Optimization- Non derivative approach_pt-2010-v2.ppt

Sept, 2010
®Copyright of Shun-Feng Su
1
Optimization:
Non-Derivative Approaches
非微分型最佳化
Offered by 蘇順豐
Shun-Feng Su,
E-mail: su@orion.ee.ntust.edu.tw
Department of Electrical Engineering,
National Taiwan University of Science and Technology

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Preface
 Optimization is central to many occasions
involving decision or finding good solutions in
various research problems.
 In this talk, I shall provide some fundamental
concepts and ideas about optimization.
 This talk will also introduce one group of
optimization techniques – non-derivative
optimization, like genetic algorithms, ant
systems, and particular swarm optimization.

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 Preface
 Fundamentals of Optimization
 Traditional Optimization
 Non-derivative Approaches
 Genetic Algorithms
 Particle Swarm Optimization
 Ant colony optimization
 Epilogue
Outline

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Fundamentals of Optimization
 Optimization is to find the best one among all
possible alternatives.
 It is easy to see that optimization is always a
good means in demonstrating your research
results.
 But, the trick is what you mean “better”?

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 Optimization is to find the best one among all
possible alternatives.
 It is easy to see that optimization is always a
good means in demonstrating your research
results.
 But, the trick is what you mean “better”?
Why the optimal one is better than the others?
In other words, based on which criterion the
evaluation is conducted?

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 The measure of goodness of alternatives is
described by an so-called objective function or
performance index.
 Thus, it is desired that when you see “optimal”, you
should first check what is the objective function used.
 Optimization then is to maximized or minimized the
objective function considered.
 Other terms used are cost function (maximized),
fitness function (minimized), etc.

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Consider an intelligent system, usually
optimization methodology is required due to :
 Better selection of applicable knowledge or
strategies can result in better performance;
 In the learning process, an optimal way of
defining the updating rule is required.

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Consider an intelligent system, usually
optimization methodology is required due to:
 Better selection of applicable knowledge or
strategies can result in better performance;
 In the learning process, an optimal way of
defining the updating rule is required.
To act as a leaning mechanism is the most popular
approach currently employed in the literature.

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In general, an optimization problem requires
finding a setting of variable vector (or
parameters) of the system such that an
objective function is optimized. Sometimes, the
variable vector may have to satisfy some
constraints.
• Alternatives are to choose among values 
Numerical approach.
• This is why optimization is considered as one part of
computational intelligence.

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 Preface
 Epilogue
Outline

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Traditional Optimization
A traditional optimization problem can be
expressed as
Min (or Max) f(x)
subject to x
f( ) is the objective function to be optimized.
If some constraint like x is specified, it is
referred to as a constrained optimization
problem; otherwise it is called unconstrained
optimization problem.

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Traditional approaches for
unconstrained optimization
If the objective function can be explicitly
expressed as a function of parameters,
traditional mathematic approaches can be
employed to solve the optimization:
Traditional optimization approaches can be
classified into two categories; direct approach
and incremental approach.

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Direct approaches can be said to find the
solution mathematically (to find the solution
with certain properties).
In a direct approach, the idea is to directly find x
such that df(x)/dx=0 or f(x)=0.
This kind of approaches is Newton kind of
approaches.
In optimization, it is
f(x)=0
Newton’s method is to find a way of solving
f(x)=0 and the used approach can also be
iterative.

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Increment approach is to find which way can
improve the current situation based on the
current error. (back forward approach)
Usually, an incremental approach is to update
the parameter vector as x(k+1)=x(k)+x.
In fact, such an approach is usually fulfilled as a
gradient approach; that is x=f(x)/x.
Need to find a relationship between the current error
and the change of the variable considered; that is
why x=f(x)/x is employed.

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constrained optimization
In general, the constraint can be written as h(x)=0.
When the constraint is not expressed as such an
equality (e.g., h(x)0),
either the constraint is not effective (not used) or
the minimier is located on the boundary (i.e.,
h(x)=0)

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constrained optimization
In general, the constraint can be written as h(x)=0.
A commonly-used approach is the Lagrange
Theorem, which is to find x and  such that
f(x)+  h(x)= 0
where  is called the Lagrange multiplier.
Then, traditional unconstrained optimization
approaches can be employed.

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Traditional optimization approaches are to
develop a formal model (objective function and
constraints) that resembles the original
problem and then to solve it by means of
traditional mathematical methods.
In other words, in order to find f(x)=0 or
f(x)/x, the objective function f( ) must be
explicitly expressed as a function of the
parameter vector x

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Traditional optimization techniques even have
some problems (like being trapped in local
optima), those approaches have been shown to
be very successful in many applications.
However, other drawbacks are found in those
traditional optimization applications.

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In real-world problems, the objective function
and/or the constraints imposed on the variables
may not be analytically treatable or even cannot
be expressed in a closed form.
Thus, either there is no way of representing the
problem considered in a form so that the
derivative of the form can be performed or
simplifications of the original problem formulation
are required.

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When there is no way of representing the problem
in a closed form, backward kinds of approaches
cannot be implemented.
When simplification is conducted, it is more than
often that the found solutions do not solve the
original problem but the simplified problem.
Thus, some approaches of use only forward are
needed. Then it is impossible to modify
candidates based on the current output.

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If only forward evaluation is used, the approach is
to find the objective function value for the current
candidate and then try another candidate.
It is more like a search algorithm.
The issue may be how to define the next
candidates.
Randomly select or
select with some guidance?

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 Preface
 Epilogue
Outline

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Non-Derivative Optimization
An important property of such algorithms is that in
the process, auxiliary forms of the objective
function, such as derivations, are not required.
 non-derivative optimization
Non-derivative optimization does not define the
relationship between the current situation (error)
and the variable considered. Thus, another way
of defining the finding the optimal solution need
to be employed.

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A search algorithm is to find the solution based
on trying possible candidates.
Since it is impossible to try all possibilities, how
to define the next one usually is the key issue
in search algorithms.
Non-derivative optimization are also called
Evolutionary Computation, Nature Inspired
Algorithm or meta-heuristic Algorithms.

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Non-Derivative Optimization approaches are to
mimic various natural phenomena, like natural
selection process or animal behaviors so as to
find the best candidate for the problem.
Those search processes are to find the next
candidates by using experience obtained from
previous search together with some random
search mechanisms.
That is to define the next candidates with
some guidance and randomness.

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1. It works with a coding of solution set, not the
solutions itself.  need to code solutions
2. It searches from a population of solutions, not a
single solution.  parallel search
3. It uses payoff information (fitness function), not
derivatives or other auxiliary knowledge.
 non-derivative optimization
4. It uses probabilistic transition rules, not
deterministic rules.  random search with
guidance (stochastic search)

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Genetic Algorithms
Genetic Algorithms (GAs) simulate the natural
evolutionary process in searching for the best
solution based on the mechanism of natural
selection and natural genetic operation.
John Holland, from the University of Michigan
began his work on GAs in the early 60s.
A first achievement was the publication of
Adaptation in Natural and Artificial System in
1975.

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Genetic Algorithms
GA encodes solutions to the problem in a structure
that can be stored in the computer.
This object is a genome (or chromosome). GA
creates a population of genomes then applies
genetic operators (crossover and mutation) to the
candidates in the population to generate new
candidates.
It uses various selection criteria so that it picks the
best candidates for mating (and subsequent
crossover). The objective function determines how
'good' each individual is.

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Genetic Algorithms
To represent solutions in terms of genes --
Representation of Candidate Solutions (CS):
– Binary encoding
– Real number encoding
– Integer or literal permutation encoding
– General data structure encoding : array, tree,
matrix, . . . , etc.

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Genetic Algorithms
A Genetic Algorithm (GA) emulates biological
evolution to solve a complex problem.
GAs rely heavily on randomness. Instead of trying
to solve the problem directly, they create
random solutions and randomly mix them up
until a good solution is found.

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Genetic Algorithms
The evolution starts from a population of
completely random candidates and searches
for the best generation by generation.
In each generation, multiple candidates are
stochastically selected from the current
population, modified (mutated or recombined)
to form a new population, which is used in the
next generation (iteration).

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Genetic Algorithms
Use of the encoding of the parameters, not the
parameters themselves.
Work on a population of points, not a unique one.
Use the only values of the function to optimize,
not their derived function or other auxiliary
knowledge.
Use probabilistic transition function not
determinist ones.

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Apply reproduction and crossover on P(t)
to yield C(t)
Apply mutation on C(t) to yield
and then evaluate D(t)
Select P(t+1) from P(t) and D(t)
based on the fitness
Initialize population P(t)
Evaluate P(t)
Stop criterion
satisfied ?
Stop
Flow chart of a
simple genetic
algorithm

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Genetic Algorithms
GA uses three basic operators to manipulate the genetic
composition (chromosomes) of a population:
 Reproduction is a process of selecting parents for
generating offspring. The most highly rated
chromosomes in the current generation are most likely
copied in the new generation.
 Crossover provides a mechanism for chromosomes to
mix and match attributes through random processes.
 Mutation is to changed attributes (genes) in the new
generation to bring new possibility. Mutation is a very
important mechanism in avoiding local minimum in
optimization search.

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Reproduction
Reproduction can be divided into two kinds of
processes:
 to select parents from the population and
 to determine who will survive in the next generation.
Both processes need to select among all candidates.
Selection methodology can be considered based
on the following foundations:
 Sampling space
 Sampling Mechanism
 Probability Selection

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Reproduction
To select parents from the population:
GA researchers have used a number of parent
selection methods. Some of the more popular
methods are:
Proportionate Selection
Linear Rank Selection
Tournament Selection
How many parents will be selected is also an
issue for designing GA.

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In Proportionate Selection, candidates are
assigned a probability of being selected based
on their fitness: pi = fi / fj,
where pi is the probability that candidate i will be
selected and fi is the fitness of candidate i.
This type of selection is also referred to as the
roulette wheel selection.
Fitness maximum problem.
If a minimum problem is consider,
some modifications are needed.

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There are a number of disadvantages associated
with using proportionate selection:
– Cannot be used on minimization problems,
– Loss of selection pressure (search direction)
as population converges,
– Susceptible to Super Individuals
– Scaling issue for fitness values
Local optima
issue

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In Linear Rank selection, candidates are assigned
subjective fitness based on the rank within the
population: sfi = (P-ri)(max-min)/(P-1) + min
where ri is the rank of individual i,
P is the population size,
Max represents the fitness to assign to the
best candidate,
Min represents the fitness to assign to the
worst candidate.

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pi = sfi / sfj Roulette Wheel Selection can be
performed using the subjective fitness values.
One disadvantage associated with linear rank
selection is that the population must be sorted
on each cycle.

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Tournament Selection
In Tournament Selection, q candidates are
randomly selected from the population and the
best of the q candidates is returned as a parent.
Selection pressure increases as q is increased
and decreases as q is decreased.

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Selecting Who Survives
An Example Genetic Algorithm
Procedure GA{
t = 0;
Initialize P(t);
Evaluate P(t);
While (Not Done)
{
Parents(t) = Select_Parents(P(t));
Offspring(t) = Procreate(Parents(t));
Evaluate(Offspring(t));
P(t+1)= Select_Survivors(P(t),Offspring(t));
t = t + 1;
}
Genetic operations:
crossover and mutation
Select who
survive
Parent
selection

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Selecting Who Survives
By itself, pick best.
Darwinian survival of the fittest.
Give more copies to better guys.
Ways to do:
–truncation
–roulette wheel
–tournament

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Selection Who Survives
Basically, there are two types of selections in GAs:
Let  = # of parents,  = # of offspring,
( +) selection: select  best out of offspring and
old parents as parents of the next generation.
(, ) selection: select  best offspring as parents
of the next generation. ( <)

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Genetic Operators
Genetic Algorithms typically use two types of
operators: Crossover and Mutation.
Crossover is usually the primary operator for
inheriting properties from parents with mutation
serving only as a mechanism to introduce
diversity in the population.
However, when designing a GA, it is possible to
develop unique crossover and mutation
operators that take advantage of the structure
of the problem.

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Crossover Operator
There are a number of crossover operators that
have been used on binary and real-coded GAs:
Single-point Crossover,
Two-point Crossover,
Uniform Crossover
How many offspring will be generated is also an
issue in designing GA.

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Crossover Operator
Given two parents, single-point crossover will
generate a cut-point and recombines the first
part of first parent with the second part of the
second parent to create one offspring.
Example:
Parent 1: X X | X X X X X
Parent 2: Y Y | Y Y Y Y Y
Offspring 1: X X Y Y Y Y Y
Offspring 2: Y Y X X X X X

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Two-Point Crossover
Two-Point crossover is very similar to single-point
crossover except that two cut-points are
generated instead of one.
Example:
Parent 1: X X | X X X | X X
Parent 2: Y Y | Y Y Y | Y Y
Offspring 1: X X Y Y Y X X
Offspring 2: Y Y X X X Y Y

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Uniform Crossover
In Uniform Crossover, a value of the first parent’s
gene is assigned to the first offspring and the
value of the second parent’s gene is to the
second offspring with probability 0.5.
With probability 0.5 the value of the first parent’s
gene is assigned to the second offspring and
the value of the second parent’s gene is
assigned to the first offspring.

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Uniform Crossover
Example:
Parent 1: X X X X X X X
Parent 2: Y Y Y Y Y Y Y
Offspring 1: X Y X Y Y X Y
Offspring 2: Y X Y X X Y X

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Real-Coded Crossover Operators
For Real-Coded representations there exist a
number of other crossover operators:
Mid-Point Crossover,
Flat Crossover (BLX-0.0),
BLX-0.5

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Mid-Point Crossover
Given two parents where X and Y represent a
floating point number:
Parent 1: X
Parent 2: Y
Offspring: (X+Y)/2
If a chromosome contains more than one gene,
then this operator can be applied to each gene
with a probability of Pmp.

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Flat Crossover (BLX-0.0)
Flat crossover was developed by Radcliffe (1991)
Parent 1: X
Parent 2: Y
Offspring: rnd(X,Y)
Of course, if a chromosome contains more than
one gene then this operator can be applied to
each gene with a probability of Pblx-0.0.

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BLX-
Developed by Eshelman & Schaffer (1992)
Given two parents where X and Y represent a floating
point number, and where X < Y:
Parent 1: X
Parent 2: Y
Let  = (Y-X), where  = 0.5
Offspring: rnd(X-, Y+ )
Of course, if a chromosome contains more than one gene
then this operator can be applied to each gene with a
probability of Pblx-.

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Mutation (Binary-Coded)
In Binary-Coded GAs, each bit in the chromosome is
mutated with probability pbm known as the mutation rate.
Parent1 1 0 0 0 0 1 0
Parent2 1 1 1 0 0 0 1
Child1 1 0 0 1 0 0 1
Child2 0 1 1 0 1 1 0
An Example of Single-point Crossover Between the
Third and Fourth Genes with a Mutation Rate of
0.01 Applied to Binary Coded Chromosomes

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Mutation (Real-Coded)
In real-coded GAs, Gaussian mutation can be
used.
For example, BLX-0.0 Crossover with Gaussian
mutation.
Parent 1: X
Parent 2: Y
Offspring: rnd(X,Y) + N(0,1)

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Advanced GA techniques
• Elitism – Carry over some portion of the best
solutions to the next generation.
• Variable operators – Create multiple types of
crossovers and mutations. Track the health of
the offspring they produce, and adjust their
usage accordingly.
• Tribes – Create separate populations that only
occasionally mix. This may help avoid
converging on local maxima.

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Advanced GA techniques
"General Structure of Hybrid Genetic Algorithms"
Begin
t←0;
initialize P(t);
evaluate P(t);
while (not termination condition) do
recombine P(t) to yield C(t);
locally climb C(t);
evaluate C(t);
selecte P(t+1) from P(t) and C(t);
t← t + 1;
end
end
Local search
mechanism: to
provide greedy
advance in candidate
in this step.
There are various approaches and
variants for this local search
mechanism.

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Genetic Algorithms
Genetic operations play a role of generating the
new chromosomes for evolution. Hopefully,
the best-fitted solution can be generated.
In the algorithm, randomness plays essential
roles in all operations.
One attractive property of GA is that the
performance of the solution is always getting
better.

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Genetic Algorithms
In fact, GA should be understood as a general
adaptable concept for problem solving rather
than a collection of related and ready-to-use
algorithms.
However, due to the nature of adaptation to the
problems, the operations of GAs must be
designed by the users.
Moreover, if the optimization is constrained, the
initial population and the generations of new
chromosomes must be carefully selected.

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Genetic Algorithms
Illegal chromosome can not be decoded to a
solution. It can not be evaluated.
To use a penalty function is usually a bad
approach to this situation.
Some repairing techniques have been proposed
to convert an illegal or infeasible chromosome
to an acceptable one.

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Genetic Algorithms -- websites
IlliGAL (http://www-illigal.ge.uiuc.edu/) - Illinois Genetic
Algorithms Laboratory - Download technical reports
and code
Golem Project (http://demo.cs.brandeis.edu/golem/) -
Automatic Design and Manufacture of Robotic
Lifeforms
Introduction to Genetic Algorithms Using
RPL2 (http://www.epcc.ed.ac.uk/computing/training/doc
ument_archive/GAs-course/main.html)
Talk.Origins FAQ on the uses of genetic algorithms, by
AdamMarczyk (http://www.talkorigins.org/faqs/genalg/g
enalg.html)
Genetic algorithm in search and optimization, by Richard
Baker (http://www.fenews.com/fen5/ga.html)

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Genetic Algorithm and Markov chain Monte Carlo: Differential
Evolution Markov chain makes Bayesian Computing easy
(http://www.biometris.nl/Markov%20Chain.pdf)
Differential Evolution using Genetic Algorithm
(http://www.icsi.berkeley.edu/~storn/code.html#hist)
Introduction to Genetic Algorithms and Neural Networks
(http://www.ai-junkie.com/) including an example windows program
Genetic Algorithm Solves the Toads and Frogs Puzzle (http://www.cut-
the-knot.org/SimpleGames/evolutions.shtml) (requires Java)
Not-So-Mad Science: Genetic Algorithms and Web Page Design for
Marketers (http://www.marketingprofs.com/4/syrett6.asp) by
Matthew Syrett

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http://www.aic.nrl.navy.mil/galist/The Genetic Algorithms Archives
(maintained by Alan C Schultz at The Navy Center for Applied
Research in Artificial Intelligence)
http://www.genetic-programming.org/(A source of information about
the field of genetic programming)
http://www.genetic-programming.com/(the home page of Genetic
Programming Inc.)
http://www.genetic-programming.com/johnkoza.html(Home Page of
Professor John R. Koza)
http://www-illigal.ge.uiuc.edu:8080/(International Society for Genetic
and Evolutionary Computation)
http://www-illigal.ge.uiuc.edu/index.php3(Illinois Genetic Algorithms
Laboratory ILLiGAL)
http://cs.felk.cvut.cz/~xobitko/ga/Introduction to Genetic Algorithms
http://ww.lalena.com/ai/tsp/Travelling Salesman Problem Using
Genetic Algorithms
http://www4.ncsu.edu/eos/users/d/dhloughl/public/stable.htmGenetic
Algorithms Online

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http://cs.gmn.edu/research/gag/George Mason University GA Group
(GAG)
http://garage.cse.msu.edu/Michigan State University - Genetic
Algorithms Research and Application Groups (GARAGe)
http://gaslab.cs.unr.edu/Genetic Adaptive Systems LAB (GASLAB)
Evoluationary Computation (Journal)
http://www.densis.fee.unicamp.br/~moscatoMemetic Algorithms – Prof
Pablo Moscato
http://www.cs.newcastle.edu.au/~mendesMemetic Algorithms –
Softwares
http://groups.yahoo.com/group/MALL/Memetic Algorithms Discussion
Group
http://www.cs.newcastle.edu.au/~nbiNewcastle Bioinformatics Group
http://webhost.ua.ac.be/eume/welcome.htm?eume.sidebar.html&0Eur
opean Chapter on Metaheuristics

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Other Non-derivation Optimization
Other often mentioned approaches are Particle
Swarm Optimization (PSO) and Ants (ACS,
ACO, etc).
The overall ideas are all similar in that they all
use fitness values to guide the search with
some random mechanisms associated with
the search process.
Usually, these approaches can have better
search performance than that of genetic
algorithms.

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Particle Swarm Optimization
Particle Swarm Optimization is an optimization
technique which provides an evolutionary based
search. This search algorithm was introduced
by R. Eberhart and J. Kennedy in Proc. 1995
IEEE Int'l. Conf. on Neural Networks IV, pp.
1942-1948.
PSO shares many similarities with evolutionary
computation techniques such as Genetic
Algorithms (GA). The system is initialized
with a population of random solutions and
searches for optima by updating generations.
However, unlike GA, PSO has no evolution
operators such as crossover and mutation.

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PSO algorithms are especially useful for
parameter optimization in continuous, multi-
dimensional search spaces.
PSO is mainly inspired by social behavior
patterns of organisms that live and interact
within large groups.
In PSO, the potential solutions, called particles,
fly through the problem space by following
the current optimum particles.

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The connection to search and optimization
problems is made by assigning direction vectors
and velocities to each particle in a multi-
dimensional search space.
Each particle then 'moves' or 'flies' through the
search space following its velocity vector, which
is influenced by the directions and velocities of
other particles in its neighborhood.
These localized interactions with neighboring
particles propagate through the entire 'swarm' of
potential solutions.

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How much influence a particular point has on
other points is determined by its 'fitness‘;
that is, a measure assigned to a potential
solution, which captures how good it is
compared to all other solution points.
Hence, an evolutionary idea of 'survival of the
fittest' comes into play, as well as a social
behavior component through a 'follow the
local leader' effect and emergent pattern
formation.

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Each particle keeps track of its coordinates in
the problem space which are associated
with the best fitnes) it has achieved so far.
This value is called pbest.
Another "best" value is obtained so far by any
particle in the neighbors of the particle. This
location is called lbest.
When a particle takes all the population as its
topological neighbors, the best value is a
global best and is called gbest.

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The particle swarm optimization concept
consists of, at each time step, changing the
velocity of (accelerating) each particle toward
its pbest and gbest (global version of PSO) or
lbest locations (local version of PSO).
Acceleration is weighted by a random term, with
separate random numbers being generated
for acceleration toward pbest and gbest (or
lbest) locations.

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PSO Algorithm
After finding the two best values, the particle
updates its velocity and positions with
following equation (a) and (b).
(a) v[ ] = v[ ] + c1 * rand() * (pbest[ ] - present[ ])
+ c2 * rand() * (gbest[ ] - present[ ])
(b) present[ ] = present[ ] + v[ ]

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PSO Algorithm
1) Initialize the population - locations and velocities
2) Evaluate the fitness of the individual particle
(pBest)
3) Keep track of the individuals highest fitness
(gBest)
4) Modify velocities based on pBest and gBest
position
5) Update the particles position
6) Terminate if the condition is met
7) Go to Step 2

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PSO Algorithm

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PSO shares many common points with GA.
However, PSO does not have genetic
operators like crossover and mutation.
Particles update themselves with the internal
velocity. They also have memory, which is
important to the algorithm.
Compared with genetic algorithms (GAs), the
information sharing mechanism in PSO is
significantly different.

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In GAs, chromosomes share information with
each other. So the whole population moves
like a one group towards an optimal area. In
PSO, only gbest (or lbest) gives out the
information to others. It is a one -way
information sharing mechanism.
The evolution only looks for the best solution.
Compared with GA, all the particles tend to
converge to the best solution quickly even in
the local version in most cases.

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Ant Colony Optimization
Ant colony optimization (ACO) is a population-
based metaheuristic that can be used to find
approximate solutions to difficult optimization
problems.
An analogy with the way ant colonies function has
suggested the definition of a new
computational paradigm.
M. Dorigo, V. Maniezzo and A. Colorni, “Ant System: Optimization by
a Colony of Cooperating Agents,” IEEE Trans. Systems, Man,
and Cybernetics, Part B: Cybernetics, vol. 26, no. 1, pp. 29-41,
1996.

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Ant Colony Optimization
In ACO, a set of agents called artificial ants search
for good solutions to a given optimization
problem.
In ACO, the optimization problem is transformed
into the problem of finding the best path on a
weighted graph.
The artificial ants incrementally build solutions by
moving on the graph.
The solution construction process is stochastic
and is biased by a pheromone model.

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What is “pheromone”
A moving ant lays some pheromone on paths on
which it traverses, thus marking the path by a
trail of this substance.
While an isolated ant moves essentially at
random,
an ant encountering a previously laid trail can
detect it and follow it with a high probability.

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An example with Real Ants
(a) Ants follow a path between A and E.
(b) An obstacle is interposed.
(c) On the shorter path more pheromone is laid down.

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An Example with Artificial Ants
(a) The initial graph with distances.
(b) At time t = 0 there is no trail on the graph edges.
(c) At time t = 1 trail is stronger on shorter edges.

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Ant System
Each ant is a simple agent with the following
characteristics:
it chooses a path to go to with a probability that is
a function of heuristics (distance) and the
amount of trail (pheromone) present on the
connecting edge.
to force an ant to make legal tours, transitions to
already visited towns are disallowed until a tour
is completed.
when it completes a tour, it lays a substance called
trail (pheromone) on each edge visited.

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Pseudo Code of Ant System
The ACO metaheuristic is:
Set parameters, initialize pheromone trails
SCHEDULE_ACTIVITIES
ConstructAntSolutions
DaemonActions {optional}
UpdatePheromones
END_SCHEDULE_ACTIVITIES

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Schedule_Activities
The Schedule_Activities does not specify how the
three algorithmic components are scheduled
and synchronized.
In most applications of ACO to NP-hard problems
however, the three algorithmic components
undergo a loop that consists in (i) the
construction of solutions by all ants, (ii) the
(optional) improvement of these solution via the
use of a local search algorithm, and (iii) the
update of the pheromones.

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At each construction step, the current partial solution
is extended by adding a feasible solution
component from the set of feasible neighbors .
The process of constructing solutions can be
regarded as a path on the construction graph
GC(V,E).
The allowed paths in GC are implicitly defined by the
solution construction mechanism that defines the
set with respect to a partial solution .

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The choice of a solution component is done
probabilistically. The rules for the probabilistic
choice of solution components vary across
different ACO variants. The best known rule is
the one of ant system (AS):
where  and  are the pheromone value and the
heuristic value associated with the component.
 and  are parameters used to representing
the importance of pheromone and heuristics.
roulette wheel
selection

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DaemonActions
Once solutions have been constructed, and before
updating the pheromone values, often some
problem specific actions may be required. These
are often called daemon actions, and can be used
to implement problem specific and/or centralized
actions, which cannot be performed by single ants.
The most used daemon action is the use of local
search to the constructed solutions: the locally
optimized solutions are then used to decide which
pheromone values to update.

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UpdatePheromones
The aim of the pheromone update is to increase the
pheromone values associated with good solutions,
and to decrease those that are associated with bad
ones (not used).
Usually, this is achieved (i) by decreasing all the
pheromone values through pheromone evaporation,
and (ii) by increasing the pheromone levels
associated with a chosen set of good solutions
Evaporation for
all edges Add those who are good.

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Main ACO Algorithms
Several special cases of the ACO metaheuristic
have been proposed in the literature.
Ant System (Dorigo 1992, Dorigo et al. 1991,
1996),
Ant Colony System (ACS) (Dorigo & Gambardella
1997), and
MAX-MIN Ant System (MMAS) (Stützle & Hoos
2000).

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Ant Systems
Ant system (AS) was the first ACO algorithm
proposed in the literature .
Its main characteristic is that the pheromone
values are updated by all ants that have
completed the tour.
When constructing solutions, ants in AS traverse
the construction graph and make a probabilistic
decision at each vertex. The transition
probability is

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Ant Colony Systems
The first major improvement over the original ant system
was ant colony system (ACS), introduced by Dorigo and
Gambardella (1997).
The main difference between ACS and AS is the decision
rule used by the ants during the construction process.
Ants in ACS use the so-called pseudorandom proportional
rule: the probability of selecting next edges depends on
a random variable q uniformly distributed over [0, 1],
and a parameter q0; if q<q0, then, among the feasible
components, the component with maximal pheromone
heurestic is chosen, otherwise the same equation as in
AS is used.

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MAX-MIN Ant Systems
MAX-MIN ant system (MMAS) is another improvement,
proposed by Stützle and Hoos (2000), over the original
ant system idea.
MMAS differs from AS in that (i) only the best ant adds
pheromone trails, and (ii) the minimum and maximum
values of the pheromone are explicitly limited (in AS
and ACS these values are limited implicitly, that is, the
value of the limits is a result of the algorithm working
rather than a value set explicitly by the algorithm
designer).

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Other Non-derivation Optimization
It is because GAs and PSO are solution-wise
search and swarm search algorithms are
component-wise search.
Also, it can be found that solution-wise search
algorithms are easier to be trapped into a
local minimum if the initial population has
some local optimum properties.
Component-wise search algorithms can easily
escape from such an initial local optimum
phenomena.

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Epilogue
Traditional optimization approaches are good but
only for the mathematical form is true and can
be manipulated.
Non derivate optimization is one nice kind of
optimization techniques, but you need to adapt
the methodology to the problem you face.
An often used idea is to adapt your problem to
those traditional NP problems, like Travel
Salesman Problem (TSP), Quadratic
Assignment Problem (QAP), etc.

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10
0
Epilogue
Non-derivative optimization cannot guarantee the
success of the search.
Most of unsuccessful cases are either the search
converges too slow or the search gets stuck in
local optima.
The guadiance is
not strong
enough
Randomness is
not sufficently
used.

Sept, 2010
10
1
Epilogue
Non-derivative optimization cannot guarantee the
success of the search.
Most of unsuccessful cases are either the search
converges too slow or the search gets stuck in
local optima.
How to strengthen and to balance those two
factors are important issues in the design of
those search approaches.

Sept, 2010
10
2
Thankyoufor yourattention!
Any Questions ?!
Shun-Feng Su,
Professor of Department of Electrical Engineering,
National Taiwan University of Science and Technology
E-mail: su@orion.ee.ntust.edu.tw,

Optimization- Non derivative approach_pt-2010-v2.ppt

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