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midterm-2016.tex
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midterm-2016.tex
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\documentclass{article}
\input{macros}
\usepackage{fancyhdr}
\usepackage{pifont}
% Number of problem sheet
\newcounter{problemSheetNumber}
\setcounter{problemSheetNumber}{1}
\newcommand{\matlabprob}{\ding{100} \ }
\newcommand{\examprob}{\ding{80} \ }
%\setcounter{section}{\theproblemSheetNumber}
%\renewcommand{\theparagraph}{(\thesection.\arabic{paragraph})}
\newcounter{problems}
\setcounter{problems}{0}
%\setlength{\parindent}{0cm}
\renewcommand{\problem}[1]{\paragraph{(\theproblems)}\addtocounter{problems}{1}\label{#1}}
\ifx\marks\undefined
\newcommand{\marks}[2][0mm]{\hspace{30mm}\mbox{}\vskip #1\hspace{-30mm}\hfill{\sf [#2 marks]}\\[-\baselineskip]}
\else
\renewcommand{\marks}[2][0mm]{\hspace{30mm}\mbox{}\vskip #1\hspace{-30mm}\hfill{\sf [#2 marks]}\\[-\baselineskip]}
\fi
\renewcommand{\solution}[1]{\paragraph{Solution (\theproblems)}\addtocounter{problems}{1}\label{#1}}
\pagestyle{fancy}
\lhead{MATH36061}
\chead{Convex Optimization}
\rhead{November 8, 2016}
\begin{document}
\begin{center}
{\Large {\bf Midterm test}}
\end{center}
\begin{center}
\emph{Closed book.\quad Attempt all questions.\quad Calculators permitted. \quad 13:00-13:50}\\
{\em Please write your name and student identity number on the front page.}
\end{center}
%--
\problem{p0} Determine the order of convergence of each of the following sequences (if they converge at all).
\begin{equation*}
\text{(a) } \ x_k = \frac{1}{k!}, \quad \text{(b) } \ x_k = 1+(0.3)^{2^k}, \quad \text{(c) } \ x_k=2^{-k}, \quad \text{(d) } \ x_k = 1/k
\end{equation*}
\marks[-6mm]{4}
\newpage
\mbox{}
\newpage
\problem{p1} Consider the function on $\R^2$, $f(\vct{x}) = (x_1^2+x_2)^2$. Show that the direction $\vct{p}=(1,-1)^{\trans}$ is a descent direction at $\vct{x}_0=(0,1)^{\trans}$, and determine a step length $\alpha$ that minimizes $f(\vct{x}_0+\alpha \vct{p})$.
\marks[-6mm]{4}
\newpage
\mbox{}
\newpage
%--
\problem{p2} Determine, with justification, which of the following sets is convex.
\begin{itemize}
\item[(a)] $\displaystyle \{(x,y) \mid x>1, \ y>\log(x)\}$;
\item[(b)] $\displaystyle \{(x,y) \mid x>0,y>0,xy<1\}$;
\item[(c)] $\displaystyle \{(x,y,1) \mid x^2+y^2\leq 2\}$;
\item[(d)] $\displaystyle \{\vct{x}\in \R^n \mid \norm{\vct{x}}_\infty+\norm{\vct{x}}_1\leq 1\}$.
\end{itemize}
Recall that $\norm{\vct{x}}_\infty = \max_{i} |x_i|$ and $\norm{\vct{x}}_1 = \sum_i |x_i|$.
\marks[-6mm]{4}
\newpage
\mbox{}
\newpage
\problem{p3} Consider the following linear programming problem
\begin{align*}
\maximize & x_1+2x_2 \\
\subjto & x_1\geq 0\\
& x_1\leq 1\\
& x_1+2x_2\leq 2\\
& x_1+x_2\geq 1
\end{align*}
\begin{itemize}
\item[(a)] Sketch the feasible set and determine the vertices of the polyhedron of feasible points from the diagram or from the inequalities;
\item[(b)] Find an optimizer and the optimal value.
\end{itemize}
\marks[-6mm]{4}
\newpage
\mbox{}
\newpage
\problem{p4} Consider the function
\begin{equation*}
f(x_1,x_2) = 100(x_2-x_1^2)^2+(1-x_1)^2.
\end{equation*}
Formulate Newton's method for finding a local minimizer. By inspecting the gradient, show that $(1,1)^{\trans}$ is the only local minimizer of this function.
\marks[-4mm]{4}
\newpage
\mbox{}
\newpage
\end{document}
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