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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

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
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
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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