Algorytmy ewolucyjne

Evolutionary Algorithms 

evolutionary operators 

Piotr Lipiński 

lipinski@ii.uni.wroc.pl 

draft 

Piotr Lipiński – Evolutionary Algorithms – p. 1/24

Examples of EAs 

SSGA 

SGA 

CGA 

PBIL 

ECGA 

GAs for graphs and objects partitioning 

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General schema of the EA 

EVOLUTIONARY-ALGORITHM(F,N) 

1 P ← RANDOM-POPULATION(N); 

2 POPULATION-EVALUATION(P,F); 

3 while not TERMINATION-CONDITION(P) 

4 do 

5 EVOLUTIONARY-OPERATOR-1(P); 



8 ... 

9 EVOLUTIONARY-OPERATOR-K(P); 


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Benchmark functions 

OneMax – the number of ones in the binary vector 

Pattern – the number of places, where the binary vector is identical 

with a given pattern 

DeceptiveOneMax – the number of ones in the binary vector, 

except the binary vector of zeros, where the value is the length of the 

vector plus one 

K-DeceptiveOneMax – the sum of DeceptiveOneMax over 

successive blocks ofK coordinates of the binary vector 

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SSGA 

SIMPLIFIED-SIMPLE-GENETIC-ALGORITHM(F,N,M) 




4 do 

5 BLOCK-SELECTION(P,M); 

6 UNIFORM-CROSSOVER(P); 


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SGA 

SIMPLE-GENETIC-ALGORITHM(F,N,M,θ C ,θ M ) 




4 do 

5 P (P) ← PARENT-SELECTION(P,M); 

6 P (C) ← CROSSOVER(P (P) ,θ C ); 

7 MUTATION(P (C) ,θ M ); 

8 REPLACEMENT(P,P (C) ); 


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Selection and Reproduction 

selection is sometimes not considered as an evolutionary 

operator, although it has a strong influence on the algorithm 

selection may be applied BEFORE as well as AFTER the other 

evolutionary operators 

when selection is applied BEFORE the other evolutionary 

operators, the process of creating the new population is 

referred to as REPRODUCTION 

generational EA vs. steady-state EA 

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example of a steady-state EA: 

- select two parent individuals, 

- produce two offspring individuals, 

- replace two worst individuals in the population with the two 

offspring individuals, 

- repeat the above process until the termination criteria, 

REMARK: only two individuals concerned in each iteration, 

not the entire population, 

PROBLEM: how to compare a steady-state EA with a 

generational EA (in terms of the computational complexity or 

time) ? 

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some individuals may be sometimes transferred to the next 

population without applying any evolutionary operators, 

generation gap is the fraction of the population replaced in 

each iteration (e.g. 1.0 when the entire population is replaced), 

the best individuals may be sometimes automatically included 

in the next population (so-called elitism), 

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Fitness Function 

LetP = {x 1 ,x 2 ,...,x N } be a population. 

One may assign to each individual x i ,i = 1,2,...,N a value 

f(x i ) = 

F(x i )−F min 

∑ N 

j=1 (F(x j)−F min ) , 

whereF min = min{F(x 1 ),F(x 2 ),...,F(x N )}, called the 

fitness value of the individual x i in the population P. 

The function f is called the fitness function. 

It is easy to see that 

0 ≤ f(x i ) ≤ 1 oraz 

N∑ 

f(x j ) = 1. 

draft 

j=1 


Selection According to the Fitness 

Function 

sometimes referred to as the roulette wheel method 

for each individual, the fitness value is evaluated and it defines 

the probability of selection for this individual 

PROBLEM: the domination of some super-individuals in the 

first iterations and a weak convergence in the next iterations. 

in order to avoid such a problem, fitness function scaling may 

be introduced 

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Fitness Function Scaling 

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Simple scaling The fitness value of the i-th individual is given 

by 

f scaled (t) = f original (t)−f worst (t), 

where t is the number of iteration and f worst (t) is the worst 

objective value found so far 

Sigma scaling The fitness value of the i-th individual is given 

by 

f scaled (t) = f original (t)−( ¯ f(t)−cσ f (t)), 

if it is a positive value or 0 otherwise, wherecis a constant 

(e.g. 2), f(t) ¯ is the average fitness in the current population and 

σ f (t) is the standard deviation. 


Fitness Function Scaling 

Power scaling The fitness value of the i-th individual is given 

by 

f scaled (t) = (f original (t)) k , 

for k > 0. 

Exponential scaling The fitness value of the i-th individual is 

given by 

f scaled (t) = exp(f original (t)/T), 

where T > 0, called the temperature, is descending to 0 in 

successive iterations. 

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Ranking methods 

Linear ranking Assuming that the population size is µ, the 

worst individual has the rang0, the best one has the rangµ−1. 

The probability of selection of the i-th individual is: 

P linear (i) = 

α+(rank(i)/(µ−1))(β −α) 

, 

µ 

where α and β are parameters, denoting the expected number 

of offspring individual produced by the worst and the best 

individual, respectively. It is easy to see that 

∑ µ−1 

i=0 P linear(i) = 1, so α+β = 2, hence1 ≤ β ≤ 2 and 

α = 2−β. 

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Ranking methods 

Power ranking C is a normalization coefficient,0 < α < β, 

P power (i) = α+(rank(i)/(µ−1))k (β −α) 

. 

C 

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Recombination for continuous representations 

no recombination – copying the entire chromosome from one 

selected parent 

Uniform Crossover 

x i = b ki 

for a randomly chosen k i ∈ {1,2,...,ρ}, u ∈ (0,1) 

Discrete Recombination 

classic crossover (like in SGA) for vectors of numbers 

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Recombination for continuous representations 

Global Intermediary Recombination each offspring gene is the 

average of all the parent genes 

ρ∑ 

x i = 1 ρ 

b ki 

k=1 

Local Intermediary Recombination each offspring gene is the 

average of two randomly chosen parents 

x i = u i b k1 i +(1−u i )b k2 i 

where k 1 ,k 2 ∈ {1,2,...,ρ}, u ∈ (0,1) are randomly chosen, 

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Recombination for discreet representation 

Multi-Point Recombination 

Global Discrete Recombination similar to uniform crossover 

for binary representation 

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Intermediary Recombination 

with two parents x 1 , x 2 , 

x ′ i = αx 1i +(1−α)x 2i , α ∈ [0,1] 

with many parents x 1 , x 2 ,...,x n , 

where ∑ i α i = 1. 

x ′ i = α 1 x 1i +α 2 x 2i +...+α n x ni 

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Other recombinations 

Heuristic recombination Letx 2 be not worse than x 1 

x ′ = u(x 2 −x 1 )+x 2 

where u ∈ [0,1] is a uniform distributed random number. 

Simplex Recombination First, randomly select a group of at 

least 2 parents. Letx 1 be the best and x 2 be the worst parent. 

Next, evaluate the centercof the group of parents and 

x ′ = c+(c−x 2 ) 

Geometric Recombination (it may be generalized to many 

parents also) 

draft 

x ′ = ( √ x 11 x 21 , √ x 12 x 22 ,...) 


Quadratic Recombination 

Letx ij be the j-th gene of the chromosome x i , for 

i = 1,2,3,j = 1,2,...,n where n is the chromosome length and 

x 4,j = 1 2 · (x2 2j −x 2 3j)f(x 1 )+(x 2 3j −x 2 1j)f(x 2 )+(x 2 1j −x 2 2j)f(x 3 ) 

(x 2j −x 3j )f(x 1 )+(x 3j −x 1j )f(x 2 )+(x 1j −x 2j )f(x 3 ) 

is the offspring individual produced by the 3 parent individuals x 1 , 

x 2 ,x 3 . 

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Hybrid EA with Local Search 

Generate the initial population of µ individuals at random. 

Perform local search for each individual. 

REPEAT 

Generate three points P 1 ,P 2 ,P 3 using Global Discrete 

Recombination 

Use Quadratic Recombination with P 1 ,P 2 ,P 3 to produce P 4 

Perform local search forP 4 

Add P 1 ,P 2 ,P 3 ,P 4 to the current population and delete 4 

selected individuals from the current population (e.g. by 

deterministic selection). 

draft 

UNTIL termination condition 


Local Search with Random Memorising 

Save the best solution in the memory. 

Take a randomly chosen saved solution (x old ) where a new 

solution x new outperforming the current population is found. 

Perform a search in the direction old → new, i.e. in the 

direction 

x new −x old 

||x new −x old || 

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Experiments 

18 benchmark functions 

Population of 30 individuals 

50 trials for each benchmark 

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