hw1: done up to 3.2

hw1/ex3.m

@@ -1,9 +1,12 @@
+% Sources:
+% https://www.youtube.com/watch?v=91RZYO1cv_o
+
 %% Exercise 3.1
 
 % f(x1, x2) = x1^2 + u * x2^2;
 
-% [x1 x2] [1 0] [x1] + [0][x1]
+% 1/2 * [x1 x2] [2  0] [x1] + [0][x1]
-%         [0 u] [x2] + [0][x2]
+%               [0 2u] [x2] + [0][x2]
 
 % A = [1 0; 0 u]; b = [0; 0]
 
@@ -40,4 +43,6 @@ for u = 1:10
     subplot(4, 5, con);
     contour(xs, ys, Z, 20);
     title(sprintf("Contour for u=%d", u));
+    
 end
+matlab2tikz('showInfo', false, './ex32.tex')

hw1/main.tex

@@ -1,6 +1,11 @@
 \documentclass{scrartcl}
 \usepackage[utf8]{inputenc}
 \usepackage{amsmath}
+\usepackage{pgfplots}
+\pgfplotsset{compat=1.17}
+\usepackage{mathrsfs}
+\usepackage{hyperref}
+\usetikzlibrary{arrows}
 
 \setlength{\parindent}{0cm}
 \setlength{\parskip}{0.5\baselineskip}
@@ -67,4 +72,25 @@ Not in general. If for example we consider A and b to be only zeros, then $J(x)
 
 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).
 
+\section{Exercise 3}
+
+\subsection{Quadratic form}
+
+$f(x,y)$ can be written in quadratic form in the following way:
+
+\[f(v) = \frac12 \langle\begin{bmatrix}2 & 0\\0 & 2\mu\end{bmatrix}v, v\rangle + \langle\begin{bmatrix}0\\0\end{bmatrix}, x\rangle\]
+
+where:
+
+\[v = \begin{bmatrix}x\\y\end{bmatrix}\]
+
+\subsection{Matlab implementation with \texttt{surf} and \texttt{contour}}
+
+The graphs generated by MATLAB are shown below:
+
+\resizebox{\textwidth}{!}{\input{ex32.tex}}
+
+Isolines get stretched along the y axis as $\mu$ increases. For a large $\mu$, points well far away from the axes could be a
+problem since picking search directions and steps using a naive gradient based method iterations will zig-zag to the minimizer reaching it slowly.
+
 \end{document}