1. Evolutionary Computation (EC)
eie426-ec ppt
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2. Contents Basic Concepts of EC Genetic Algorithms An Example
Chromosome Representation
Stopping Criteria
Initial Population
Selection Mechanisms
Crossover and Mutation
Fitness Functions
Another Example
Application: Routing Optimization
Advantages and Disadvantages of EC
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3. Evolution and Search
Evolution - search through the enormous genetic parameter space for the best genetic make-up.
Borrow ideas from nature to help us solve problems that have equally large search spaces or similarly changing environment.
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4. Natural Evolution and Evolutionary Computation
Individual
Fitness
Environment
Evolutionary Computing
Candidate Solution
Quality
Problem
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5. Different ECs Several classes of EC algorithms have been developed:
- Genetic algorithms (GA’s): model genetic evolution
- Genetic programming: based on GA’s, but individuals are programs (represented as trees)
- Evolutionary programming: from the simulation of adaptive behavior in evolution (phenotype evolution)
- Evolution strategies: model the strategic parameters that control variation in evolution, i.e., the evolution of evolution
- Culture evolution: models the evolution of culture of a population and how the culture influences the evolution of individuals.
- Co-evolution: individuals evolve through cooperation, or in competition with one other.
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6. Basic Concepts Chromosome: individual Population: many individuals
Gene: each characteristics of chromosome (one parameter)
Allele: the value of a gene
Crossover: generate offspring by combining parts of the parents.
Mutation: introduce new genetic material into an existing individual.
Fitness: the survival strength of an individual
Culling (removing) and elitism (copying)
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7. Evolutionary Computation
Selection
Parents
Recombination
(Crossover)
Population
Mutation
The evolutionary cycle
Replacement
Offspring
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8. Genetic Algorithms
The GA was the first EC paradigm developed and applied (Holland 1975).
The features of the original GA’s:
A bit string representation
Proportional selection
Cross-over as the primary method to produce new individuals.
Several changes have been made:
Different representation schemes
Different selection methods
Different GA operators (cross-over, mutation and elitism)
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9. Random Search The GA is a search procedure.
Random search is possibly the simplest search procedure. Its training time may be very long before an acceptable solution is obtained.
Procedure:
Start from an initial search point or a set of initial points.
Random perturbations to the points
Repeat until an acceptable solution is reached or a maximum number of iterations is exceeded.
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10. General Genetic Algorithm
Let g = 0.
Initialize the initial generation Cg .
While no stopping criterion is satisfied
(a) Evaluate the fitness of each individual in Cg .
(b) g  g+1.
(c) Select parents from Cg-1.
(d) Recombine selected parents through cross-over to form offspring Og (with a probability pc).
(e) Mutate offspring in Og (with a probability pm).
(f) Select the new generation Cg from (the previous generation Cg-1, e.g., the best individuals are copied) and the offspring Og.
g: generation
Note: The things in () might or might not be carried out.
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11. An Example
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13. Use a genetic algorithm to solve the problem:
Coding (chromosome representation of a solution)
Generation of initial population (solutions)
Fitness calculation
Genetic operation
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14. Coding (chromosome representation):
Use a binary string to represent x. If the solution is to be precise to 10-6, then the interval (2-(-1)) = 3 should be divided into 3× 106. At least 22 bits should be used because
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15. Decoding
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17. Generation of initial population
A set of N 22-bit binary strings can be randomly generated as the initial population.
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18. Fitness calculation
Since f(x) > 0 in the interval, we can directly use f(x) as a fitness function:
f(s) = f(x)
e.g.,
s1 = < >, f(s1)=
s2 = < >, f(s2)=
s3 = < >, f(s3)=
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19. (1) Selection: based on the fitness of individuals
Genetic operation
(1) Selection: based on the fitness of individuals
e.g., roulette wheel selection (fitness proportionate selection)
(2) Crossover (with a probability pc)
e.g.,
s2 = <00000 | >, f(s2)=
s3 = <11100 | >, f(s3)=
After the crossover operation:
s’2 = <00000 | >, f(s’2)=
s’3 = <11100 | >, f(s’3)=
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20. (3) Mutation (with a probability pm) e.g.,
s3 = < >
f(s3)=
After the mutation operation:
s’3 = < >
f(s’3)= or
s3 = < >
s”3 = < >
f(s”3)=
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21. Simulation results:
N = 50, pc = 0.25, pm = 0.01, at 89 generations, the best individual was obtained:
smax = < >
xmax =
f(xmax) =
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22. The chromosome of the best individual
The best individual at each iteration (up to 150 generations)
Generation
The chromosome of the best individual x
fitness 1 11 17 54 71 89
150
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23. Summary on Basic Concepts
Evolution is an optimization process, where the aim is to improve the ability of individuals to survive.
An evolutionary algorithm (EA) is a stochastic search for an optimal solution to a given problem.
Evolution - search through the enormous genetic parameter space for the best genetic make-up.
Borrow ideas from nature to help us solve problems that have an equally large search spaces or similarly changing environment.
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24. Genotype: describes the genetic composition of an individual
Phenotype: the expressed behavioral traits of an individual in a specific environment.
Selection: use the fitness evaluations to decide which are the best parents to reproduce.
Crossover: generate offspring by combining parts of the parents
Mutation: introduce new genetic material into an existing individual.
Coding: phenotype  genotype
Decoding: genotype  phenotype
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25. Simple Genetic Algorithm (SGA)
Representation
Binary strings
Recombination
(crossover)
N-point (commonly used 1-point and 2-point) or uniform;
pc typically in range (0.6, 0.9)
Mutation
Bitwise bit-flipping with fixed probability pm (typically between 1/pop_size and 1/ chromosome_length)
Parent selection
Fitness-Proportionate
Survivor selection
All children replace parents
Speciality
Emphasis on crossover
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26. Chromosome Representation
Genotype space = {0,1}I
I-bit binary strings
Phenotype space
Coding (encoding or
(representation)
Decoding
(inverse representation)
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27. Binary-valued variables: no extra coding required
Nominal-valued variables
D-bit with 2D discrete nominal values
e.g., four colors: red (00), blue (01), green (10), yellow (11)
Continuous-valued variables
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28. Other Representations
Gray coding of integers (still binary chromosomes)
Gray coding is a mapping that means that small changes in the genotype cause small changes in the phenotype (unlike binary coding, e.g., 0111(7) and 1000 (8)).
It is generally accepted that it is better to encode numerical variables directly as
Integers
Floating point variables
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29. Stopping Criteria The maximum number of generation is exceeded
An acceptable best fit individual has evolved
The average and/or maximum fitness value do not change significantly over the past g generations.
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30. Initial Population
The standard way of generating the initial population is to choose gene values randomly from the allowed set of values.
The goal of random selection is to ensure that the initial population is a uniform representation of the entire search space.
A large population covers a larger area of the search space, and may require less generations to converge.
In the case of a small population, the EA can be forced to explore a large search space by increasing the rate of mutation.
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31. Selection Mechanisms Selection operators Random selection
Proportional selection: roulette wheel selection
Tournament selection
Rank-based selection
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32. Random Selection
Individuals are selected randomly with no reference to fitness at all. All the individuals, good or bad, have an equal chance of being selected.
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33. Proportional Selection: Roulette Wheel Selection
Individual
Chromosome
Fitness fi
Selection probability, Pi
Accumulated probability 1 8 2 5 3 4 10 7 6 12 19 9 14
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34. Roulette Wheel Selection
It can be visualized as the spinning of the wheel and testing which slide ends up at the top. Fitness values are usually normalized to [0,1].
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35. Assume that the following random number sequence is generated:
Compared the random number sequence with the accumulated probabilities, we select the individuals: 1, 8, 9, 6, 7, 5, 8, 4, 6, 10. Individuals 2 and 3 were removed and replaced with individuals 8 and 6. The individuals with high fitness tends to survive but those with low fitness may be removed.
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36. The pseudocode:
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37. Tournament Selection A group of k individuals is randomly selected.
The individual with the best fitness is selected from the group.
The advantage: the worse individuals of the population will not be selected and the best individuals will not dominate in the reproduction process.
For crossover, two tournaments are held to select each of two parents. It is possible that
(1) A parent can be selected to reproduce more than once; and
(2) One individual can combine with itself to reproduce offspring.
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38. Rank-Based Selection
The rank ordering of the fitness values is used to determine the probability of selection and not the fitness values itself.
Non-deterministic linear sampling
Individuals are sorted in decreasing fitness value.
(1) Let n = random(random(N))
where N is the total number of individuals and random(N) return a number between 1 and N.
(2) Return n as the selected individual
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39. Elitism
Elitism involves the selection of a set of individuals from the current generation to survive to the next generation.
The number of individuals to survive to the next generation, without being mutated, is referred to as the generation gap.
Generation gap = k
k best individuals or
k individuals selected using any selection operator
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40. Crossover
Crossover
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41. Uniform Crossover
A mask (vector) of length I (I-bit binary string) is created at random for each pair of individuals selected for reproduction. A bit with value of 1 indicates that the corresponding allele (bit) has to be swapped.
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42. Parent1
Parent 2
Mask
Offspring 1
Offspring 2 1
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43. One-point Crossover
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44. Parent1
Parent 2
Mask
Offspring 1
Offspring 2 1
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45. Two-point Crossover
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46. Parent1
Parent 2
Mask
Offspring 1
Offspring 2 1
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47. Arithmetic Crossover
Arithmetic crossover can be used in the case of continuous-valued genes.
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48. Mutation
Alter each gene independently with a probability pm (the mutation rate)
Real-valued representations, mutation occurs by adding a random value (usually sampled from a Gaussian distribution ) to allele. The variance is usually a function of the fitness of the individual. Individuals with a good fitness value will be mutated less, while a bad fitness value will lead to large mutations.
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49. 1 1
Mutation (fox)
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50. The Evolution Mechanism
Increasing diversity by using genetic operators
mutation
crossover
Decreasing diversity by selection of
parents
things to kill
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51. Fitness Functions Common fitness functions:
Use the objective function f(x) directly
(1) For a maximization problem, Fit(f(x)) = f(x)
e.g.,
(2) For a minimization problem, Fit(f(x)) = -f(x)
e.g., solution for x2 + x = 2, to minimize f(x) = x2 + x - 2
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52. Clipping
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53. Mapping
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54. Linear transformation of fitness functions
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55. Fig. ft1
Fig. ft2
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56. Genetic algorithms: case study
To find the maximum of the “peak” function of two variables x and y:
-3 ≤ x, y ≤ 3
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57. Chromosome representation
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58. Initial population
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59. The first generation
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60. Local maximum
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61. Global maximum
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62. Evolutionary Computation: Applications
Robotics
Control
Design
Scheduling/Routing/Resource Allocation
Machine Learning
Pattern Recognition
Market forecasting
Data Mining
Game Playing
- Robocode
- Backgammon
- Chess
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63. Robocode
An Open Source educational game started by Mathew Nelson (originally provided by IBM). Currently contributions are being made by various people; officially Flemming N. Larsen is working on Robocode to keep it current and fix the bugs. The game is designed to help people learn to program in Java and enjoy the experience.
Genetic Programming (GP) Robocode
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64. Application: Routing Optimization
The problem: Given a network of M switches, an origin and a destination switch, the objective is to find the best route to connect a call between the origin and destination switches (Sevenster and Engelbrecht 1996).
PSTN: Public Switch Telephone Network
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65. Chromosome representation: - variable length
- each gene representing one switch
- integer values for switch numbers
- the first gene and last gene representing the origin and last switches, respectively
Examples: ( )
( ) = ( )
Duplicate switches are ignored
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66. Initialization of population:
- randomly generated with the restriction that the first gene represents the origin switch and the last gene the destination switch
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67. Fitness function: a multi-criteria objective function was applied.
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68. Selection: any selection operator Crossover: any crossover operator
Mutation: replacing selected genes with a uniformly random selected switch in the range [1, M].
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69. Real World EC Tends include:
More complex representations and operators
Use of problem specific knowledge for seeding the initial population and creating heuristic operators
Hybridisation with other methods
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70. Advantages of EC Handles huge search spaces
Balances exploration and exploitation
Easy to try - not knowledge intensive
Easy to combine with other methods
Provides many alternative solutions
Can continually evolve solutions to fit with a continually changing problem
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71. Disadvantages of EC
No guarantee for optimal solution within finite time
Weak theoretical basis
May need extensive parameter tuning
Often computationally expensive, i.e., slow
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