# Csls workshop on optimization of eigenvalues

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## Workshop overview

A wealth of interesting problems in engineering, control, finance, and statistics can be formulated as optimization problems involving theeigenvalues of a matrix function. These very challenging problems cannot usually be solved via traditional techniques for nonlinearoptimization. However, they have been addressed in recent years by a combination of deep, elegant mathematical analysis and ingeniousalgorithmic and software development. In this workshop, three leading experts will discuss applications along with the theoretical andalgorithmic aspects of this fascinating topic.

Remark: This workshop was held on October 7, 2004 as part of the Computational Sciences Lecture Series (CSLS) at the University of Wisconsin-Madison.

## Semidefinite programming

By Prof. Stephen Boyd (Stanford University, USA)

Slides of talk [PDF] (Not yet available.) | Video [WMV] (Not yet available.)

ABSTRACT: In semidefinite programming (SDP) a linear function is minimized subject to the constraint that the eigenvalues of asymmetric matrix are nonnegative. While such problems were studied in a few papers in the 1970s, the relatively recent development ofefficient interior-point algorithms for SDP has spurred research in a wide variety of application fields, including control system analysisand synthesis, combinatorial optimization, circuit design, structural optimization, finance, and statistics. In this overview talk I willcover the basic properties of SDP, survey some applications, and give a brief description of interior-point methods for their solution.

## Eigenvalue optimization: symmetric versus nonsymmetric matrices

By Prof. Adrian Lewis (Cornell University, USA)

Slides of talk [PDF] (Not yet available.) | Video [WMV] (Not yet available.)

ABSTRACT: The eigenvalues of a symmetric matrix are Lipschitzfunctions with elegant convexity properties, amenable to efficient interior-point optimization algorithms. By contrast, for example, thespectral radius of a nonsymmetric matrix is neither a convex function, nor Lipschitz. It may indicate practical behaviour much less reliablythan in the symmetric case, and is more challenging for numerical optimization (see Overton's talk). Nonetheless, this function doesshare several significant variational-analytic properties with its symmetric counterpart. I will outline these analogies, discuss thefundamental idea of Clarke regularity, highlight its usefulness in nonsmooth chain rules, and discuss robust regularizations of functionslike the spectral radius. (Including joint work with James Burke and Michael Overton.)

## Local optimization of stability functions in theory and practice

By Prof. Michael Overton (Courant Institute of Mathematical Sciences New York University,USA)

Slides of talk [PDF] (Not yet available.) | Video [WMV] (Not yet available.)

ABSTRACT: Stability measures arising in systems and control are typically nonsmooth, nonconvex functions. The simplest examples arethe abscissa and radius maps for polynomials (maximum real part, or modulus, of the roots) and the analagous matrix measures, the spectralabscissa and radius (maximum real part, or modulus, of the eigenvalues). More robust measures include the distance to instability(smallest perturbation that makes a polynomial or matrix unstable) and the $\epsilon$ pseudospectral abscissa or radius of a matrix (maximumreal part or modulus of the $\epsilon$\-pseudospectrum). When polynomials or matrices depend on parameters it is natural to consideroptimization of such functions. We discuss an algorithm for locally optimizing such nonsmooth, nonconvex functions over parameter spaceand illustrate its effectiveness, computing, for example, locally optimal low-order controllers for challenging problems from theliterature. We also give an overview of variational analysis of stabiity functionsin polynomial and matrix space, expanding on some of the issues discussed in Lewis's talk. (Joint work with James V. Burke and AdrianS. Lewis.)

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
how did you get the value of 2000N.What calculations are needed to arrive at it
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