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IN THE NAME OF A
THE MOST BENEFTHE MOST MER
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qsmzeeshan@yahoo.com ; 0321-9
_________________PhD, FLIGHT VEHICLE DESIGNBEIJING UNIVERSITY OF AERONAUTICS AND ASTRONAUTICS, BUAA, P.R.CHINA, 2009
MS, FLIGHT VEHICLE DESIGNBEIJING UNIVERSITY OF AERONAUTICS AND ASTRONAUTICS, BUAA, P.R.CHINA, 2006
BE, MECHANICAL ENGINEERINGNATIONAL UNIVERSITY OF SCIENCE AND TECHNOLOGY, NUST, PAKISTAN, 2000
EMAIL: qsmzeeshan@yahoo.com
qasim.zeeshan@ist.edu.pk
TEL: +92-320-9595510
WEB: www.ist.edu.pk/qasim-zeeshan LINKEDIN: pk.linkedin.com/pub/qasim-zeeshan/67/554/ba7
Dr Qasim Zeeshan
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MULTIDISCIPLISY
DOPTIMIZA
LECTURE # 7
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STATUS
PHASE-I
Introduction to Multidisciplinary System Design Optimizatio
Terminology and Problem Statement
Introduction to Optimization
Classification of Optimization Problems
Numerical/ Classical Optimization
MSDO Architectures
Practical Applications: Structure, Aero etc
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STATUS
PHASE-II WEEK 8: Genetic Algorithm
WEEK 9: Particle Swarm Optimization
WEEK 10: Simulated Annealing
WEEK 11: MID TERM
WEEK 12:
Ant Colony Optimization, Tabu Search, Pattern Search
WEEK 13:
LAB, Practical Applications
00
20
40
60
-0.5
0
0.5
1
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STATUS
PHASE-III WEEK 14: Design of Experiments, Meta-modeling, and Ro
WEEK 15: Multi-objective Optimization
Hybrid Optimization & Hyper Heuristic Optimiz
WEEK 16: Post Optimality Analysis/ Revision & Discussion
WEEK 17: END TERM/ Paper Presentations ?
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In this LECTURE
GENETIC ALGORITHM HISTORY
TERMINOLOGY/ VOCABULARY
ALGORITHM
APPLICATION IN MATLAB
EXAMPLES
ASSIGNMENT
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M
GENETIC ALGO
Dr. Qasim ZeeshanLECTURE # 7
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WHAT IS OPTIMIZATION?
● “Making things better”
● “Generating more profit”
● “Determining the best”
● “Do more with less”
● “The determination of values for design variables wh
(maximize) the objective, while satisfying all constraints”
Principles of Optimal Design: Modelin
2d Ed. by Panos Y. Papalambros and Douglass J. Wilde, Cambridge University Press, New
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Search for a doc
(Search)Strategies are of an disordered world
(Search)Strategies need apredictable order of the world
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RECAP
Local
Multi-Objec
Un-Constrain
Non-GradientGradient Based
Constrained
Single-Objective
Global
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EVOLUT
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Evolution
Evolution (also known asbiological or organic evolution) isthe change over time in one ormore inherited traits found inpopulations of organisms
Adaptation is the process that
makes organisms better suited totheir habitat
Adaptation may cause either thegain of a new feature, or the lossof an ancestral feature
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Evolution means
climbing a fitness-hill
F i t
n e s s
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Evolution in Biology
Organisms produce a number of offspring similar to themselves butvariations due to:
Mutations (random changes)
Sexual reproduction (offspring have combinations of features inherited
parent)
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Evolution in Biology
Some offspring survive, and produce next generations, and some d The organisms adapted to the environment better have higher
survive
Over time, the generations become more and more adapted befittest organisms survive
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GENALGORI
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Survival of the Fittest
The main principle of evolution used in GAis “survival of the fittest”
The good solution survive, while bad ones die!
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Genetic Algorithms are good at taking large,
huge search spaces and navigating them,
optimal combinations of things, solutions yo
otherwise find in
- Salva
Computer De
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What are Genetic Algorithms?
Search algorithms based on mechanics of natural selection Based on genetic processes of biological organisms
Result is search algorithm with innovative flair of human search
Efficiently exploit historical information to speculate on new searchexpected improved performance
Bottom line: GAs are Intelligent exploitation of a random search
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What are Genetic Algorithms?
GA are adaptive methods which may be used to solve search and opproblems
Based on genetic processes of biological organisms
Combine survival of fittest with structured yet randomized information exchange
Result is search algo
Over many generations, populations evolve according to principles of natural selethe fittest => a Darwin type of approach
GAs simulate those processes in natural populations which are essential to
Metaphor underlying GA is that of natural evolution
each species faces search for beneficial adaptations to complicated and changing
“knowledge” each species gain is embodied in the makeup of the chromosomes of
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Conventional search methods, such as the Descent Method, aincapable of optimizing non-linear multimodal functions. So search method is required.
A GA is a directed random search technique, which can findoptimal solution in complex multi-dimensional search space.
A GA is modeled on natural evolution in that the operators iare inspired by the natural evolution process.
These operators manipulate individuals in a population overgenerations to improve their fitness gradually.
What are Genetic Algorithms?
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Individuals in a population are likened to chromoso
chromosome has a fitness value associated with it.
GAs do not use much knowledge about the prob
optimized and do not deal directly with the parameproblem. They manipulate codes which represent the pa
What are Genetic Algorithms?
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Biology vs Optimization
Candidate solutions to the optimization problem play the role oin a population (or chromosomes)
Cost/fitness/objective function determines the environment withsolutions “live”
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The Metaphor26
NatuGenetic Algorithm Environment Optimization problem
Individuals living in
environment
Feasible solutions
Individual’s degree
to its surrounding enSolutions quality (fitness function)
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The Metaphor (cont)27
NatuGenetic Algorithm A population of org(species)
A set of feasible solutions
Selection, recombina
mutation in nature’s process
Stochastic operators
Evolution of populatheir environment
Iteratively applying a set ofstochastic operators on a set of
feasible solutions
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GENETIC ALG
HISTORICAL PERSP
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GA
Invented by John Holland 1975
Made popular by John Koza 199
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HISTORICAL PERSPECTIVE
Charles Darwin (1809-1882)
Controversial and very influential book (1859) On the
origin of species by means of natural selection, or the
preservation of favored races in the struggle for life
Observations:
Species are continually developing Homo sapiens and apes have common ancestors
Variations between species are enormous
Huge potential for production of offspring, but only asmall/moderate percentage survives to adulthood
Evolution = natural selection of inheritable variations
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HISTORICAL PERSPECTIVE
Gregor Mendel (1822-1884) Investigated the inheritance of
characteristics (“traits”)
Conducted extensive experimentswith pea plants
Examined hybrids from differentstrains of plant
Character (gene) for tallness isdominant
Character (gene) for shortness isrecessive
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HISTORICAL PERSPECTIVE
Rechenberg (1960)
Idea of evolutionary computing was introduced in the1960s by I. Rechenberg in his work "Evolution strategies"(Evolutions strategie in original).
German computer scientist and professor (born January 20,1934 in Berlin). Rechenberg is a pioneer of the fields ofevolutionary computation and artificial evolution. In the1960s and 1970s he invented a highly influential set ofoptimization methods known as evolution strategies (fromGerman Evolutions strategie).
http://www.bionik.tu-berlin.de/institut/xn2rechenb.html
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HISTORICAL PERSPECTIVE
Rechenberg (1960)
His group successfully applied the new algorithms tochallenging problems such as aerodynamic wing design.These were the first serious technical applications ofartificial evolution
His idea was then developed by other researchers.
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HISTORICAL PERSPECTIVE
John Holland (1970)
Genetic Algorithms (GAs) were invented by JohnHolland and developed by him and his students andcolleagues.
This lead to Holland's book "Adaption in Natural and
Artificial Systems" published in 1975
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HISTORICAL PERSPECTIVE
John Holland (1970)
Developed by John Holland, his colleagues andstudents at University of Michigan (1970’s)
To understand processes in natural systems
To design artificial systems retaining the robustness
and adaptation properties of natural systems Goals
To abstract and rigorously explain the adaptiveprocess of the natural system
Design artificial systems software that retains the
important mechanisms of natural systems
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HISTORICAL PERSPECTIVE
John Holland (1970)
Holland’s original GA is known as the simple
genetic algorithm(SGA)
Provide efficient techniques for optimization andmachine learning applications
Widely used in business, science and engineering
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HISTORICAL PERSPECTIVE
KOZA (1992)
John R. Koza received his Ph.D. in computer science from theUniversity of Michigan in 1972, working under John Holland. From1973 through 1987, he was co-founder, chairman, and CEO ofScientific Games Inc. where he co-invented the rub-off instant lotteticket used by state lotteries.
In 1992 John Koza has used genetic algorithm to evolve programs toperform certain tasks.
He called his method "genetic programming" (GP).
LISP programs were used, because programs in this language canexpressed in the form of a "parse tree", which is the object the GA woon.
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HISTORICAL PERSPECTIVE
David Edward Goldberg (born 1953) is an American computer scientist, andprofessor at the department of Industrial and Enterprise Systems Engineering(IESE) at the University of Illinois at Urbana-Champaign and is most noted for hisseminal works in the field of genetic algorithms. He is the director of Illinois genealgorithms laboratory (IlliGAL) and also the chief scientist of Nextumi Inc.
He is also the author of "Genetic Algorithms for Search, Optimization, and MachineLearning", which is one of the most cited books in computer science.
David E. Goldberg received a PhD degree in civil engineering in 1983 fromUniversity of Michigan and his advisors were E. Benjamin Wylie and John HenHolland. He is one of the most connected scientists in the evolutionary computatiofield, having collaborated, among others, with Kalyanmoy Deb and Jeff Horn.
In 2003 David Goldberg was appointed as the first holder of Jerry S. DobrovolnProfessorship in Entrepreneurial Engineering at the University of Illinois at Urbana
Champaign
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HISTORICAL PERSPECTIVE
Central theme is ROBUSTNESS
Balance between efficiency and efficacy needed for survival in manyenvironments
Robustness
If you look at biological systems one can only marvel at its robustnessflexibility
Includes
self repair
self guidance
reproduction
These features barely exist in sophisticated artificial systems
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GENETIC ALGOR
_______________________VOCABU
TERMINO
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Terminology
CHROMOSOME (genotypes): a string representing a solution
problem.
GENES (phenotypes): a coding representing the chromosome.
ALLELES : the various values that a gene can take.
LOCI : the position of a given allele in the chromosome.
POPULATION (gene-pool): the set of chromosomes used in a ggeneration (iteration).
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Some Terminology
CHILDREN: the generated chromosomes from the current pop
PARENTS : the chromosomes that form the children (current chr
FITNESS : a value of the function to optimize under a given ch
OPERATORS : transformations that generate new solutions bas
current ones; the way of reproduction of new solutions. Selection, crossover, mutation, and inversion
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GA: VOCABULARY
CHROMOSOME All living organisms consist of cells. In each cell there is
the same set of chromosomes. Chromosomes are strings of DNA and serves as a
model for the whole organism. A chromosome consist of genes, blocks of DNA. Each
gene encodes a particular protein. Basically 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.
Each gene has its own position in the chromosome. Thisposition is called locus.
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GA: VOCABULARY
Complete set of genetic material (all chromosomes) is
called genome.
Particular set of genes in genome is called genotype.
The genotype is with later development after birthbase for the organism's phenotype, its physical and
mental characteristics, such as eye color, intelligenceetc.
GA VOCABULARY
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GA: VOCABULARY
REPRODUCTION
During reproduction, first occurs recombination (orCrossover).
Genes from parents form in some way the whole newchromosome.
The new created offspring can then be mutated.
Mutation means, that the elements of DNA are a bitchanged. This changes are mainly caused by errors incopying genes from parents.
The fitness of an organism is measured by success ofthe organism in its life.
GA VOCABULARY
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GA: VOCABULARY
SEARCH SPACE
If we are solving some problem, we are usually looking for some solution, best among others. The space of all feasible solutions (it means objects amdesired solution is) is called search space (also state space).
Each point in the search space represent one feasible solution. Each feasib"marked" by its value or fitness for the problem. We are looking for our so
one point (or more) among feasible solutions - that is one point in the sear
GA VOCABULARY
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GA: VOCABULARY SEARCH SPACE
The looking for a solution is then equal to a looking for some extreme (minmaximum) in the search space..
The problem is that the search can be very complicated. One does not knofor the solution and where to start.
There are many methods, how to find some suitable solution (ie. not necess
solution), for example hill climbing, tabu search, simulated annealing and g
S h S
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Search Space51
If we are solving some problem, we are usually looking for some solution, which will others. The space of all feasible solutions (it means objects among those the desiredsearch space (also state space). Each point in the search space represent one feasibfeasible solution can be "marked" by its value or fitness for the problem.
Initialization: Initially many individual solutions are randomly generated to form an incovering the entire range of possible solutions (the search space)Each point in the search space represents one possible solution marked by its value(
Selection: A proportion of the existing population is selected to bread a new bread
Reproduction: Generate a second generation population of solutions from those seleoperators: crossover and mutation.
Termination: A solution is found that satisfies minimum criteria Fixed number of generations found Allocated budget (computation, time/money) reached The highest ranking solution’s fitness is reaching or has reached
GA VOCABULARY
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GA: VOCABULARY
Should be obvious by now that GAs use vocabulary borrowed from
Genotype
Individuals in population
also called structures, strings, or chromosomes
Genes
Unit or section of genotype (chromosome) Arranged in linear succession
also called features, characters, or decoders
GA VOCABULARY
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GA: VOCABULARY
Loci
String positions
Locations of genes
Allele
Value of a particular loci in a gene
GA VOCABULARY
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GA: VOCABULARY
Population of individuals
The individuals are solutions to problem
Population dynamics
Births
Deaths
“Survival of the fittest” Only the fittest survive to mate
Inheritance
Features of fittest inherited by offspring
GA VOCABULARY
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GA: VOCABULARY
Population of individuals
Genetic representation for potential solutionsWay to create initial population of potential solution
Population dynamicsTraits of “good” individuals passed on Traits of “bad” individuals die off
“Survival of fittest” Evaluation process to measure each individualSelection process
InheritanceNeed to pass on traits they were selected for
Genetic operators that alter composition of children
GA Elements
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GA Elements
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GENETIC ALGO
______________________POPULATION OPERA
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GA: Population operators
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GA: Population operators
REPRODUCTION:
Exact copy/copies of individual
CROSSOVER:
Randomly exchange genes of different parents
Many possibilities: how many genes, parents, children …
MUTATION: Randomly flip some bits of a gene string
Used sparingly, but important to explore new designs
GA: Population operators
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GA: Population operators
CROSSOVER:
1 1 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1
Parent 1 Parent 2
0 1 1 0 0 0 1 0 1 1 0 0 1 0 1
Child 1 Child 2
1 1 0 1 1 0 0 1 0 1
● MUTATION:
1 1 0 1 0 0 1 0 1 1 0 0 1 0 1
1 1 0 1 0 1 1 0 1 1 0 0 1 0 1
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GENETIC ALGOR _______________________
ALGOR
The GA Algorithm
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The GA Algorithm
1. Initialize a population of chromosomes (a set of solutions).2. Evaluate each chromosome in the population.
3. Create new chromosomes by mating current chromosomes usi
operators.
4. Delete some old chromosomes to maintain the size of the pop
5. Evaluate the new chromosomes and insert them into the popu
6. If certain stopping criteria are met, stop; otherwise go to ste
GA: Basic Algorithm
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GA: Basic Algorithm
REPRODUCTION
Algorithm is started with a set of solutions (represented by chromosopopulation.
Solutions from one population are taken and used to form a new pop
This is motivated by a hope, that the new population will be better tha
Solutions which are selected to form new solutions (offspring) are selto their fitness - the more suitable they are the more chances they hav
This is repeated until some condition (for example number of popula
improvement of the best solution ) is satisfied.
GA: Basic Algorithm
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GA: Basic Algorithm
[Start] Generate random population of n chromosomes (suitable solutions
[Fitness] Evaluate the fitness f(x) of each chromosome x in the population [New population] Create a new population by repeating following steps
population is complete [Selection] Select two parent chromosomes from a population according to the
fitness, the bigger chance to be selected) [Crossover] With a crossover probability cross over the parents to form a new
If no crossover was performed, offspring is an exact copy of parents. [Mutation] With a mutation probability mutate new offspring at each locus (po
chromosome). [Accepting] Place new offspring in a new population
[Replace] Use new generated population for a further run of algorithm [Test] If the end condition is satisfied, stop, and return the best solution in [Loop] Go to step 2
GA flowchart
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GA flowchart
Create initial
population Evaluate fitnessof all individuals
Test termincriteria
Select individualsfor reproduction
Create new population
Crossover Mutation Reproduction
Termination criteria can be fixed number of generations, a certain required fitness level is reached, no
ti
Flowchart of a simple GA
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Flowchart of a simple GA
Initial population
Evaluation
Selection
Crossover
Mutation/ Inversion
GA: Basic Algorithm
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GA: Basic Algorithm
[Start] Generate random population of n chromosomes (suitable solutions
[Fitness] Evaluate the fitness f(x) of each chromosome x in the population [New population] Create a new population by repeating following steps
population is complete [Selection] Select two parent chromosomes from a population according to the
fitness, the bigger chance to be selected) [Crossover] With a crossover probability cross over the parents to form a new
If no crossover was performed, offspring is an exact copy of parents. [Mutation] With a mutation probability mutate new offspring at each locus (po
chromosome). [Accepting] Place new offspring in a new population
[Replace] Use new generated population for a further run of algorithm [Test] If the end condition is satisfied, stop, and return the best solution in [Loop] Go to step 2
Methodology Associated with GA
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Begin
Initialize Population
Optimum
Solution?
T=T+1
(go to next step)
Se
Cro
Mu
N
Evaluate Solutions
Y
Stop
T =0 (first step)
Creating a GA on Computer
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Simple_Genetic_Algorithm()
{ Initialize the Population;
Calculate Fitness Function;
While(Fitness Value != Optimal Va
{
Selection;//Natural Selection, Surv
Crossover;//Reproduction, PropagMutation;//Mutation
Calculate Fitness Fun
}
}
Nature Vs Computer - Mapping
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73
p pp g
Nature
Computer
Population
Individual
Fitness
Chromosome
Gene
Reproduction
Set of solutions.
Solution to a problem.
Quality of a solution.
Encoding for a Solution.
Part of the encoding of a solution.
Crossover
Nature Vs Computer - Mapping
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p pp g
Classical Algorithm Genetic Algorith
Generates a single point at eachiteration.
The sequence of pointsapproaches an optimal solution.
Selects the next point in the sequenceby a deterministic computation.
Generates a population ofeach iteration.
The best point in thepopulation approaches an
solution.
Selects the next populationcomputation which uses rannumber generators.
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GENETIC ALGOR _______________________
ENCOD
ENCODING
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The process of representing the solution in the form of aconveys the necessary information.
Just as in a chromosome, each gene controls a particularcharacteristic of the individual, similarly, each element in
represents a characteristic of the solution.
Encoding Methods
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Binary Encoding – Most common method of encoding. Chromoso
strings of 1s and 0s and each position in the chromosome represparticular characteristic of the problem.
Permutation Encoding – Useful in ordering problems such as theSalesman Problem (TSP). Example. In TSP, every chromosome is anumbers, each of which represents a city to be visited.
11111110000000011111Chromosome B
10110010110011100101Chromosome A
8 5 6 7 2 3 1 4 9Chromosome B
1 5 3 2 6 4 7 9 8Chromosome A
g
Encoding Methods
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Value Encoding – Used in problems where complicated valuesnumbers, are used and where binary encoding would not suffic
Good for some problems, but often necessary to develop crossover and mutation techniques for these chromosomes.
(left), (back), (left), (right), (forwChromosome B
1.235 5.323 0.454 2.321 2Chromosome A
g
GA Elements
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Citation:
http://ocw.mit.edu/NR/rdonlyres/Aeronautics-and-Astronautics/16-888Spring-2004/D66C4396-90C8-49BE-BF4A-4EBE39CEAE6F/0/M
Encoding Methods (contd.)
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Tree Encoding –is used mainly for evolving programs or expressions, i.e. for Genetic p
Tree Encoding - every chromosome is a tree of some objects, such as values/arith
commands in a programming language.
( + x ( / 5 y ) ) ( do_until step wall )
Characterizing a GA Via an Examp
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g p
Explain the main elements that form a GA using an illustratmaximize F ( x) = x2
s.t. x
[0, 31]
Main elements of GAs
Coding a chromosome (representation)
Creation of initial population
Genetic operators
Control parameters
Representation (Coding a Chromosome
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p g
The parameters to be optimized are usually represented in a string
genetic operators are suitable for this type of representation.
The method of representation has a major impact on the perfor
GA.
Two common representation methods for numerical optimization pr
The preferred method is the binary string representation method
The second is to use a vector of integers or real numbers, with ea
real number representing a single parameter.
Representation (Coding a Chromosome
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When a binary representation scheme is employed, an impo
to decide the number of bits used to encode the parameters.
The length of the coding string that needs to be used has to b
Each parameter should be encoded with the optimal number of bits
possible solutions in the solution space. When too few or too many bits are used the performance can be a
affected.
Binary String Representation of an Integer Nu
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Any integer number can be written in decimal system x = 2,765 can be written as
2,765 = 2.103 + 7.102 + 6.101 +5.100
It is also possible to code a number in binary form
x = 39 = 1.25 + 0.24 + 0.23 + 1.22 + 1.21 + 1.20
or simply x can be represented by a string of 6 bits as
x = (100111)
Binary String Representation of an Integer Nu
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Estimation on the length of the binary string for an integer numberan integer variable x [a, b]
Length of binary string > log2(b-a)
For example, x [0, 31]
len> log2(31-0) = log231 = 5
Binary String Representation of a ContinuousFunction
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222 1
b a
For functional optimization
Maximize F ( x) where x [a, b]
Generate a bit string of length k, say 22.
For instance, this gives x' = (01011 . . . 0110), hence
x' [0, 222 -1]
Translate x' into x [a, b]
precision or accuracy=12
22
ab
xa x
Binary String Representation of a ContinuousFunction
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Accuracy estimation
A continuous variable x [a, b]Length of binary string = m
Accuracy = (b-a)/(2m-1)
Estimation on the length of the binary string
For example, x [4.1, 6.8], accuracy required=10-4
len> log2[(6.8-4.1)/10-4 ]= log217000 = 14.1
2log accuracy requir
b alength
Binary String Representation of a ContinuousFunction
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In the example of the continuous function optimizatiomaximize F ( x) = x2 where x [0, 31]
we use a binary coding, set accuracy=1, then the stri
length is 5.
maximum x = 31 can be represented by (11111
31= 24 + 23 + 22 + 21 + 20
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GENETIC ALGOR _______________________
INITIALIZAT
INITIALIZATION
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90
Start with a population of randomly generated individuals, o
A previously saved population
A set of solutions provided by a human expert
A set of solutions provided by another heuristic algorithm
POPULATION
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A population is all the organisms that both belong to the same spe
the same geographical area. The area that is used to define the population is such that inter-bre
possible between any pair within the area and more probable thabreeding with individuals from other areas.
Normally breeding is substantially more common within the area thborder
CREATION OF INITIAL POPULA
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Two ways of forming an initial population
Using a random number generator to produce solutions randomly
preferred for problems about which no a prior knowledge exists or for performance of an algorithm.
Employing a prior knowledge about the given problem to obtain a requirements, and solutions satisfying those requirements are collect
initial population.
the GA starts the optimization with a set of approximately known solutiotherefore converges to an optimal solution in short time.
GENETIC OPERATORS
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Three common genetic operators: selection, crossover and mutation.
An additional reproduction operator: inversion.
It is not necessary to employ all of these operators in a GA becausfunctions independently of the others.
The choice or design of operators depends on the problem and
representation scheme employed. For instance, operators designed for binary strings cannot be directly used
coded with integers or real numbers.
SELECTION
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Natural selection is the process by which biologic traits b
more or less common in a population due to consistent e
upon the survival or reproduction of their bearers.
It is a key mechanism of evolution.
SELECTION
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The aim of the selection mechanism is to reproduce more copies of
individuals whose fitness values are higher than those whose fitnessare low.
The selection procedure has a significant influence on driving the setowards a promising area and finding good solutions in a short tim
There are several selection mechanisms. Two ways used extensively
RANKING-BASED SELECTION
PROPORTIONAL SELECTION
METHODS of SELECTION
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There are many different techniques which a genetic algorithm can useindividuals to be copied over into the next generation, but listed below
most common methods. Some of these methods are mutually exclusive, and often are used in combination. Elitist selection
Fitness-proportionate selection
Roulette-wheel selection
Scaling selection
Tournament selection
Rank selection
Generational selection
Steady-state selection
Hierarchical selection
METHODS of SELECTION
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Elitist selection: The most fit members of each generation are guaranteed t(Most GAs do not use pure elitism, but instead use a modified form where a few of the best, individuals from each generation are copied into the nein case nothing better turns up.)
Fitness-proportionate selection: More fit individuals are more likely, but not selected.
Roulette-wheel selection: A form of fitness-proportionate selection in which
individual's being selected is proportional to the amount by which its fitnesthan its competitors' fitness. (Conceptually, this can be represented as a gaeach individual gets a slice of the wheel, but more fit ones get larger sliceThe wheel is then spun, and whichever individual "owns" the section on whictime is chosen.)
METHODS of SELECTION
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Scaling selection: As the average fitness of the population increases, the strselective pressure also increases and the fitness function becomes more dismethod can be helpful in making the best selection later on when all indivirelatively high fitness and only small differences in fitness distinguish one f
Tournament selection: Subgroups of individuals are chosen from the larger members of each subgroup compete against each other. Only one individusubgroup is chosen to reproduce.
Rank selection: Each individual in the population is assigned a numerical rafitness, and selection is based on this ranking rather than absolute differenadvantage of this method is that it can prevent very fit individuals from gaearly at the expense of less fit ones, which would reduce the population's and might hinder attempts to find an acceptable solution.
METHODS of SELECTION
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Generational selection: The offspring of the individuals selected from each become the entire next generation. No individuals are retained between g
Steady-state selection: The offspring of the individuals selected from each gback into the pre-existing gene pool, replacing some of the less fit membegeneration. Some individuals are retained between generations.
Hierarchical selection: Individuals go through multiple rounds of selection eaLower-level evaluations are faster and less discriminating, while those that
levels are evaluated more rigorously. The advantage of this method is thatcomputation time by using faster, less selective evaluation to weed out the individuals that show little or no promise, and only subjecting those who surto more rigorous and more computationally expensive fitness evaluation.
Ranking-based selection
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The diversity of the population must be maintained to avoid p
convergence and to reach the global optimal solution.
One way to prevent premature is to limit the number of offsprings individual, so that no individual generates too many offsprings.
Ranking-based selection chooses a certain number of parents
the ranks of their fitness values, and not on the magnitudes. This number can be fixed a priori or variable depending on the po
average fitness.
Roulette Wheel selection
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This is a way of choosing members from the population of
chromosomes in a way that is proportional to their fitness. It does not guarantee that the fittest member goes through
to the next generation, merely that it has a very goodchance of doing so. It works like this:
Imagine that the population’s total fitness score is
represented by a pie chart, or roulette wheel. Now youassign a slice of the wheel to each member of thepopulation.
The size of the slice is proportional to that chromosomesfitness score. i.e. the fitter a member is the bigger the sliceof pie it gets. Now, to choose a chromosome all you haveto do is spin the ball and grab the chromosome at thepoint it stops.
The Roulette Wheel Selection
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Proportional selection is usually called “roulette wheel” selecti
its mechanism is reminiscent of the operation of a roulette wh
Fitness values of individuals represent the widths of slots on t
wheel.
After a random spinning of the wheel to select an individuanext generation, individuals in slots with large widths represe
high fitness values will have a higher chance to be selected.
The Roulette Wheel Selection
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For each parent k, the probability of being selected can be cF( x)/F where x is the corresponding value for parent k.
The values for all the parents are arranged in a roulette wheparent to choose is recorded according to the outcome.
These values can be put in a list from 0 to 1, a random number in [0
and for whichever range contains such a number, the associated paselected.
This process is repeated until the new generation is completed.
The Roulette Wheel Selection
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String No. Current
Population
x value F (x ) P[select] Expec
count
1 01101 13 169 0.14 0.58
2 11000 24 576 0.49 1.97
3 01000 8 64 0.06 0.22
4 10011 19 361 0.31 1.23
sum 1170 1.00 4.00
Average(F) 293 0.25 1.00 Maximum 579 0.49 1.97
P[select] = F (x )/ Sum
Expected count = F (x )/F
Actual count = number found by roulette wheel or simply the nearest integer of the expected count.
Red denotes the fittest chromosome.
The Roulette Wheel Selection
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Chromosomes that fail to pass the test (an expected count
least 1), say Nf , will be removed from the current populati
More formally the number of dropped chromosomes can b
expressed as Nd = Max {K 0, Nf }
K 0 is the minimum number that has to be dropped from one gen
the next. Nf can be found as |{k Population st: P[select k] < , say = 0.
where | E | denotes the cardinality of the set E.
The Roulette Wheel Selection
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The threshold can vary from problem to problem
For instance, parent 3 can be removed and replaced by a new
which can be formed from the current remaining parents or s
generated completely afresh.
The new parent is generated via some operators inclucrossover, mutation and inversion.
Local Tournament Selection107
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Extracts k individuals from the population with uniform pr
(without re-insertion) and makes them play a “tournamen
the probability for an individual to win is generally propor
fitness
Selection pressure is directly proportional to the number
participants
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GENETIC ALG
___________________CRO
109
CROSSOVER
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Main idea:
combine genetic material ( bits ) of 2 “parent” chrom
( solutions ) and produce a new “child” possessing
characteristics of both “parents”.
How it works ?
Several methods ….
CROSSOVER
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Crossover operator is used to create two new individuals
(children) from two existing individuals (parents) pickedfrom the current population by the selection operation.
Crossover process can be repeated until a suitable
stopping criterion is met. The selection of parents for
mating is critical to the success of GA.
Some common crossover operations: one-point crossover,
two-point crossover, cycle crossover, and uniform crossover.
GA: Population operators
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CROSSOVER:
1 1 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1 Parent 1 Parent 2
0 1 1 0 0 0 1 0 1 1 0 0 1 0 1
Child 1 Child 2
1 1 0 1 1 0 0 1 0 1
Why does crossover work?
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A lot of theory about this and some controversy
Holland introduced “Schema” theory
The idea is that crossover preserves “good bits” from different
combining them to produce better solutions
A good encoding scheme would therefore try to preserve “goo
during crossover and mutation
CROSSPVER: TECHNIQUES
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ONE-POINT CROSSOVER
A single crossover point on both parents' organism stringsis selected. All data beyond that point in either organismstring is swapped between the two parent organisms. Theresulting organisms are the children:
TWO-POINT CROSSOVER Two-point crossover calls for two points to be selected on
the parent organism strings. Everything between the twopoints is swapped between the parent organisms,rendering two child organisms:
CROSSPVER: TECHNIQUES
U if C d H lf U if C
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Uniform Crossover and Half Uniform Crossover The Uniform Crossover uses a fixed mixing ratio between two parents. Unlike, on
crossover, the Uniform Crossover enables the parent chromosomes to contribute rather than the segment level. If the mixing ratio is 0.5, the offspring has approthe genes from first parent and the other half from second parent, although crobe randomly chosen as seen below
The Uniform Crossover evaluates each bit in the parent strings for exchange wit0.5. Even though the uniform crossover is a poor method, empirical evidence sugmore exploratory approach to crossover than the traditional exploitative appro
maintains longer schemata. This results in a more complete search of the design maintaining the exchange of good information.
One-point Crossover
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Two individuals are randomly selected as parents from the p
individuals formed by the selection procedure and cut at a rachosen point. The tails, which are the parts after the cutting pswapped and two new individuals (children) are produced.
Child 1 and child 2 are different from their parents and thesmore diversification leading to new solutions.
Parent 1: 10001|001111
Parent 2: 01101|100011
Child 1: 10001100011
Child 2: 01101001111
One-point Crossover
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The one-point crossover operator may have no effect on the s
Each child was found to be exactly the same as his or her pathe genes associated with both parents after the crossover po
to be the same.
Parent 1: 10001|001111
Parent 2: 01101|001111
Child 1: 10001001111
Child 2: 01101001111
One-point Crossover
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The other parents unfortunately lost their best genes to
produce less fit children.
Those two less fit chromosomes will die away and will be
replaced by two other ones, which are either selected randomly
or copied directly from the best current chromosomes.
By keeping the best fit chromosomes in the population, we
are hoping to make even fitter chromosomes.
Two-point Crossover
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Parent 1: 101|0001|1010
Parent 2: 011|0111|1011
Child 1: 10101111010
Child 2: 01100011011
Crossover (contd.)
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Crossover between 2 good solutions MAY NOT Ayield a better or as good a solution.
Since parents are good, probability of the childgood is high.
If offspring is not good (poor solution), it will be in the next iteration during “SELECTION”.
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GENETIC ALGO _____________________
E
122
ELITISM
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Main idea: copy the best chromosomes (solutions) population before applying crossover and mutation
When creating a new population by crossover or mutation the bchromosome might be lost.
Forces GAs to retain some number of the best individuals at eac
Has been found that elitism significantly improves performanc
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GENETIC ALGOR
MUTA
MUTATION
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In molecular biology and genetics, mutations are changes in
genomic sequence: the DNA sequence of a cell's genome or t
or RNA sequence of a virus.
They can be defined as sudden and spontaneous changes in
Mutations are caused by radiation, viruses, and mutagenic chas well as errors that occur during meiosis or DNA replication
MUTATION
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All individuals are checked bit by bit and the bit values are rando
according to a specified rate. Unlike crossover, a child string is prosingle parent string.
The mutation operator forces the algorithm to search new areas, h
premature convergence and find the global optimal solution.
Parent 1 : 1 1 0 0 0 1 0 1 1 1 0
child 1 : 1 1 0 0 1 1 0 1 1 1 0
MUTATION
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For each bit in each string a random number is generated, sa
If < (fixed acceptance probability value, say = 0.005the value of that bit (allele) may mutate as this will take a va0 or 1 randomly.
Note that the value of that bit may not necessarily change although
passed the first probability test. If > this particular bit is kept unchanged and the next bi
string is tested and the process is repeated.
MUTATION
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The new population has a different spread of solutions.
According to these results both the third and the fourth chrom
will die away and be replaced by two other ones.
The old second chromosome has lost some of its fitness, where
old fifth chromosome has increased its fitness by inverting theat loci 3 of this string from 0 to 1.
GENETIC ALGOR _______________________
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CROSSO
MUTA
n-point crossover
Choose n random crossover points
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Choose n random crossover points
Split along those points
Glue parts, alternating between parents
Generalisation of 1 point (still some positional bias)
Uniform crossover
Assign 'heads' to one parent, 'tails' to the other
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Assign heads to one parent, tails to the other
Flip a coin for each gene of the first child
Make an inverse copy of the gene for the second child Inheritance is independent of position
CROSSOVER OR MUTATION?
Decade long debate which one is better / necessary / main backg
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Decade long debate: which one is better / necessary / main-backg
Answer (at least, rather wide agreement):
it depends on the problem, but
in general, it is good to have both
both have another role
mutation-only-EA is possible, crossover-only-EA would not work
Exploration Discovering promising areas in the search spac
Crossover OR mutation? (cont’d)
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Exploration: Discovering promising areas in the search spac
information on the problemExploitation: Optimising within a promising area, i.e. using
There is co-operation AND competition between them
Crossover is explorative, it makes a big jump to an area
“in between” two (parent) areas Mutation is exploitative, it creates random small diversio
staying near (in the area of ) the parent
Crossover OR mutation? (cont
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Only crossover can combine information from two paren
Only mutation can introduce new information (alleles)
Crossover does not change the allele frequencies of the
(thought experiment: 50% 0’s on first bit in the populati
after performing n crossovers)
To hit the optimum you often need a ‘lucky’ Mutation
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GENETIC ALGO _____________________
INVE
Inversion works on a single chromosome
INVERSION
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Inversion works on a single chromosome
It generates two positions in the string (locus) randomly or vand then all the elements between those two points will hav
inverted (from 0 to 1 and vice versa).
Parent 1 : 1 0 0 1 0 0 1 1
Child 1 : 1 0 0 0 1 1 0 1
INVERSION
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Populati on After
Reproduction
I nversion Points
(Chosen Randomly )
New Populati on x Value F (x ) P[s
0 1 1 0 1 2 3 0 0 0 0 1 1 1 0.0
1 1 0 0 0 3 4 1 1 1 1 0 30 900 0.8
0 1 0 0 0 2 3 0 0 1 0 0 4 16 0.0
1 0 0 1 1 1 2 0 1 1 0 0 12 144 0.1
Sum 1061 1.0
Average (F ) 265 0.2
Maximum 900 0.8
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GENETIC ALGOR _______________________
CONTROL PARAME
CONTROL PARAMETERS
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Important control parameters of a simple GA include
POPULATION SIZE
CROSSOVER RATE
MUTATION RATE
Population Size
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A large population size means the simultaneous handlin
solutions and increases the computation time per iteratio
Since many samples from the search space are used, th
probability of convergence to a global optimal solution
than when using a small population size.
Cross Over Rate
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The crossover rate determines the frequency of the crossover
operation.
It is useful at the start of optimization to discover a promising regio
A low crossover frequency decreases the speed of convergence to
such an area.
If the frequency is too high, it leads to saturation around one solutio
MUTATION RATE
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A high mutation rate introduces high diversity in the pop
and might cause instability.
It is usually very difficult for a GA to find a global opti
solution with too low a mutation rate.
Research in GA concentrates on several sensitive parts o
Future Research in GA
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Research in GA concentrates on several sensitive parts o
how to better represent a solution of a given problem
the development of new operators
how to integrate operators with the choice of the par
mating
Hybridization with other search algorithms
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GENETIC ALGOR _______________________
MULTI ISLAND
MIGA
Genetic algorithms work well because they incorporate randomness in thei
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A more proficient GA called “Multi-island Genetic Algorithm (MIGA)” is us
research.
MIGA originated from the traditional genetic algorithm (GA), which involveGA
It gives the algorithm the ability to correct deterministic search bottlenecksby the reasoning in the “space sampling” methods like Simulated Annealing
gradient methods
MIGA algorithm divides the population into several islands, performs tradioperations on each island separately, and then migrates individuals betwesearches many designs and multiple locations of the design space.
MIGA
Individual
Parameter
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Migration
Individual Island
Population
Traditional GA
Operations on
each island
Size of sub-
population
Number of
Population
Number of
generations
Total individ
Rate of cros
Rate of mut
Rate of mig
MIGA
MIGA allows preservation of the best individuals from the previous generalt ti " liti "
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alteration, elitism .
Elitism guarantees that the best genetic material is carried over to the child
The selection operation in MIGA employs the "tournament selection" schemindividuals are selected not from the whole population, but rather from a srandomly selected individuals.
Main feature of MIGA that distinguishes it from traditional GAs is the fact
population of individuals is divided into several subpopulations called "isla
All traditional genetic operations are performed separately on each sub-p
MIGA The exchange of individual information, termed “migration”, is carried out peri
sub-populations.
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Migration is performed according to a migration policy, which defines where aindividuals move.
Simple migration policy consists of a migration interval, which is the number of occur between migrations, and a migration topology, which determines where imigrate.
The following experimental design decisions have to be made to define the isla
Number of Sub populations: 2, 3, 4 .. Size of Sub Population: uniform or non uniform
Connectivity Topology: ring, star, fully connected, random
Migration Mechanism
MIGA
The motivation for using MIGA is two fold
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Firstly, we need to improve the speed of evolutionary processes by con
concurrent evaluations of individuals in a population.
Secondly, we need to improve the problem solving process by overcomsuch as premature convergence.
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CONCLUD
REMA
ISSUES
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GA: COMPUTATIONAL ISSUES
Population size
Representation of individual
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Representation of individual
Phenotypic: problem specific; real representation Genotypic: domain independent; encoding
How to produce offspring Mutation Crossover
Selection Worst Preserve the best Fitness proportional Block or group selection / deletion
GA: COMPUTATIONAL ISSUES
Population size
correlation between population size and behavior of GA
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correlation between population size and behavior of GA
Phenotype
The observable physical or biochemical characteristics of an organism, aboth genetic makeup and environmental influences
Genotype
The genetic constitution of an organism or a group of organisms.
Allele
One member of a pair or series of genes that occupy a specific position chromosome.
GA: COMPUTATIONAL ISSUES Measuring evaluation and fitness
Measure each individual separately
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Context sensitive - compare against others How to handle noise and constraints
Use as problem solvers Have new populations generating but how to use them
Population 1 Population 2 Population 3 Popu
Generation 1 Generation 2 Generation 3 Gen
Concluding Remarks
Differ from traditional search/optimization methods:
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/ p
GAs search a population of points in parallel, not only a single
GAs use probabilistic transition rules, not deterministic ones
GAs work on an encoding of the design variable set rather thavariables themselves
GAs do not require derivative information or other auxiliary knonly the objective function and corresponding fitness levels infl
Concluding Remarks
stochastic, directed and highly parallel search technique basprinciples of population genetics
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p p p p g
Difference with traditional search techniques: Coding of the design variables as opposed to the design va
themselves, allowing both discrete and continuous variables
Works with population of designs as opposed to single desireducing the risk of getting stuck at local minima
Only requires the objective function value, not the derivativeaspect makes GAs domain-independent
GA is a probabilistic search method, not deterministic, makinsearch highly exploitative.
COMMENTS on GA
Can successfully deal with a wide range of problem areas
Including those which are difficult for other methods to solve
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Including those which are difficult for other methods to solve
Not guaranteed to find the global optimum solutions
Common with all heuristics
Generally good at finding “acceptably good” solutions “acceptab
Do not require much in the way of mathematical requirements
Quite easy to “hybridize” Will cover such approaches in later lectures
COMMENTS on GA
Any population-based model that uses selection and recombinationgenerate new sample points in a search space
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GAs work with a coding of the solutions (versus the actual solution) GAs search using a population of solutions (versus a single point)
GAs do not derivative for the search
GAs use probabilistic transition rules (versus deterministic)
Advantages and disadvantag159
Advantages:
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Always an answer; answer gets better with time Good for “noisy” environments
Inherently parallel; easily distributed
Issues:
Performance
Solution is only as good as the evaluation function Termination Criteria
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GENETIC ALGO _____________________
M
MATLAB
SYNTAX
x = ga(fitnessfcn nvars)
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x = ga(fitnessfcn,nvars)
x = ga(fitnessfcn,nvars,A,b)
x = ga(fitnessfcn,nvars,A,b,Aeq,beq)
x = ga(fitnessfcn,nvars,A,b,Aeq,beq,LB,UB)
x = ga(fitnessfcn,nvars,A,b,Aeq,beq,LB,UB,nonlcon)
x = ga(fitnessfcn,nvars,A,b,Aeq,beq,LB,UB,nonlcon,options)
MATLAB
Fitnessfcn Fitness functionNvars Number of design variables
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Aineq A matrix for linear inequality constraintsBineq b vector for linear inequality constraintsAeq A matrix for linear equality constraintsBeq b vector for linear equality constraintsLb Lower bound on xUb Upper bound on x
Nonlcon Nonlinear constraint functionRandstate Optional field to reset rand stateRandnstate Optional field to reset randn stateSolver 'ga'Options Options structure created using gaoptimset
MATLAB nvars = 17; % NUMBER OF VARIA
fitnessFunction = @MY_FUNCTION; options = gaoptimset; % %Start with default options %Modify some param
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options = gaoptimset(options,'PopInitRange' ,[lb;ub]); % options = gaoptimset(options,'InitialPop options = gaoptimset(options,'PopulationSize' ,100); % POPU
options = gaoptimset(options,'Generations' ,100); % NUMB options = gaoptimset(options,'StallGenLimit' ,50); % STALL options = gaoptimset(options,'StallTimeLimit' ,200000000); % STALL
options = gaoptimset(options,'TolFun' ,1e-9); % FUNCT options = gaoptimset(options,'TolCon' ,1e-9); % CONS options = gaoptimset(options,'CrossoverFcn' ,@crossovertwopoint); % CROSS options = gaoptimset(options,'CrossoverFraction' ,0.8); % CROSS options = gaoptimset(options,'SelectionFcn' ,{ @selectiontournament 4 }); % SELEC options = gaoptimset(options,'MutationFcn' ,{ @mutationuniform 0.25641 }); % MUTA options = gaoptimset(options,'Display' ,'iter'); % DISPLA [X,FVAL,REASON,OUTPUT,POPULATION,SCORES] = ga(fitnessFunction,nvars,options); %Run GA
MATLABThe genetic algorithm uses the following conditions to determine when to stop:
Generations — The algorithm stops when the number of generations reaches thGenerations.
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Time limit — The algorithm stops after running for an amount of time in seconds Fitness limit — The algorithm stops when the value of the fitness function for th
current population is less than or equal to Fitness limit.
Stall generations — The algorithm stops when the weighted average change in tvalue over Stall generations is less than Function tolerance.
Stall time limit — The algorithm stops if there is no improvement in the objectiveinterval of time in seconds equal to Stall time limit.
Function Tolerance — The algorithm runs until the weighted average change in tover Stall generations is less than Function tolerance.
Nonlinear constraint tolerance — The Nonlinear constraint tolerance is not usedIt is used to determine the feasibility with respect to nonlinear constraints
MATLABRastrigin's Function
This section presents an example that shows how to find the minimum
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function, a function that is often used to test the genetic algorithm. For two independent variables, Rastrigin's function is defined as
MATLAB
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MATLAB
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GENETIC ALGOR
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_______________________
EVOLUTIONARY STRAT
Evolution-Strategy
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ES-]),/(,/[
Wright Haldane Fisher ' = Number of offsp
' = Number of pop
' = Number of pare
= Number of pare
= Number of offsp = Generations of
' = Mixing number
= Mixing number
carnation
EVOLUTIONARY STRATEGY
Evolution-strategic optimization is based on the hypothesis that
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biological evolution the laws of heredity have been developedphylogenetic adaptation.
Evolution-Strategies (ES) imitate, in contrast to the genetic algoeffects of genetic procedures on the phenotype.
The presumption for coding the variables in the ES is the realiz
sufficient strong causality (small changes of the cause must creachanges of the effect).
EVOLUTIONARY STRATEGY
The (environmental) selection in evolution strategies is deterministic and onthe fitness rankings, not on the actual fitness values. The resulting algorithm
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invariant with respect to monotonic transformations of the objective functioevolution strategy operates on a population of size two: the current point the result of its mutation. Only if the mutant's fitness is at least as good asone, it becomes the parent of the next generation. Otherwise the mutant iThis is a (1 + 1)-ES. More generally, λ mutants can be generated and comparent, called (1 + λ )-ES. In (1 , λ)-ES the best mutant becomes the parengeneration while the current parent is always disregarded.
Contemporary derivatives of evolution strategy often use a population ofalso recombination as an additional operator, called ( μ / ρ+, λ )-ES. This mprone to get stuck in local optima
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Elementary Evolution-Strategic Algorit
(1 + 1)-ES
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D ARWINs theory at the
level of maximum abstraction
(1 , )-ES
= 6
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Evolution Strategy
with more than one offspring
( , )-ES
= 7 = 2
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Evolution Strategy with
more parents and more offspring
( , )-ES
= 8
= 2 = 2
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Evolution Strategy
with mixing of variables
ES]),(,[
1
2 1
5 4
New founder popula
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The Nested
Evolution Strategy
The notation
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will be an algebraic scheme
When to Use GA’s vs. ES’s
Genetic Algorithms
Evolution Strategie “Good enough” solu
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More important to findoptimal solution (GA’s more
likely to find globalmaximum; usually slower)
Problem parameters can be
represented as bit strings(computational problems)
acceptable (ES’s usufaster; can readily fmaximum)
Problem parameters
numbers (engineerinproblems)
GENETIC ALGOR
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_______________________
APPLICAT
GENETIC ALGORITHM
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Question: ‘If GAs are so smart, why ain’t they rich?’
Answer: ‘Genetic algorithms are rich - rich in application alarge and growing number of disciplines.’
- David E. Goldberg, Genetic Algorithms in Searc
Optimization and Machine
Some GA Application TypesDomain Application Types
Control gas pipeline, pole balancing, missile evasion, pu
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Design semiconductor layout, aircraft design, keyboardcommunication networks
Scheduling manufacturing, facility scheduling, resource allo
Robotics trajectory planning
Machine Learning designing neural networks, improving classificatclassifier systems
Signal Processing filter design
Game Playing poker, checkers, prisoner’s dilemma
Combinatorial Optimization set covering, travelling salesman, routing, bin pacolouring and partitioning
PROS AND CONS OF GABENEFITS ISSUES
Applicable on mixed, discrete, continuous problems It is still and art and requires signi
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Require little information about problem Can be computationally expensive
No gradients required Convergence behavior dependentpopulation size, selection, crossoveSimple to understand setup and implement
Always an answer; gets better with time (Robust) Termination criteria
Support multi-objective
Good for noisy environments
Easy to improve after learning about problem
Sustainable history and range of use
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REFEREN
REFERENCE BOOKS
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ASSIGNM
ASSIGNMENT No. 2 (Individual Study any paper on implementation of SQP
Identify Problem Statement
Id if P bl F l i
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Identify Problem Formulation Identify Design Variables
Identify Design Objectives
Identify Design Constraints
Explain Results
Solve any CONSTRAINED optimization problem ofusing SQP fmincon in MATLAB
USE AIAA Paper format please
SUBMISSION DEADLINE: 4 MAY 2013
ASSIGNMENT No. 3 (Individual Study any paper on implementation of GA
Identify Problem Statement
Id if P bl F l i
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Identify Problem Formulation Identify Design Variables
Identify Design Objectives
Identify Design Constraints
Explain Results
Solve any CONSTRAINED optimization problem ofusing GA Toolbox in MATLAB
USE AIAA Paper format please
SUBMISSION DEADLINE: 4 MAY 2013
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THANK YOU FOR YOUR INT