71 lines
2.4 KiB
TeX
71 lines
2.4 KiB
TeX
\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}
|