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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

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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Source:  OpenStax, Computational sciences lecture series at uw-madison. OpenStax CNX. May 01, 2005 Download for free at http://cnx.org/content/col10277/1.5
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