2.1-Genetic Algorithms

8/7/2019 2.1-Genetic Algorithms

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Genetic Al orithms

ENETIC LGORITHMS


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Introduction

Genetic algorithms are inspired by Darwin's theoryof evolution (i.e., survival of the fittest);

The new breeds of classes of living things come into existence

through the process of reproduction, crossover, and mutation

among existing organisms.

been translated into an algorithm to search for

solutions to problems in a more natural way.


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In the process of improving performance as muchas possible via tuning and specializing the geneticalgorithm operators, new and important findingsregarding the generality, robustness, andapplicability of genetic algorithms becameavailable.

Following the last couple of years of furiousdevelo ment of enetic al orithms in the sciences,

engineering and the business world, thesealgorithms in various guises have now beensuccessfull a lied to o timization roblemsscheduling, data fitting and clustering, trend

spotting and path finding.


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Biological Background

Chromosome: are strings of DNA (genes) andserve as a model for the whole organism

ac gene enco es a par cu ar pro e n. as ca y,

it can be said that each gene encodes a trait, for

example color of eyes.

Possible settings for a trait (e.g. blue, brown) are

called alleles.

ac gene as ts own pos t on n t echromosome. This position is called locus.


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Complete set of genetic material (all chromosomes)is called genome.

Particular set of genes in genome is called

The genotype is the after birth base for the

organism'sphenotype, its physical and mentalcharacteristics, such as eye color, intelligence, etc.


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http://gslc.genetics.utah.edu/units/basics/tour/


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During reproduction, recombination(orcrossover)first occurs. Genes from parents combine to form a

The newly created offspring can then be mutated.u a on means a e e emen s o are a

changed. This changes are mainly caused by errorsin copying genes from parents.

The fitnessof an organism is measured by successof the or anism in its life survival .


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Biological Metaphor

Algorithm begins with a set of solutions (represented by

chromosomes) called population.

goodness (fitness) and used to form a new population,(reproduction). This is motivated by a hope, that the newpopulation will be better than the old one.

Solutions which are then selected to form new solutions.(Offspring) are selected according to their fitness - the

reproduce.

This is repeated until some condition (for examplenum er o popu a ons or mprovemen o e essolution) is satisfied.


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Flowchart

1. Generate random population ofNchromosomes (feasiblesolutions for the problem)

START

FITNESS

2. va ua e e ness x o eacchromosome x in the population

3. Create a new population by

GENERATE NEW

POPULATION

the new population is completeby means of selection andcrossover or mutation

REPLACE

4. Replace unfit individuals in oldpopulation by new off springs

5. If the end condition is satisfied,END

CONDITION

SATISFIED

NO

,in current population

else Go to step 2STOPYES


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An Illustrative Example

How a line may be fit through a given data set using agenetic algorithm?

ons er a nown mo e structure, a near t

y= C1x+ C2

Encode C and C in 6-bit strin each L = 6

E.g., 0 0 0 1 1 1 0 1 0 1 0 0 (genotype)

C1 = 7 C2 = 20

Initial population = 4

Assume the minimum ofC1 and C2 are Cmin = -2; and the

max mum o 1 an 2 are max = .

Evaluate the performance from the given data set Cut-off value = 0.8 (relative fitness)


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Mappingfrom genotype to phenotype Ci= Cmin,i+ b / (2

L-1) (Cmax,i- Cmin,i)

corresponding more copies in the new generation (i.e.,

strings with lower fitness values are eliminated) (i.e.,

Crossover: strings are able to mix and match theirdesirable qualities in a random fashion (i.e., one-point

crossover Mutation: helps to increase the searching space,

allowing a vital bit of information to vary (from 1 o 0 orfrom 0 to 1) (i.e., bit-flip mutation). Mutation takes place

very rarely (e.g., 0.005/bit/generation)


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Another Illustrative Example

Given some functional mapping for a system, some membershipfunctions and their shapes are assumed for the various fuzzy

variables defined for a problem.

These membership functions are them coded as bit strings that are

then concatenated.

each set of membership functions.

Single-input (x) single-output (y) system:

y 1 4 9 16 25

Functional mapping in fuzzy logic for the system:

y S (small) VL (very large) [ ]250y


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The only parameter needed to described themembership functions (shape and position) is the

x

.S L

6-bit string (binary) to define the base 24-bit string

Base 1 Base 2 x

c romosome

Initial population = 4


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Design Parameters

1. Design a representation2. Decide how to initialize a population

. es gn a mec an sm o map e p eno ype ogenotype and vice versa

4. Design a way of evaluating an individual5. Design suitable mutation operators

6. Design suitable crossover operators

. 8. Decide how to select individuals to be replaced

9. Decide when to sto the al orithm

10. Decide how to manage the population


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1. Designing a Representation

We have to come up with a method of representingan individual as a genotype.

The way we choose to do it must be relevantto the

.

When choosing a representation, we have to bear in

mind how the genotypes will be evaluated and whatthe genetic operators might be.


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Forn-city Traveling Salesman Problem,

how should we desi n a re resentation(phenotype vs. genotype)?

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0 0.1 0.2 0. 3 0.4 0.5 0.6 0.7 0.8 0.9 1

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.


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Binary Valued Representation

A chromosome should in some way containinformation about solution that it represents. Themost used way of encoding is a binary string. Ac romosome en cou oo e s:

Chromosome 1 1101 1001 0011 0110 Chromosome 2 1101 1110 0001 1110

string. Each bit in the string can represent somecharacteristics of the solution. Another possibility isthat the whole strin can re resent a number


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Examples

PHENOTYPE

Integer

Real Number

8 bit GENOTYPE

.

.

ny ng

WHAT CAN IT REPRESENT ???


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Phenotype can be an integer number

=

0*22+1*21+1*20

= 128+32+2+1 = 163


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Phenotype can be areal number

e.g. a num er etween

2.5 and 20.5 using 8

binar di its

X= 2.5 + (163/256) (20.5-

2.5) = 13.9609


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Job Timestep

Phenotype can

be a schedule

1 2

2 1

e.g. 8 jobs , 2

time steps4 1

5 1

6 1

7 2


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Real Valued Representation

A very natural encoding if the solution we are lookingfor is a list of real-valued numbers, then encode as a

-

0s)

Lots of applications , e.g. parameter optimization


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Individuals are represented as a tuple ofn realvalued numbers:

The fitness function maps tuples of real numbers to

f:Rn R


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Binary vs. Real Representation

Original genetic algorithm is based on binary coding, due to

biological evidence. But it has some drawbacks when applied to

multidimensional, high-precision numerical problems.

There are three advantages of real representation:

First, the real representation proves adequate precision so that good values

are presentable in the search space. If a parameter is coded in binary form,ere s a ways e anger a one s a owe enoug prec s on o represen

parameter values that produce the best solution values.

Second, with the same digits, the real representation has larger range and

does not have to be a power of two.

Third, the real-coded genes have the ability to exploit the gradualness ofcontinuous variables (the gradualness means that small changes in the

variables correspond to small changes in the function).

disadvantages of binary one, there are still some weak points like

inherent premature convergence and leak mountain climbing.


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Order Based Representation

Individuals are represented as permutations Used for ordering, sequencing problems

Famous example: traveling salesman problem where

every city gets assigned a unique number from 1 to

. , , , , .

Famous example: N-queen problem (orN2-queen)

Needs special operators to make sure the individuals

stay in validpermutations.


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NQueens Problem

In chess, a queen can move as far

as she pleases, horizontally,vertically, or diagonally. A chess

.The standard 8 by 8 Queen'sproblem asks how to place 8

queens on an ordinary chess board

other in one move.

One solution - the prettiest in myopinion - is given in the figure

.12 essentially distinct solutions.(Two solutions are not essentiallydistinct if you can obtain one fromano er y ro a ng your c essboard, or placing in in front of a

mirror, or combining these twooperations.)


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Constraint SatisfactionProblem


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n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Unique

solution

1 0 0 1 2 1 6 12 46 92 341 1787 9233 45752 285053

Distinct

solution

1 0 0 2 10 4 40 92 352 724 2680 14200 73712 365596 2279184

* Distinct solutions are derived from unique solutions by rotations,

reflections and transpositions


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Variants: Super Queen orN2 Queens (3-D version) problem

e requ res p ac ng queens n an

cube so that no two queens threaten each other. There can

only be one queen in any row parallel to any axis, or in any

. , ,

cube, so that Sijcontains a height, k, then S is clearly a

Latin square. A solution to 3DTN2QP is also a pandiagonal

Latin s uare or Knut-Vik desi n.

Demonstration (N: 4-60)

http://www.apl.jhu.edu/~hall/NQueens.html


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Tree Based Representation

Individuals in the population are trees Any expression can be drawn as a tree of function

an erm na s

Functions : sine, cosine, add, sub, if-then-else

, , . , , , ,


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Tree Based: Closure and Sufficiency

In the tree based representation we need to specifya function set and a terminal set. It is very desirable

By closure we mean that each of the functions in theunc on se s a e o accep as s argumen s anyvalue and data type that may possibly be returned bysome other function or terminal.

By sufficiencywe mean that there should be asolution in the s ace of all ossible ro ramsconstructed from the specified function and terminal

sets.


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2. Initialization of a Population

Usually, at random

Uniformly on the search space . ,if possible

Binary representation: 0 and 1 with probability of 0.5

-

(OK for bounded values only)

Seed the population with previously known values orthose from heuristics. With care:

Possible loss of enetic diversit

Possible unrecoverable bias


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an effective strate to initialize thepopulation, given the way how the solution is

coded?


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Randomly initialized path

er space- ng curve

Fast closest ath


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3. Mapping a Genotype to Phenotype

Sometimes producingthe phenotype from the Genotype Problem

Dataprocess.

Other times the

of parameters to somealgorithm, which work

Growt

Function

produce the phenotype.

Phenot e


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it should be strai htforward for the ma infrom genotype to phenotype.


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4. Evaluating an Individual

By far the most costlystep for real applications Do not re-evaluate unmodified individuals

It mi ht be a subroutine a black-box simulator or anexternal process (e.g., robot experiment).

The effectiveness of the process depends on the

choice of the fitness function. Your could use approximate fitness, but not far too long

(fitness inheritance orfitness approximation).

Constraint handlin - what if the henot e breaks

some constraint of the problem: Penalize the fitness

Specific evolutionary method

Multiobjective evolutionary optimization gives a set of

compromise solutions


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what would be the cost function?


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5. Mutation Operators

The mutation operator should allow every part of the

search space to be reached.

The size of mutation is important and should be

.

Mutation should produce validchromosomes.


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how should we desi n the mutation o erator?

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0 0. 1 0. 2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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1 1 1 1 1 1 1e ore

Grey Elephant

mutated gene

Mutation probabilitypm, for each gene


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Perturb values by adding some random noise

Often, a Gaussian/normal distribution N(0,) is used,w ere

is the standard deviation

and xi = xi + N(0,i) for each parameter


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Randomly select two different genes and swap them.

7 83 41 2 6 5

7 83 46 2 1 5

bit swapping bit reversal

a sequence of bits is reversedsomewhere in the chromosome


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Single point mutation selects one node and replaces

it with a similar one.

* *

2 * *r r r r


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Annealing Procedure Conjecture

Two fundamental parameters to be controlled: mutation rate and

degrees of variations

In early stages, allow mutation to occur frequent and

Toward the later stages (convergence), limit

occurrence in mutation and with small jumps for

exp o tat on


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6. Recombination/Crossover Operators

The child should inherit something from both.

mutation operator.

The recombination/crossover operator should bedesigned in conjunction with the representation sothat recombination is not always catastrophic.

Recombination should produce validchromosomes.


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how should we desi n the crossover o erator?


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. . .Whole Po ulation:

1 1 1 1 1 1 1 0 0 0 0 0 0 0parents

offspring1 1 1 0 0 0 0 0 0 0 1 1 1 1

1-point crossover


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Uniform crossover: given two parents one child iscreated as follows

a db fc e g hg

a c eH


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Intermediate recombination (arithmetic crossover):given two parents one child is created as follows

a c e

FD ECBA

(a+A)/2 (b+B)/2 (c+C)/2 (e+E)/2(d+D)/2 (f+F)/2

o1 =p1*bias + p2*(1-bias)

* *2 1 - 2

bias toward the centerbetween two parents


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Blend (BLX-) Crossover:

picking a random number between min(p1,p2)- * and

*1, 2 , 1 2 ,

positive number (typically 0.5), and is |p1 p2|.

This formulation allows for the BLX operator to choose a

two parents. This allows for better placement of children

than the Arithmetic Crossover; however, the children will still

be biased to lie on baselines between sets of arents.

Therefore, children could still lack diversity.

iiii dxxX = ),min(211

21xxd =

iiii dxxX += ),max(212


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Unimodal Normally Distributed (UNDX) Crossover: It first selects three parents at random from the population.

Next it finds the mid oint of the first two arents and calls it xp.Then, it finds the difference vector of the first two parents as d= x1

x2. The line containing the first two parents is called the primary

search line, and the value D is computed as the distance from the

.

combined to form a child xc:

=

++=1

1

n

i

ii

ceDdmx

i

space and and are standard deviations determinedempirically.

This crossover operator has the advantage of preserving the mean

vector and covariance matrix of the parent population thereby

maintaining a similar distribution to the parent population in child

generations.


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Choose an arbitrary part from the first parent and

copy this to the first child.

Copy the remaining genes that are not in the copied

part to the first child: starting right from the cut point of the copied part

using the order of genes from the second parent

wra in around at the end of the chromosome

Repeat this process with the parent roles reversed.


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7 83 41 2 6 5 78 16 5234

Parent 1 Parent 2

7, 3, 4, 6, 5

81 2 4, 3, 6, 7, 5

Child 1

7 85 41 2 3 6


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*

*

*

+ 2 * (r * r )

r / * r + r

1 r

wo su - rees are se ec e

for swapping.


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+

*

*

*

r / r r*

1 r* 2 +

2 * r /

r r 1 r

expressions


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Crossover vs. Mutation

Crossover modifications depend on the whole population

decreasing effects with convergence

exploitation operator

Mutation mandatory to escape local optima

exploration operator

emp as ze crossover; w e anemphasize mutation


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Exploration vs. Exploitation

Exploration = sample unknown regions

Too much exporation = random search, no

convergence

- -individuals

Too much expoitation = local search only

convergence to a local optimum


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7. Selection Strategy

We want to have some way to ensure that better

fitted individuals have a better chance of being

.

This will give us selection pressure which will drive

the o ulation forward. On the other hand, we have to be careful to give less

good individuals at least some chance of being

-

material.

Risk of loss diversit


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how should we desi n the selection o erator?


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Roulette Wheel Selection

Parents are selected according to their

fitness. The better the chromosomes are,the more chances they are to be selected.

The size of the section in the roulette

wheel is proportional to the value of thefitness function of every chromosome - thebigger the value is, the larger the sectionis.

Better (fitter) individuals have: more space more chances to be selected


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Fitness Proportionate Selection

Disadvantages:

Danger of premature convergence becauseoutstanding individuals take over the entire

o ulation ver uickl .

Low selection pressure when fitness values are near.

Behave differentl on trans osed versions of thesame function.


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Fitness Scaling

At the start, a few extraordinary individuals will dominate the

evolution process premature convergence Later on, the population average fitness may be closed to the

population best fitness random walk among the mediocre

a cure for the problem issue

Start with the raw fitness function

Standardize to ensure

Lower fitness is better fitness

Optimal fitness equals to 0

Fitness ranges from 0 to 1 Normalize to ensure

The sum of the fitness values equals to 1

Linear scaling: f = af+ b

a and b are chosen such that favg = favg


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Tournament Selection

Select k random individuals, without replacement

a e t e est

k is called the size of the tournament


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Ranked Based Selection

Individuals are sorted on their fitness value from best

to worse. e p ace n t s sorte st s ca e ran .

,rank is used by a function to select individuals from

this sorted list. The function is biased towards

individuals with a high rank (= good fitness).


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Fitness: f(A) = 5, f(B) = 2, f(C) = 19

Rank: r(A) = 2, r(B) = 3, r(C) = 1

h(x) = min + (max min)* (r(x) 1)/(n 1)

unc on: = , = , =

If applied, proportion on the roulette wheel:= = =. , . , .


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8. Replacement Strategy

The selection pressure is also affected by the way in

which we decide which members of the population to

individuals.

e can use e s oc as c se ec on me o s nreverse, or there are some deterministic replacementstrategies.

We can decide never to replace the best in theo ulation: elitism.


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how should we desi n the re lacementstrategy?


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Elitism

Should fitness constantly improves?

Re-introduce in the population previous best-so-far (elitism)or

Keep best-so-far in a safe place (preservation)

Theory GA: preserve mandatory ES: no elitism sometimes is better

Application: avoid users frustration


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Bottomline is survival of the fittest

Generational Replacement Genetic Algorithm (GRGA).

approach of Holland. If the original population size is N, then Noffspring

are generated, and they are combined with the parents to get 2N

individuals which are then sorted and the best Nindividuals are accepted

for the next generation. This approach is rather simple, and places the

entire burden of selection and extinction upon the reproductive

operations.

ea y a e ene c gor m

uses a fixed population wherein a finite fraction of the population isextinguished at every generation. This approach separates natural

,

replacement strategy. Whereas there is no choice of replacement

strategy for the GRGA (because everyone is replaced), there are choices

for replacement strategy in the SSGA.


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Replace Worstreplaces the worst existing chromosome in the

population with probability equal to one by the newly created child

chromosome in the population by the newly created child to

preserve diversity

Replace Random replaces a random existing chromosome in the

population.

,

of parents at random and replace the worst, or pick two parents atrandom and replace the worst of the two, with or without some

attached robabilit check. This can also be combined with roulette

wheel selection where the two candidates are chosen by roulette

wheel probabilities based upon absolute fitness, and then proceed

with the tournament. This is known as a Stochastic Tournament.


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Population Decimation All chromosomes below the cut-off

threshold do not survive to the next generation. Their replacementsare formed by mating the remaining individual chromosomes.

Mass Extinction Extinction allows the repopulation of niches and

gives room for new adaptations. In mass extinction models, major

parts of the population are occasionally replaced, as opposed to agradual substitution of single individuals that one might see in a

classical GA. Most of the time there are no extinctions, sometimes

a few individuals die, and very rarely a large proportion of the

population is affected. However, the current best individual (an elite

class of size one) always survives extinction events, and is oftenused to repopulate the extinguished individuals in the extinction

zones y mu a e e ana ogy n na ure s e re-co on za on o a

niche by fit individuals that quickly explore the environment and

adapt to it.


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9. Stopping Criteria

The optimum is reached !!!!!

m on resources

Maximum number of evolution generations

Maximum number of fitness evaluations

m t on t e user s pat ence

After some generations without improvement


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what would be a reasonable sto in criterion?


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In general, generation number, evolution time,

fitness threshold, fitness convergence are indicators.

Population convergence The population is deemed

as conver ed when the avera e fitness across thecurrent population is less than a user-specified

percentage away from the best fitness of the current

.

Gene convergence This is a method that stops theevolution when a user-specified percentage of the

genes are deemed converged.


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10. Managing the Population

In general, fix population is always used for simplicity.

If the predetermined population size is too small, there will not be-

solution. If the population size is too large, it may require

unnecessarily large computational resources and result into an

extremely long running time. Optimal population size?

For MOP, a fixed population size will have great difficulty in

obtainin a Pareto front with a desired resolution because the

size and shape of the true Pareto front is unknown in apriorifor

most of the MOPs

A d namic o ulation size will be more reasonable for MOEA if

the computational effort can be adaptively adjusted based on the

complexity of the problem


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Performance

Neverdraw any conclusion from a single run

Use statistical measure (average, median) (Box plot)

rom a su c en num er o n epen en runs - m n mum

From the application point of view

Design perspective

Find a very goodsolution at least once

Production ers ective

Find a goodsolution at almost every run

,

performance on toy data and expect it to work with real

data


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Box Plot

In descriptive statistics, a boxplot (alsoknown as a box-and-whisker diagram or

plot) is a convenient way of graphicallydepicting groups of numerical data through

e r ve-num er summar es e sma esobservation, lower quartile (Q1), median(Q2), upper quartile (Q3), and largestobservation). A boxplot may also indicate

, ,considered outliers. The boxplot wasinvented in 1977 by the Americanstatistician John Tukey.

Box lots can be useful to dis ladifferences between populations without

making any assumptions of the underlyingstatistical distribution. The spacingsbetween the different parts of the box helpn ca e e egree o spers on spreaand skewness in the data, and identifyoutliers. Boxplots can be drawn either

horizontally or vertically.


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For this data set:

" "- lower (first) quartile (Q1,x.25) = 7

median(second quartile) (Med,x.5) = 8.5 u er third uartile 3 x = 9 .

largest non-outlier observation = 10 (right "whisker")

the value 3.5 is a "mild" outlier, between

the value 0.5 is an "extreme" outlier,

the data are skewed to the left (negatively skewed)

the mean value of the data can also be labeled with a point.


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Flowchart

1. Generate random population of

Nchromosomes (feasiblesolutions for the problem)

START

FITNESS

2. va ua e e ness x o eacchromosome x in the population

3. Create a new population byGENERATE NEW

POPULATION

the new population is completeby means of selection andcrossover or mutation

REPLACE

4. Replace unfit individuals in old

population by new off springs5. If the end condition is satisfied,

END

CONDITION

SATISFIED

NO

,in current population

else Go to step 2STOP

YES


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Genetic Algorithm for TSP

Chromosome representation: (order-based)

Population size & initialization: (50, at random)

Fitness function: (distance traveled)

Mutation: (5%)

ecom na on: (order-based)

Selection: (rank-based)

Stopping criteria: (number of generations- 200)


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Demonstration

C S d 101 i TSP bl ( il101)


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Case Study: 101-city TSP problem (eil101)

Total Distance:678.854309.

Population:50

Generations:200

C St d 535 i t bl ( li535)


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Case Study: 535-airport problem (ali535)

Total Distance:

2433.995486

Population:50

Generations:

C St d N Q P bl


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Case Study: N Queens Problem

In chess, a queen can move as faras she pleases, horizontally,vertically, or diagonally. A chess

.The standard 8 by 8 Queen'sproblem asks how to place 8queens on an ordinary chess board

other in one move.

One solution - the prettiest in myopinion - is given in the figure

.12 essentially distinct solutions.(Two solutions are not essentiallydistinct if you can obtain one fromano er y ro a ng your c essboard, or placing in in front of amirror, or combining these two

operations.)


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Applications in

Traffic control

VLSI design

Deadlock prevention

Parallel memory storage schemes

Sensor deployment

.

Proble Iss es in GA


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Problem Issues in GA

Population size

Binary representation vs. real representation Population initialization

Noisy fitness function Stochastic fitness function (or dynamic environment) Fitness inheritance and fitness approximation

Selection/rankin strate Crossover/Recombination operator

Mutation operator

Replacement strategy

Elitism strategy Benchmark test functions

Ex loration vs. ex loitation dilemma

Constraint handling Diversity promotion

Population management

Homework #3 due 11/7/2009


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Homework #3- due 11/7/2009

Problem #1 (Combinatorial Optimization)

Develop a genericgenetic algorithm to solve the traveling

within a unit square in a 2-dimensional plane. The coordinates

of 20 cities are given below in a matrix:

=95,0.2867.7148,0.239,0.2184,09697,0.597,0.8274,0.740,0.50290.9500,0.6

67,0.8148.6091,0.877,0.9536,00398,0.136,0.2124,0.695,0.59060.6606,0.9cities

36,0.8668.1247,0.169,0.9464,07471,0.544,0.3429,0.041,0.02130.3876,0.7

Show the best route you find and the associated distance with

46,0.6768.4351,0.862,0.8191,08192,0.939,0.3025,0.296,0.16490.8200,0.3

attached computer coding (with documentation- show your

recipe). An example is given below for reference.


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Problem 2 (Scalability)

Extend your genetic algorithm to solve the benchmark 101-city

Eilson).The benchmark problem can be found from TSPLIB

archive at

http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/

o nee o urn n e co es. n y e es rou e oun .e., s

configuration and distance in the form of above figure) is to beturned in. This problem is to test if your algorithm can scale up

.


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Problem 3 (Numerical Optimization)

Apply the genetic algorithm to De Jongs Rosenbrock's Valley

where global minimum is at the origin. This function is known

2

1

2

2

2

121 )1()(100),( xxxxxF +=

. .global optimum is at the tip of the sharp ridge, which is

parabolic shaped flat valley. The difficulty of this problem is to

conver e to the lobal o timum.


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Problem 4 (Multimodel Function)

Apply the genetic algorithm to De Jongs Shekels Foxholes

( ) =

+=25

1

5

1002.0

j jfF x ( )

=

+=2

1

6

,

i

jiij axjf

=

321601632321601632321601632321601632321601632a

This function, a multimodel function, which has many local

323232323216161616160000016161616163232323232

2,1,536.16536.16where = ixi

optimal (in this case 25). This function is a challenge to many

standard optimization algorithms because they can get stuck ina local optimal. Note that all 25 peaks have the same base

, , , , ... , - , -

32).


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Documents

2.1-Genetic Algorithms