<< Chapter < Page Chapter >> Page >

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

Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
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
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
hi
Loga
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Software receiver design' conversation and receive update notifications?

Ask