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To apply steepest descent to the minimization of the polynomial J ( x ) in [link] , suppose that a current estimate of x is available at time k , which is denoted x [ k ] . A new estimate of x at time k + 1 can be made using

x [ k + 1 ] = x [ k ] - μ d J ( x ) d x x = x [ k ] ,

where μ is a small positive number called the stepsize, and where the gradient (derivative) of J ( x ) is evaluated at the current point x [ k ] . This is then repeated again and again as k increments. This procedure isshown in [link] . When the current estimate x [ k ] is to the right of the minimum, the negative of the gradient points left. When the current estimate is to the left of the minimum, thenegative gradient points to the right. In either case, as long as the stepsize is suitably small, the newestimate x [ k + 1 ] is closer to the minimum than the old estimate x [ k ] ; that is, J ( x [ k + 1 ] ) is less than J ( x [ k ] ) .

Steepest descent finds the minimum of a function by always pointing in the direction that leads downhill.
Steepest descent finds the minimum of a function by always pointing in the direction that leads downhill.

To make this explicit, the iteration defined by [link] is

x [ k + 1 ] = x [ k ] - μ ( 2 x [ k ] - 4 ) ,

or, rearranging,

x [ k + 1 ] = ( 1 - 2 μ ) x [ k ] + 4 μ .

In principle, if [link] is iterated over and over, the sequence x [ k ] should approach the minimum value x = 2 . Does this actually happen?

There are two ways to answer this question. It is straightforward to simulate the process. Here is some M atlab code that takes an initial estimate of x called x(1) and iterates [link] for N=500 steps.

N=500;                          % number of iterations mu=.01;                         % algorithm stepsizex=zeros(1,N);                   % initialize x to zero x(1)=3;                         % starting point x(1)for k=1:N-1   x(k+1)=(1-2*mu)*x(k)+4*mu;    % update equationend
polyconverge.m find the minimum of J ( x ) = x 2 - 4 x + 4 via steepest descent (download file)

[link] shows the output of polyconverge.m for 50 different x(1) starting values superimposed; all converge smoothly to the minimum at x = 2 .

The program polyconverge.m  attempts to locate the smallest value of J(x)=x^2-4x+4 by descending the gradient. Fifty different starting values all converge to the same minimum at x=2.
The program polyconverge.m attempts to locate the smallestvalue of J ( x ) = x 2 - 4 x + 4 by descending the gradient. Fifty different starting values all converge to thesame minimum at x = 2 .

Explore the behavior of steepest descent by running polyconverge.m with different parameters.

  1. Try mu = -.01, 0, .0001, .02, .03, .05, 1.0, 10.0. Can mu be too large or too small?
  2. Try N= 5, 40, 100, 5000. Can N be too large or too small?
  3. Try a variety of values of x(1) . Can x(1) be too large or too small?

As an alternative to simulation, observe that the process [link] is itself a linear time invariant system, of the general form

x [ k + 1 ] = a x [ k ] + b ,

which is stable as long as | a | < 1 . For a constant input, the final value theorem of z-Transforms (see [link] ) can be used to show that the asymptotic (convergent)output value is lim k x k = b 1 - a . To see this withoutreference to arcane theory, observe that if x k is to converge, then it must converge to some value, say x * . At convergence, x [ k + 1 ] = x [ k ] = x * , and so [link] implies that x * = a x * + b , which implies that x * = b 1 - a . (This holds assuming | a | < 1 .) For example, for [link] , x * = 4 μ 1 - ( 1 - 2 μ ) = 2 , which is indeed the minimum.

Thus, both simulation and analysis suggest that the iteration [link] is a viable way to find the minimum of the function J ( x ) , as long as μ is suitably small. As will become clearer in later sections, suchsolutions to optimization problems are almost always possible—as long as the function J ( x ) is differentiable. Similarly, it is usually quite straightforward to simulate thealgorithm to examine its behavior in specific cases, though it is not always so easy to carry out a theoretical analysis.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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.
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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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