\resizebox{\textwidth}{\textwidth}{\includegraphics{1-4-ys-nonlog.jpg}}
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\caption{Objective function values \\(zoomed, y axis is linear for this plot)}
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\end{subfigure}
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\begin{subfigure}{0.45\textwidth}
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\resizebox{\textwidth}{\textwidth}{\includegraphics{1-4-grad.jpg}}
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\caption{Gradient norms (zoomed)}
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\end{subfigure}
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\begin{subfigure}{0.45\textwidth}
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\resizebox{\textwidth}{\textwidth}{\includegraphics{1-4-ys.jpg}}
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\caption{Objective function values (zoomed)}
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\end{subfigure}
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\begin{subfigure}{0.45\textwidth}
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\resizebox{\textwidth}{\textwidth}{\includegraphics{1-4-grad-large.jpg}}
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\caption{Gradient norms}
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\end{subfigure}
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\begin{subfigure}{0.45\textwidth}
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\centering
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\resizebox{\textwidth}{\textwidth}{\includegraphics{1-4-ys-large.jpg}}
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\caption{Objective function values}
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\end{subfigure}
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\caption{Gradient norms and objective function values (y-axes) w.r.t. iteration numbers (x-axis) for Newton and GD methods (y-axis is log scaled, points at $y=0$ not shown due to log scale)}\label{fig:gsppn}
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\end{figure}
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\section{Exercise 1.5}
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TBD
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\section{Exercise 2}
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\subsection{Exercise 2.1}
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Please consult the MATLAB implementation in the file \texttt{BGFS.m}.
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\subsection{Exercise 2.2}
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Please consult the MATLAB implementation in the file \texttt{main.m} in section 2.2.
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\subsection{Exercise 2.3}
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Please find the requested plots in figure \ref{fig:3}. The code used to generate these plots can be found in section 2.3 of \texttt{main.m}.
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\begin{figure}[h]
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\centering
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\resizebox{.6\textwidth}{.6\textwidth}{\input{ex2-3}}
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\caption{Steps in the energy landscape for BGFS method}\label{fig:3}
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\end{figure}
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\subsection{Exercise 2.4}
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Please find the requested plots in figure \ref{fig:4}. The code used to generate these plots can be found in section 2.4 of \texttt{main.m}.
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\begin{figure}[h]
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\begin{subfigure}{0.5\textwidth}`
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\resizebox{\textwidth}{\textwidth}{\input{ex2-4-grad}}
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\caption{Gradient norms}
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\end{subfigure}
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\begin{subfigure}{0.5\textwidth}
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\resizebox{\textwidth}{\textwidth}{\input{ex2-4-ys}}
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\caption{Objective function values}
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\end{subfigure}
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\caption{Gradient norms and objective function values (y-axes) w.r.t. iteration numbers (x-axis) for BFGS method (y-axis is log scaled, points at $y=0$ not shown due to log scale)}\label{fig:4}
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\end{figure}
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\subsection{Exercise 2.5}
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TBD
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\end{document}
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