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hw2.tex
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\documentclass{article}
\usepackage[a4paper, total={6in, 10in}]{geometry}
\usepackage[fleqn]{mathtools}
\usepackage{amssymb}
\usepackage{relsize}
\allowdisplaybreaks
\usepackage{enumitem}
\usepackage{changepage}
\usepackage{heuristica} \usepackage[heuristica,vvarbb,bigdelims]{newtxmath}
%\usepackage{epigrafica} \usepackage[LGR,OT1]{fontenc}
\usepackage[T1]{fontenc}
\renewcommand*\oldstylenums[1]{\textosf{#1}}
\setlength\parindent{0pt}
\newcommand{\bs}{\hfill$\blacksquare$}
\newcommand{\ws}{\hfill$\square$}
\newcommand{\claimproof}[2]{
\textbf{claim}: #1 \\
\textbf{pf}: #2 \bs
}
\newcommand{\mws}{\tag*{\hfill$\square$}}
\newcommand{\indented}[1]{%
\begin{adjustwidth}{3em}{0em}%
#1
\end{adjustwidth}
}
\newcommand{\qgroup}[1]{
{\sffamily
\textbf{#1}\vspace{-5pt} \\
\makebox[\linewidth]{\rule{\linewidth}{2pt}}
}
}
\newcommand{\tr}[1]{\ensuremath{#1^{\mathsmaller T}}}
\newcommand{\R}{\ensuremath{\mathbb{R}}}
\newcommand{\Sym}{\ensuremath{\mathbb{S}}}
\begin{document}
\setlength{\abovedisplayskip}{0pt}
\setlength{\belowdisplayskip}{0pt}
\setlength{\abovedisplayshortskip}{0pt}
\setlength{\belowdisplayshortskip}{0pt}
{\sffamily
\textbf{Convex Optimization}\hfill\textbf{Homework 2}\vspace{-5pt}
\makebox[\linewidth]{\rule{\linewidth}{2pt}} \\
\textbf{Neil Vyas}\hfill nv4594\\
}
\qgroup{Written Questions}
2. (from Boyd and Vanderberghe, Ex. 2.10) Consider the set
$$ C = \{ x \in \R^n \mid \tr{x}Ax + \tr{b}x + x + c \leq 0 \} $$
where $A\in\Sym^n, b\in\R^n, c\in\R$.
\begin{enumerate}[label=(\alph*)]
\item Show that if $A\in\Sym^n_+$, then $C$ is convex. \\
\item Consider the set obtained by intersecting $C$ with a hyperplane:
$$ C_1 = C \cap \{x \mid \tr{g}x + h = 0 \} $$
Show that $C_1$ is convex if $\exists \lambda\in\R $ such that $A + \lambda g\tr{g} \in \Sym^n_+$. \\
\end{enumerate}
3. (B\&V, Ex. 2.21) For disjoint, convex sets $C,D \subset \R^n$, let
$$ S = \{ (a,b) \mid \forall x\in C, \tr{a}x \leq b, \forall x \in D, \tr{a}x \geq b \} $$
be the set of separating hyperplanes. Show that $S$ is convex.
7. Let $v_1, v_2 \in \R^n$. Then prove $\exists c,d \in \R^n$ such that
$$ \{ x \mid ||x-v_1|| \leq ||x-v_2|| \} = \{x \mid \tr{c}x \leq d\} $$
and construct $c,d$.
8. Let $A \in \R^{n \times m}, B \in \R^{k \times m}$. Suppose $\forall x\in\R^m, Ax = 0 \Rightarrow Bx =0$. Then show
$$\exists C\in\R^{k \times n} \text{ s.t } CA = B $$
Let $y \in \R^n$.
\end{document}