Initial commit

.gitignore

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+*.ijvm
+*.mic1
+
+# ---> TeX
+## Core latex/pdflatex auxiliary files:
+*.aux
+*.lof
+*.log
+*.lot
+*.fls
+*.out
+*.toc
+
+## Intermediate documents:
+*.dvi
+*-converted-to.*
+# these rules might exclude image files for figures etc.
+# *.ps
+# *.eps
+# *.pdf
+
+## Bibliography auxiliary files (bibtex/biblatex/biber):
+*.bbl
+*.bcf
+*.blg
+*-blx.aux
+*-blx.bib
+*.brf
+*.run.xml
+
+## Build tool auxiliary files:
+*.fdb_latexmk
+*.synctex
+*.synctex.gz
+*.synctex.gz(busy)
+*.pdfsync
+
+## Auxiliary and intermediate files from other packages:
+
+
+# algorithms
+*.alg
+*.loa
+
+# achemso
+acs-*.bib
+
+# amsthm
+*.thm
+
+# beamer
+*.nav
+*.snm
+*.vrb
+
+#(e)ledmac/(e)ledpar
+*.end
+*.[1-9]
+*.[1-9][0-9]
+*.[1-9][0-9][0-9]
+*.[1-9]R
+*.[1-9][0-9]R
+*.[1-9][0-9][0-9]R
+*.eledsec[1-9]
+*.eledsec[1-9]R
+*.eledsec[1-9][0-9]
+*.eledsec[1-9][0-9]R
+*.eledsec[1-9][0-9][0-9]
+*.eledsec[1-9][0-9][0-9]R
+
+# glossaries
+*.acn
+*.acr
+*.glg
+*.glo
+*.gls
+
+# gnuplottex
+*-gnuplottex-*
+
+# hyperref
+*.brf
+
+# knitr
+*-concordance.tex
+*.tikz
+*-tikzDictionary
+
+# listings
+*.lol
+
+# makeidx
+*.idx
+*.ilg
+*.ind
+*.ist
+
+# minitoc
+*.maf
+*.mtc
+*.mtc[0-9]
+*.mtc[1-9][0-9]
+
+# minted
+_minted*
+*.pyg
+
+# morewrites
+*.mw
+
+# mylatexformat
+*.fmt
+
+# nomencl
+*.nlo
+
+# sagetex
+*.sagetex.sage
+*.sagetex.py
+*.sagetex.scmd
+
+# sympy
+*.sout
+*.sympy
+sympy-plots-for-*.tex/
+
+# TikZ & PGF
+*.dpth
+*.md5
+*.auxlock
+
+# todonotes
+*.tdo
+
+# xindy
+*.xdy
+
+# WinEdt
+*.bak
+*.sav
+
+# Mac stupid tmp files
+.DS_Store
+
+!*.pdf
+*.zip
+
+**/*-figure*.pdf
+
+*~

hw1/ex3.m

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+%% Exercise 3.1
+
+% f(x1, x2) = x1^2 + u * x2^2;
+
+% [x1 x2] [1 0] [x1] + [0][x1]
+%         [0 u] [x2] + [0][x2]
+
+% A = [1 0; 0 u]; b = [0; 0]
+
+%% Exercise 3.2
+
+
+for u = 1:10
+    A = [1 0; 0 u];
+
+    xs = -10:1:10;
+    ys = xs;
+
+    Z = zeros(size(xs, 2), size(ys, 2));
+
+    for i = 1:size(xs, 2)
+        for j = 1:size(ys, 2)
+            vec = [xs(i); ys(j)];
+            Z(i, j) = vec' * A * vec;
+        end
+    end
+
+    if u <= 5
+        fig = u;
+        con = 5 + u;
+    else
+        fig = 10 + u - 5;
+        con = 15 + u - 5;
+    end
+    
+    subplot(4, 5, fig);
+    h = surf(xs, ys, Z);
+    set(h,'LineStyle','none');
+    title(sprintf("Surf for u=%d", u));
+    subplot(4, 5, con);
+    contour(xs, ys, Z, 20);
+    title(sprintf("Contour for u=%d", u));
+end

hw1/main.tex

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+\documentclass{scrartcl}
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath}
+
+\setlength{\parindent}{0cm}
+\setlength{\parskip}{0.5\baselineskip}
+
+\title{Optimisation methods -- Homework 1}
+\author{Claudio Maggioni}
+
+\begin{document}
+
+\maketitle
+
+\section{Exercise 1}
+
+\subsection{Gradient and Hessian}
+
+The gradient and the Hessian for $f$ are the following:
+
+\[\nabla f = \begin{bmatrix}2x_1 + x_2 \cdot \cos(x_1) \\ 9x^2_2 + \sin(x_1)\end{bmatrix} \]
+
+\[H_f = \begin{bmatrix}2 - x_2 \cdot \sin(x_1) & \cos(x_1)\\\cos(x_1) & 18x_2\end{bmatrix} \]
+
+\subsection{Taylor expansion}
+
+\[f(h) = 0 + \langle\begin{bmatrix}0 + 0 \\ 0 + 0\end{bmatrix},\begin{bmatrix}h_1\\h_2\end{bmatrix}\rangle + \frac12 \langle\begin{bmatrix}2 - 0 & 1\\1 & 0\end{bmatrix} \begin{bmatrix}h_1 \\ h_2\end{bmatrix}, \begin{bmatrix}h_1 \\ h_2\end{bmatrix}\rangle + O(\|h\|^3)\]
+\[f(h) = \frac12 \langle\begin{bmatrix}2h_1 + h_2 \\ h_1\end{bmatrix}, \begin{bmatrix}h_1 \\ h_2\end{bmatrix}\rangle + O(\|h\|^3)\]
+\[f(h) = \frac12 \left(2h^2_1 + 2 h_1h_2\right) + O(\|h\|^3)\]
+\[f(h) = h^2_1 + h_1h_2 + O(\|h\|^3)\]
+
+\section{Exercise 2}
+
+\subsection{Gradient and Hessian}
+
+For $A$ symmetric, we have:
+
+\[\frac{d}{dx}\langle b, x\rangle = \langle b,\cdot \rangle = b\]
+\[\frac{d}{dx}\langle Ax, x\rangle = 2\langle Ax,\cdot \rangle = 2Ax\]
+
+Then:
+
+\[\nabla J = Ax - b\]
+\[H_J = \frac{d}{dx} \nabla J = A\]
+
+\subsection{First order necessary condition}
+
+It is a necessary condition for a minimizer $x^*$ of $J$ that:
+
+\[\nabla J (x^*) = 0 \Leftrightarrow Ax^* = b\]
+
+\subsection{Second order necessary condition}
+
+It is a necessary condition for a minimizer $x^*$ of $J$ that:
+
+\[\nabla^2 J(x^*) \geq 0 \Leftrightarrow A \text{ is positive semi-definite}\]
+
+\subsection{Sufficient conditions}
+
+It is a sufficient condition for $x^*$ to be a minimizer of $J$ that the first necessary condition is true and that:
+
+\[\nabla^2 J(x^*) > 0 \Leftrightarrow A \text{ is positive definite}\]
+
+\subsection{Does $\min_{x \in R^n} J(x)$ have a unique solution?}
+
+Not in general. If for example we consider A and b to be only zeros, then $J(x) = 0$ for all $x \in \!R^n$ and thus $J$ would have an infinite number of minimizers.
+
+However, for if $A$ would be guaranteed to have full rank, the minimizer would be unique because the first order necessary condition would hold only for one value $x^*$. This is because the linear system $Ax^* = b$ would have one and only one solution (due to $A$ being full rank).
+
+\end{document}