Evolutionary Computational
Intelligence
Lecture 4:
Real valued GAs and
ES
Ferrante Neri
University of Jyväskylä
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Real valued GAs and ES
Real valued problems
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Many problems occur as real valued problems, e.g.
continuous parameter optimisation f :  n  
Illustration: Ackley’s function (often used in EC)
Real valued GAs and ES
Mapping real values on bit strings
z  [x,y]   represented by {a1,…,aL}  {0,1}L
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[x,y]  {0,1}L must be invertible (one phenotype per
genotype)
: {0,1}L  [x,y] defines the representation
 ( a 1 ,..., a L )  x 
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y x
2 1
L
L 1
 ( a L j  2 )  [ x, y ]
j
j0
Only 2L values out of infinite are represented
L determines possible maximum precision of solution
High precision  long chromosomes (slow
evolution)
Real valued GAs and ES
Floating point mutations 1
General scheme of floating point mutations
x  x1 , ..., x l  x   x1 , ..., x l
x i , x i  LB i , UB i 
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Uniform mutation:
x i drawn randomly
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(uniform)
from
LB i , UB i 
Analogous to bit-flipping (binary) or random resetting
(integers)
Real valued GAs and ES
Floating point mutations 2
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Non-uniform mutations:
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Many methods proposed,such as time-varying
range of change etc.
Most schemes are probabilistic but usually only
make a small change to value
Most common method is to add random deviate
to each variable separately, taken from N(0, )
Gaussian distribution and then curtail to range
Standard deviation  controls amount of change
(2/3 of deviations will lie in range (-  to + )
Real valued GAs and ES
Crossover operators for real valued GAs
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Discrete:
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each allele value in offspring z comes from one of its parents
(x,y) with equal probability: zi = xi or yi
Could use n-point or uniform
Intermediate
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exploits idea of creating children “between” parents (hence
a.k.a. arithmetic recombination)
zi =  xi + (1 - ) yi
where  : 0    1.
The parameter  can be:
• constant: uniform arithmetical crossover
• variable (e.g. depend on the age of the population)
• picked at random every time
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Real valued GAs and ES
Single arithmetic crossover
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Parents: x1,…,xn  and y1,…,yn
Pick a single gene (k) at random,
child1 is:
x1 , ..., x k ,   y k  (1   )  x k , ..., x n
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reverse for other child. e.g. with  = 0.5
Real valued GAs and ES
Simple arithmetic crossover
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Parents: x1,…,xn  and y1,…,yn
Pick random gene (k) after this point mix values
child1 is:
x , ..., x ,   y
 (1   )  x
, ...,   y  (1   )  x
1
k
k 1
k 1
n
n
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reverse for other child. e.g. with  = 0.5
Real valued GAs and ES
Whole arithmetic crossover
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Parents: x1,…,xn  and y1,…,yn
child1 is:
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reverse for other child. e.g. with  = 0.5
a  x  (1  a )  y
Real valued GAs and ES
Box crossover
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Parents: x1,…,xn  and y1,…,yn
child1 is:
a  x  (1  a )  y
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Where α is a VECTOR of numers [0,1]
Real valued GAs and ES
Comparison of the crossovers
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Arithmetic crossover
works on a line which
connects the two parents
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Box crossover works in a
hyper-rectangular where
the two parents are
located in the vertexes
Real valued GAs and ES
Evolution Strategies
Ferrante Neri
University of Jyväskylä
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Real valued GAs and ES
ES quick overview
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Developed: Germany in the 1970’s
Early names: I. Rechenberg, H.-P. Schwefel
Typically applied to:
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Attributed features:
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fast
good optimizer for real-valued optimisation
relatively much theory
Special:
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numerical optimisation
self-adaptation of (mutation) parameters standard
Real valued GAs and ES
ES technical summary tableau
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Representation
Real-valued vectors
Recombination
Discrete or intermediary
Mutation
Gaussian perturbation
Parent selection
Uniform random
Survivor selection
(,) or (+)
Specialty
Self-adaptation of mutation
step sizes
Real valued GAs and ES
Representation
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Chromosomes consist of three parts:
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Object variables: x1,…,xn
Strategy parameters:
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Mutation step sizes: 1,…,n
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Rotation angles: 1,…, n
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Not every component is always present
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Full size:  x1,…,xn, 1,…,n ,1,…, k 
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where k = n(n-1)/2 (no. of i,j pairs)
Real valued GAs and ES
Parent selection
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Parents are selected by uniform random
distribution whenever an operator needs
one/some
Thus: ES parent selection is unbiased - every
individual has the same probability to be
selected
Note that in ES “parent” means a population
member (in GA’s: a population member
selected to undergo variation)
Real valued GAs and ES
Mutation
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Main mechanism: changing value by adding
random noise drawn from normal distribution
x’i = xi + N(0,)
Key idea:
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 is part of the chromosome  x1,…,xn,  
 is also mutated into ’ (see later how)
Thus: mutation step size  is coevolving with
the solution x
Real valued GAs and ES
Mutate  first
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Net mutation effect:  x,     x’, ’ 
Order is important:
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Rationale: new  x’ ,’  is evaluated twice
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first   ’ (see later how)
then x  x’ = x + N(0,’)
Primary: x’ is good if f(x’) is good
Secondary: ’ is good if the x’ it created is good
Reversing mutation order this would not work
Real valued GAs and ES
Mutation case 0: 1/5 success rule
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z values drawn from normal distribution N(,)
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 is varied on the fly by the “1/5 success rule”:
This rule resets  after every k iterations by
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mean  is set to 0
variation  is called mutation step size
 =  / c if ps > 1/5
 =  • c if ps < 1/5
=
if ps = 1/5
where ps is the % of successful mutations, 0.8  c <
1
Real valued GAs and ES
Mutation case 1:
Uncorrelated mutation with one 
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Chromosomes:  x1,…,xn,  
’ =  • exp( • N(0,1))
x’i = xi + ’ • N(0,1)
Typically the “learning rate”   1/ n½
And we have a boundary rule ’ < 0  ’ =
0
Real valued GAs and ES
Mutants with equal likelihood
Circle: mutants having the same chance to be created
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Real valued GAs and ES
Mutation case 2:
Uncorrelated mutation with n ’s
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Chromosomes:  x1,…,xn, 1,…, n 
’i = i • exp(’ • N(0,1) +  • Ni (0,1))
x’i = xi + ’i • Ni (0,1)
Two learning rate parameters:
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’ overall learning rate
 coordinate wise learning rate
  1/(2 n)½ and   1/(2 n½) ½
And i’ < 0  i’ = 0
Real valued GAs and ES
Mutants with equal likelihood
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Ellipse: mutants having the same chance to be created
Real valued GAs and ES
Mutation case 3:
Correlated mutations
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Chromosomes:  x1,…,xn, 1,…, n ,1,…, k
where k = n • (n-1)/2
and the covariance matrix C is defined as:
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cii = i2
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cij = 0 if i and j are not correlated
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cij = ½ • ( i2 - j2 ) • tan(2 ij) if i and j are
correlated
Note the numbering / indices of the ‘s
Real valued GAs and ES
Correlated mutations
The mutation mechanism is then:
 ’i = i • exp(’ • N(0,1) +  • Ni (0,1))
 ’j = j +  • N (0,1)
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x ’ = x + N(0,C’)
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  1/(2 n)½ and   1/(2 n½) ½ and   5°
i’ < 0  i’ = 0 and
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| ’j | >   ’j = ’j - 2  sign(’j)
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x stands for the vector  x1,…,xn 
C’ is the covariance matrix C after mutation of the  values
Real valued GAs and ES
Mutants with equal likelihood
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Ellipse: mutants having the same chance to be created
Real valued GAs and ES
Recombination
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Creates one child
Acts per variable / position by either
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From two or more parents by either:
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Averaging parental values, or
Selecting one of the parental values
Using two selected parents to make a child
Selecting two parents for each position anew
Real valued GAs and ES
Names of recombinations
zi = (xi + yi)/2
Two fixed
parents
Two parents
selected for each
i
Local
intermediary
Global
intermediary
Local
zi is xi or yi
chosen randomly discrete
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Global
discrete
Real valued GAs and ES
Survivor selection
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Applied after creating  children from the 
parents by mutation and recombination
Deterministically chops off the “bad stuff”
Basis of selection is either:
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The set of children only: (,)-selection
The set of parents and children: (+)-selection
Real valued GAs and ES
Survivor selection
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(+)-selection is an elitist strategy
(,)-selection can “forget”
Often (,)-selection is preferred for:
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Better in leaving local optima
Better in following moving optima
Using the + strategy bad  values can survive in
x, too long if their host x is very fit
Selection pressure in ES is very high
Real valued GAs and ES
Survivor Selection
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On the other hand, (,)-selection can lead
to the loss of genotypic information
The (+)-selection can be preferred when
are looking for “marginal” enhancements
Real valued GAs and ES