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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 ) ,


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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
is it a question of log
Commplementary angles
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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how do you translate this in Algebraic Expressions
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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abeetha Reply
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I'm interested in nanotube
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
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how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
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I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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?
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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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