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By their nature, steepest descent and hill climbing methods use only local information. This isbecause the update from a point x [ k ] depends only on the value of x [ k ] and on the value of its derivative evaluated at that point. This can be a problem,since if the objective function has many minima, the steepest descent algorithm may become “trapped” at a minimum that is not (globally)the smallest. These are called local minima. To see how this can happen, consider the problem of finding the value of x that minimizes the function

J ( x ) = e - 0 . 1 | x | sin ( x ) .

Applying the chain rule, the derivative is

e - 0 . 1 | x | cos ( x ) - 0 . 1 e - 0 . 1 | x | sin ( x ) sign ( x ) ,

where

sign ( x ) = 1 x > 0 - 1 x < 0

is the formal derivative of | x | . Solving directly for the minimum point is nontrivial (try it!). Yet implementing a steepest descentsearch for the minimum can be done in a straightforward manner using the iteration

x [ k + 1 ] = x [ k ] - μ e - 0 . 1 | x [ k ] | · ( cos ( x [ k ] ) - 0 . 1 sin ( x [ k ] ) sign ( x ) ) .

To be concrete, replace the update equation in polyconverge.m with

x(k+1)=x(k)-mu*exp(-0.1*abs(x(k)))*(cos(x(k))...            -0.1* sin(x(k))*sign(x(k)));

Implement the steepest descent strategy to find the minimum of J ( x ) in [link] , modeling the program after polyconverge.m . Run the program for different values of mu , N , and x(1) , and answer the same questions as in Exercise [link] .

One way to understand the behavior of steepest descent algorithms is to plot the error surface , which is basically a plot of the objective as a function of the variablethat is being optimized. [link] (a) displays clearly the single global minimum of the objective function [link] while [link] (b) shows the many minima of the objective function defined by [link] . As will be clear to anyone who has attempted Exercise  [link] , initializing within any one of the valleys causes the algorithmto descend to the bottom of that valley. Although true steepest descent algorithms can never climb over a peak to enter another valley(even if the minimum there is lower) it can sometimes happen in practice when there is a significant amount of noise inthe measurement of the downhill direction.

Essentially, the algorithm gradually descends the error surface by moving in the (locally)downhill direction, and different initial estimates may lead to different minima. Thisunderscores one of the limitations of steepest descent methods—if there are many minima, then it is important to initialize near an acceptable one. In someproblems such prior information may easily be obtained, while in others it may be truly unknown.

The examples of this section are somewhat simple because they involve static functions. Most applications incommunication systems deal with signals that evolve over time, and the next section applies thesteepest descent idea in a dynamic setting to the problem of Automatic Gain Control (AGC). The AGC provides a simple settingin which all three of the major issues in optimization must be addressed: setting the goal, choosing a method of solution, andverifying that the method is successful.

Error surfaces corresponding to (a) the objective function Equation 13 and (b) the objective function Equation 18.
Error surfaces corresponding to (a) the objective function [link] and (b) the objective function [link] .

Questions & Answers

How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
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
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
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
Abhijith Reply
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3
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