\documentclass[12pt]{article}

\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{amsfonts}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]
\def\Z{\mathbb Z}
\def\CC{\mathbb C}
\def\rank{\hbox{\rm \ rank}}
\begin{document}
%Uhlig
%Frank","uhligfd@auburn.edu","
\section{ Convex and Non-convex Optimization Problems for the Field of Values of a Matrix} By {\sl Frank Uhlig, \ Department of Mathematics and Statistics, \  Auburn University,\  Auburn, AL  36849--5310, USA;  \  uhligfd@auburn.edu  } 

\medskip  \noindent We introduce and study numerical algorithms that compute the minimal and maximal distances between $0 \in \CC$  and  points in the field of values $F(A) = \{ x^*Ax \mid x \in \CC^n \ , \ \|x\|_2 = 1\} \subset \CC$ for a complex matrix $A_{n,n}$. Finding the minimal distance from $0 \in \CC$ to $F(A)$ is a convex optimization problem if $0 \notin F(A)$ and thus it has a unique solution, called the Crawford number whose magnitude relates information on the stability margin of the associated system. If $0 \in F(A)$, this is a non-convex optimization problem and consequently there can be multiple solutions or local minima that are not so globally. Non-convexity also holds for the maximal distance problem between points in $F(A)$ and $0 \in \CC$. This maximal distance is commonly called the numerical radius $numrad(A)$ for which the inequality $\rho(A) \leq numrad(A) \leq \|A\|$ is well established.  \\ Both cases can be solved efficiently numerically by using ideas from geometric computing, eigenanalyses of linear combinations of the hermitean and skew-hermitean parts of $A$ and the rotation method introduced by  C. R. Johnson in the 1970s to compute the boundary of the field of values.
%field of values, quadratic form, Crawford number, numerical radius, geometric computing, eigenvalue, convexity, convex optimization, non-convex optimization, efficiency","65F30","15A60, 1","","10:20:14","Thu Mar 13 2008","131.204.45.199"
\end{document}
 zzzz
