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Genetic Algorithms

Sources

- Jaap Hofstede
- Beasly, Bull, Martin

Introduction to Evolutionary Computation

- Evolutionary Computation is the field of study

devoted to the design, development, and analysis

is problem solvers based on natural selection

(simulated evolution). - Evolution has proven to be a powerful search

process. - Evolutionary Computation has been successfully

applied to a wide range of problems including - Aircraft Design,
- Routing in Communications Networks,
- Tracking Windshear,
- Game Playing (Checkers Fogel)

Introduction to Evolutionary Computation(Applicat

ions cont.)

- Robotics,
- Air Traffic Control,
- Design,
- Scheduling,
- Machine Learning,
- Pattern Recognition,
- Job Shop Scheduling,
- VLSI Circuit Layout,
- Strike Force Allocation,

Introduction to Evolutionary Computation(Applicat

ions cont.)

- Theme Park Tours (Disney Land/World)

http//www.TouringPlans.com - Market Forecasting,
- Egg Price Forecasting,
- Design of Filters and Barriers,
- Data-Mining,
- User-Mining,
- Resource Allocation,
- Path Planning,
- Etc.

Example of Evolutionary Algorithm

- An Example Evolutionary Computation
- Procedure EC
- t 0
- Initialize Pop(t)
- Evaluate Pop(t)
- While (Not Done)
- Parents(t) Select_Parents(Pop(t))
- Offspring(t) Procreate(Parents(t))
- Evaluate(Offspring(t))
- Pop(t1) Replace(Pop(t),Offspring(t))
- t t 1

Candidate Solutions CS

- In an Evolutionary Computation, a population of

candidate solutions (CSs) is randomly generated. - Each of the CSs is evaluated and assigned a

fitness based on a user specified evaluation

function. - The evaluation function is used to determine the

goodness of a CS. - A number of individuals are then selected to be

parents based on their fitness. - The Select_Parents method must be one that

balances the urge for selecting the best

performing CSs with the need for population

diversity.

Parents and Generations

- The selected parents are then allowed to create a

set of offspring which are evaluated and assigned

a fitness using the same evaluation function

defined by the user. - Finally, a decision must be made as to which

individuals of the current population and the

offspring population should be allowed to

survive. - Typically, in EC , this is done to guarantee that

the population size remains constant. - The study of ECs with dynamic population sizes

would make an interesting project for this course

Selecting and Stopping

- Once a decision is made the survivors comprise

the next generation (Pop(t1)). - This process of selecting parents based on their

fitness, allowing them to create offspring, and

replacing weaker members of the population is

repeated for a user specified number of cycles. - Stopping conditions for evolutionary search could

be - The discovery of an optimal or near optimal

solution - Convergence on a single solution or set of

similar solutions, - When the EC detects the problem has no feasible

solution, - After a user-specified threshold has been

reached, or - After a maximum number of cycles.

A Brief History of Evolutionary Computation

- The idea of using simulated evolution to solve

engineering and design problems have been around

since the 1950s (Fogel, 2000). - Bremermann, 1962
- Box, 1957
- Friedberg, 1958
- However, it wasnt until the early 1960s that we

began to see three influential forms of EC emerge

(Back et al, 1997) - Evolutionary Programming (Lawrence Fogel, 1962),
- Genetic Algorithms (Holland, 1962)
- Evolution Strategies (Rechenberg, 1965

Schwefel, 1968),

A Brief History of Evolutionary

Computation(cont.)

- The designers of each of the EC techniques saw

that their particular problems could be solved

via simulated evolution. - Fogel was concerned with solving prediction

problems. - Rechenberg Schwefel were concerned with solving

parameter optimization problems. - Holland was concerned with developing robust

adaptive systems.

A Brief History of Evolutionary

Computation(cont.)

- Each of these researchers successfully developed

appropriate ECs for their particular problems

independently. - In the US, Genetic Algorithms have become the

most popular EC technique due to a book by David

E. Goldberg (1989) entitled, Genetic Algorithms

in Search, Optimization Machine Learning. - This book explained the concept of Genetic Search

in such a way the a wide variety of engineers and

scientist could understand and apply.

A Brief History of Evolutionary

Computation(cont.)

- However, a number of other books helped fuel the

growing interest in EC - Lawrence Davis, Handbook of Genetic

Algorithms, (1991), - Zbigniew Michalewicz book (1992), Genetic

Algorithms Data Structures Evolution

Programs. - John R. Kozas Genetic Programming (1992), and
- D. B. Fogels 1995 book entitled, Evolutionary

Computation Toward a New Philosophy of Machine

Intelligence. - These books not only fueled interest in EC but

they also were instrumental in bringing together

the EP, ES, and GA concepts together in a way

that fostered unity and an explosion of new and

exciting forms of EC.

A Brief History of Evolutionary ComputationThe

Evolution of Evolutionary Computation

- First Generation EC
- EP (Fogel)
- GA (Holland)
- ES (Rechenberg, Schwefel)
- Second Generation EC
- Genetic Evolution of Data Structures

(Michalewicz) - Genetic Evolution of Programs (Koza)
- Hybrid Genetic Search (Davis)
- Tabu Search (Glover)

A Brief History of Evolutionary ComputationThe

Evolution of Evolutionary Computation (cont.)

- Third Generation EC
- Artificial Immune Systems (Forrest)
- Cultural Algorithms (Reynolds)
- DNA Computing (Adleman)
- Ant Colony Optimization (Dorigo)
- Particle Swarm Optimization (Kennedy Eberhart)
- Memetic Algorithms
- Estimation of Distribution Algorithms
- Fourth Generation ????

Introduction to Evolutionary ComputationA

Simple Example

- Lets walk through a simple example!
- Lets say you were asked to solve the following

problem - Maximize
- f6(x,y) 0.5 (sin(sqrt(x2y2))2 0.5)/(1.0

0.001(x2y2))2 - Where x and y are take from -100.0,100.0
- You must find a solution that is greater than

0.99754, and you can only evaluate a total of

4000 candidate solutions (CSs) - This seems like a difficult problem.
- It would be nice if we could see what it looks

like! - This may help us determine a good algorithm for

solving it.

Introduction to Evolutionary ComputationA

Simple Example

- A 3D view of f6(x,y)

Introduction to Evolutionary ComputationA

Simple Example

- If we just look at only one dimension f6(x,1.0)

Introduction to Evolutionary ComputationA

Simple Example

- Lets develop a simple EC for solving this

problem - An individual (chromosome or CS)
- ltxi,yigt
- fiti f6(xi,yi)

Introduction to Evolutionary ComputationA

Simple Example

- Procedure simpleEC
- t 0
- Initialize Pop(t) / of P individuals /
- Evaluate Pop(t)
- while (t lt 4000-P)
- Select_Parent(ltxmom,ymomgt) / Randomly /
- Select_Parent(ltxdad,ydadgt) / Randomly /
- Create_Offspring(ltxkid,ykidgt)
- xkid rnd(xmom, xdad) Nx(0,?)
- ykid rnd(ymom, ydad) Ny(0,?)
- fitkid Evaluate(ltxkid,ykidgt)
- Pop(t1) Replace(worst,kid)Pop(t)-worst?

kid - t t 1

Introduction to Evolutionary ComputationA

Simple Example

- To simulate this simple EC we can use the applet

at - http//www.eng.auburn.edu/gvdozier/GA.html

Introduction to Evolutionary ComputationA

Simple Example

- To get a better understanding of some of the

properties of ECs lets do the in class lab

found at http//www.eng.auburn.edu/gvdozier/GA_L

ab.html

Hill climbing

Introduction 1

- Inspired by natural evolution
- Population of individuals
- Individual is feasible solution to problem
- Each individual is characterized by a Fitness

function - Higher fitness is better solution
- Based on their fitness, parents are selected to

reproduce offspring for a new generation - Fitter individuals have more chance to reproduce
- New generation has same size as old generation

old generation dies - Offspring has combination of properties of two

parents - If well designed, population will converge to

optimal solution

Algorithm

- BEGIN
- Generate initial population
- Compute fitness of each individual
- REPEAT / New generation /
- FOR population_size / 2 DO
- Select two parents from old generation
- / biased to the fitter ones /
- Recombine parents for two offspring
- Compute fitness of offspring
- Insert offspring in new generation
- END FOR
- UNTIL population has converged
- END

Example of convergence

Introduction 2

- Reproduction mechanism has no knowledge of the

problem to be solved - Link between genetic algorithm and problem
- Coding
- Fitness function

Basic principles 1

- Coding or Representation
- String with all parameters
- Fitness function
- Parent selection
- Reproduction
- Crossover
- Mutation
- Convergence
- When to stop

Basic principles 2

- An individual is characterized by a set of

parameters Genes - The genes are joined into a string Chromosome
- The chromosome forms the genotype
- The genotype contains all information to

construct an organism the phenotype - Reproduction is a dumb process on the

chromosome of the genotype - Fitness is measured in the real world (struggle

for life) of the phenotype

Coding

- Parameters of the solution (genes) are

concatenated to form a string (chromosome) - All kind of alphabets can be used for a

chromosome (numbers, characters), but generally a

binary alphabet is used - Order of genes on chromosome can be important
- Generally many different codings for the

parameters of a solution are possible - Good coding is probably the most important factor

for the performance of a GA - In many cases many possible chromosomes do not

code for feasible solutions

Example of coding for TSP

- Travelling Salesman Problem
- Binary
- Cities are binary coded chromosome is string of

bits - Most chromosomes code for illegal tour
- Several chromosomes code for the same tour
- Path
- Cities are numbered chromosome is string of

integers - Most chromosomes code for illegal tour
- Several chromosomes code for the same tour
- Ordinal
- Cities are numbered, but code is complex
- All possible chromosomes are legal and only one

chromosome for each tour - Several others

Reproduction

- Crossover
- Two parents produce two offspring
- There is a chance that the chromosomes of the two

parents are copied unmodified as offspring - There is a chance that the chromosomes of the two

parents are randomly recombined (crossover) to

form offspring - Generally the chance of crossover is between 0.6

and 1.0 - Mutation
- There is a chance that a gene of a child is

changed randomly - Generally the chance of mutation is low (e.g.

0.001)

Crossover

- One-point crossover
- Two-point crossover
- Uniform crossover

One-point crossover 1

- Randomly one position in the chromosomes is

chosen - Child 1 is head of chromosome of parent 1 with

tail of chromosome of parent 2 - Child 2 is head of 2 with tail of 1

Randomly chosen position

Parents 1010001110 0011010010 Offspring

0101010010 0011001110

One-point crossover 2

Two-point crossover

- Randomly two positions in the chromosomes are

chosen - Avoids that genes at the head and genes at the

tail of a chromosome are always split when

recombined

Randomly chosen positions

Parents 1010001110 0011010010 Offspring

0101010010 0011001110

Uniform crossover

- A random mask is generated
- The mask determines which bits are copied from

one parent and which from the other parent - Bit density in mask determines how much material

is taken from the other parent (takeover

parameter)

Mask 0110011000 (Randomly

generated) Parents 1010001110 0011010010 Offsp

ring 0011001010 1010010110

Problems with crossover

- Depending on coding, simple crossovers can have

high chance to produce illegal offspring - E.g. in TSP with simple binary or path coding,

most offspring will be illegal because not all

cities will be in the offspring and some cities

will be there more than once - Uniform crossover can often be modified to avoid

this problem - E.g. in TSP with simple path coding
- Where mask is 1, copy cities from one parent
- Where mask is 0, choose the remaining cities in

the order of the other parent

Fitness Function

- Purpose
- Parent selection
- Measure for convergence
- For Steady state Selection of individuals to die
- Should reflect the value of the chromosome in

some real way - Next to coding the most critical part of a GA

Parent selection

- Chance to be selected as parent proportional to

fitness - Roulette wheel
- To avoid problems with fitness function
- Tournament
- Not a very important parameter

Roulette wheel

- Sum the fitness of all chromosomes, call it T
- Generate a random number N between 1 and T
- Return chromosome whose fitness added to the

running total is equal to or larger than N - Chance to be selected is exactly proportional to

fitness - Chromosome 1 2 3 4 5 6
- Fitness 8 2 17 7 4 11
- Running total 8 10 27 34 38 49
- N (1 ? N ? 49) 23
- Selected 3

Tournament

- Binary tournament
- Two individuals are randomly chosen the fitter

of the two is selected as a parent - Probabilistic binary tournament
- Two individuals are randomly chosen with a

chance p, 0.5ltplt1, the fitter of the two is

selected as a parent - Larger tournaments
- n individuals are randomly chosen the fittest

one is selected as a parent - By changing n and/or p, the GA can be adjusted

dynamically

Problems with fitness range

- Premature convergence
- ?Fitness too large
- Relatively superfit individuals dominate

population - Population converges to a local maximum
- Too much exploitation too few exploration
- Slow finishing
- ?Fitness too small
- No selection pressure
- After many generations, average fitness has

converged, but no global maximum is found not

sufficient difference between best and average

fitness - Too few exploitation too much exploration

Solutions for these problems

- Use tournament selection
- Implicit fitness remapping
- Adjust fitness function for roulette wheel
- Explicit fitness remapping
- Fitness scaling
- Fitness windowing
- Fitness ranking

Fitness scaling

- Fitness values are scaled by subtraction and

division so that worst value is close to 0 and

the best value is close to a certain value,

typically 2 - Chance for the most fit individual is 2 times the

average - Chance for the least fit individual is close to 0
- Problems when the original maximum is very

extreme (super-fit) or when the original minimum

is very extreme (super-unfit) - Can be solved by defining a minimum and/or a

maximum value for the fitness

Example of Fitness Scaling

Fitness windowing

- Same as window scaling, except the amount

subtracted is the minimum observed in the n

previous generations, with n e.g. 10 - Same problems as with scaling

Fitness ranking

- Individuals are numbered in order of increasing

fitness - The rank in this order is the adjusted fitness
- Starting number and increment can be chosen in

several ways and influence the results - No problems with super-fit or super-unfit
- Often superior to scaling and windowing

Other parameters of GA 1

- Initialization
- Population size
- Random
- Dedicated greedy algorithm
- Reproduction
- Generational as described before (insects)
- Generational with elitism fixed number of most

fit individuals are copied unmodified into new

generation - Steady state two parents are selected to

reproduce and two parents are selected to die

two offspring are immediately inserted in the

pool (mammals)

Other parameters of GA 2

- Stop criterion
- Number of new chromosomes
- Number of new and unique chromosomes
- Number of generations
- Measure
- Best of population
- Average of population
- Duplicates
- Accept all duplicates
- Avoid too many duplicates, because that

degenerates the population (inteelt) - No duplicates at all

Example run

- Maxima and Averages of steady state and

generational replacement

Introduction to Evolutionary ComputationReading

List

- Bäck, T., Hammel, U., and Schwefel, H.-P. (1997).

Evolutionary Computation Comments on the

History and Current State, IEEE Transactions on

Evolutionary Computation, VOL. 1, NO. 1, April

1997. - Spears, W. M., De Jong, K. A., Bäck, T., Fogel,

D. B., and de Garis, H. (1993). An Overview of

Evolutionary Computation, The Proceedings of the

European Conference on Machine Learning, v667,

pp. 442-459. (http//www.cs.uwyo.edu/wspears/pape

rs/ecml93.pdf) - De Jong, Kenneth A., and William M. Spears

(1993). On the State of Evolutionary

Computation, The Proceedings of the Int'l

Conference on Genetic Algorithms, pp. 618-623.

(http//www.cs.uwyo.edu/wspears/papers/icga93.pdf

)

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