<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[11pt,onecolumn]{article}
\usepackage{amssymb, amsmath, amsthm,graphicx, paralist,algpseudocode,algorithm,cancel,url,color}
\usepackage{sectsty}
\usepackage{fancyvrb}
\usepackage{mathrsfs}
\usepackage{multirow}
\usepackage{hhline}
\usepackage{booktabs}
\usepackage[table]{xcolor}
\usepackage{tikz}
% \usepackage[framed,numbered,autolinebreaks,useliterate]{mcode}
\usepackage{listings}
\usepackage{enumitem}

\newcommand{\bvec}[1]{\mathbf{#1}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Rn}{\R^{n\times n}}
\newcommand{\Rmn}{\R^{m\times n}}
\newcommand{\Cn}{\C^{n\times n}}
\newcommand{\Cmn}{\C^{m\times n}}
\newcommand{\cO}{\mathcal{O}}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\trace}{trace}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\vspan}{span}
\sectionfont{\Large\sc}
\subsectionfont{\sc}
\usepackage[margin=1 in]{geometry}
\begin{document}
\noindent
\textsc{\Large Numerical analysis: Homework 5}\\
Instructor: Anil Damle\\
Due: April 26, 2024\\

\subsection*{Policies}
You may discuss the homework problems freely with other students, but please
refrain from looking at their code or writeups (or sharing your own).
Ultimately, you must implement your own code and write up your own solution to
be turned in. Your solution, including plots and requested output from your code
should be typeset and submitted via the Gradescope as a pdf file. This file must be self contained for grading. Additionally, please
submit any code written for the assignment as zip
file to the separate Gradescope assignment for code.\\
\hrule
\vspace{1cm}
\noindent

\subsection*{Question 1:}
Consider the penalized formulation for solving a quadratic problem with equality constraints
\begin{equation}
\min_x \frac{1}{2}x^THx + x^Tc + \frac{1}{2\mu}\|Ax-b\|_2^2,
\label{eqn:1}
\end{equation}
where $H\in\Rn$ is symmetric, $c\in\R^n,$ $\mu &gt; 0,$ and $A\in\Rmn$ with $m&lt;n$ and full row-rank. Let $Z\in\R^{n\times (n-m)}$ be a matrix with orthonormal columns whose range is the null space of $A.$ Show that if $Z^THZ$ is positive definite, then for all sufficiently small $\mu$~\eqref{eqn:1} has a unique solution and that solution is a strict local minimizer.

\end{document}</pre></body></html>