Evolution strategies (ES)
Chapter 4
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Evolution strategies
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Overview of theoretical aspects
Algorithm
– The general scheme
– Representation and operators
Example
Properties
Applications
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
ES quick overview (I)
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Developed: Germany in the 1970’s
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Early names: Ingo Rechenberg, Hans-Paul Schwefel and and Peter
Bienert (1965), TU Berlin
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In the beginning, ESs were not devised to compute minima or
maxima of real-valued static functions with fixed numbers of variables
and without noise during their evaluation. Rather, they came to the
fore as a set of rules for the automatic design and analysis of
consecutive experiments with stepwise variable adjustments
driving a suitably flexible object / system into its optimal state in spite
of environmental noise.
Search strategy
– Concurrent, guided by absolute quality of individuals
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A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
ES quick overview (II)
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Typically applied to:
– application concerning shape optimization: a slender 3D body in
a wind tunnel flow into a shape with minimal drag per volume.
– numerical optimisation;
– continuous parameter optimisation
– computational fluid dynamics: the design of a 3D convergentdivergent hot water flashing nozzle.
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ESs are closer to Larmackian evolution (which states that
acquired characteristics can be passed on to offspring).
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The difference between GA and ES is the Representation and
Survival selection mechanism, that imply survival in the new
population of part from the old population
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
ES quick overview (III)
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Attributed features:
– fast
– good optimizer for real-valued optimisation (real-valued vectors
are used to represent individuals)
– relatively much theory
Strong emphasis on mutation for creating offspring
Mutation is implemented by adding some random noise drawn from
Gaussian distribution
Mutation parameters are changed during a run of the algorithm
In the ES the control parameter are included in the chromosomes
and co-evolve with the solutions.
Special:
– self-adaptation of (mutation) parameters standard
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
ES Algorithm - The general scheme
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An Example Evolution Strategy
Procedure ES{
t = 0;
Initialize P(t);
Evaluate P(t);
While (Not Done)
{
Parents(t) = Select_Parents(P(t));
Offspring(t) = Procreate(Parents(t));
Evaluate(Offspring(t));
P(t+1)= Select_Survivors(P(t),Offspring(t));
t = t + 1;
}
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The differences between GA and ES consists in representation
and survivors selection (in the new population will survive the best
of parents and offspring unlike generational genetic algorithms where
children replaced the parents).
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
ES technical summary tableau
Representation
Real-valued vectors
Encoding also the mutation rate
Recombination
Discrete or intermediary
Mutation
Gaussian perturbation
Parent selection
Uniform random
Survivor selection
(,) or (+)
Specialty
Self-adaptation of mutation
step sizes
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Evolution Strategies
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There are basically 4 types of ESs
– The Simple (1+1)-ES (In this strategy the aspect of collective
learning in a population is missing. The population is
composed of a single individual).
– The (+1)-ES (The first multimember ES.  parents give birth to 1
offspring)
For the next two ESs  parents give birth to  offspring
– The (+)-ES. P(t+1) = Best  of the + individuals
– The (,)-ES. P(t+1) = Best  of the  offspring.
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
(1+1) - Evolution Strategies (two
membered Evolution Strategy)
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Before the (1+1)-ES there were no more than two rules:
– 1. Change all variables at a time, mostly slightly and at random.
– 2. If the new set of variables does not diminish the goodness of
the device, keep it, otherwise return to the old status.
The Simple (1+1)-ES (In this strategy the aspect of collective learning
in a population is missing. The population is composed of a single
individual).
(1+1)-ES is a stochastic optimization method having similarities
with Simulated Annealing.
Represents a local search strategy that perform the current solution
exploitation.
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
(1+1) - Evolution Strategies features
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the convergence velocity, the expected distance traveled into the
useful direction per iteration, is inversely proportional to the number of
variables of the objective function;
linear convergence order can be achieved if the mutation strength
(or mean step-size or standard deviation of each component of the
normally distributed mutation vector) is adjusted to the proper order of
magnitude, permanently;
the optimal mutation strength corresponds to a certain success
probability that is independent of the dimension of the search
space and is the range of one fifth for both model functions (sphere
model and corridor model).
the convergence (velocity) rate of a ES (1 +1) is defined as the ratio
of the Euclidean Distance (ED) traveled towards the optimal point and
the number of generations required for running this distance.
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Introductory example
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n
Task: minimise f : R  R
Algorithm: “two-membered ES” using
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Vectors from Rn directly as chromosomes
Population size 1
Only mutation creating one child
Greedy selection
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Standard deviation. Normal distribution
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Consider X =  x1, x2, …,xn  n-dimensional random
variable.
The mean (μ) M(X)=(x1+ x2,+…+xn )/n.
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The square of standard deviation (also called variance):
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2 = M(X-M(X))2=(xk - M(X))2/n
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Normal distribution:
N(μ,) =
The distribution with μ = 0 and σ 2 = 1 is called the standard
normal.
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Illustration of normal distribution
http://fooplot.com/
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Introductory example: pseudocode
Minimization problem
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Set t = 0
Create initial point xt =  x1t,…,xnt 
REPEAT UNTIL (TERMIN.COND satisfied) DO
Draw zi from a normal distribution for all i = 1,…,n
yit = xit + zi or yit = xit + N(0, )
IF f(xt) < f(yt) THEN xt+1 = xt
ELSE xt+1 = yt
endIF
– Set t = t+1
endDO
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Introductory example: mutation
mechanism
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z values drawn from normal distribution N(μ,)
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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 μ is set to 0
Standard deviation  is called the mutation step size
 =  / c if Ps > 1/5
 =  • c if Ps < 1/5
=
if Ps = 1/5
(Foot of big hill  increase σ)
(Near the top of the hill  decrease σ)
where Ps is the % of successful mutations (those in
which the child is fitter than parents), 0.8  c  1,
usualy c=0.817
Mutation rule for object variables x (xit) is additive, while
the mutation rule for dispersion () is multiplicative.
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
The Rechenberg’s 1/5th - succes rule
• The 1/5th rule of success is a mechanism that ensures efficient
heuristic search with the price of decreased robustness.
• The ratio of successful mutations and other mutations must
be the fifth (1/5).
• IF this ratio is greater than 1/5 the dispersion must be
increased (accelerates convergence).
ELSE
• IF this ratio is less than 1/5 the dispersion must be
decreased.
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
The implementation of the Rechenberg’s
1/5th -rule
1. perform the (1 + 1)-ES for a number G of generations:
− keep σ constant during this period
− count the number Gs of successful mutations during this period
2. determine an estimate of the success probability Ps by
Ps := Gs/G
3. change σ according to
σ := σ / c, if Ps > 1/5
σ := σ · c, if Ps < 1/5
σ := σ,
if Ps = 1/5
4. goto 1.
The optimal value of the factor c depends on the objective function to be optimized,
the dimensionality N of the search space, and on the number G. If N is sufficiently
large N ≥ 30, G = N is a reasonable choice. Under this condition Schwefel (1975)
recommended using 0.85 ≤ c < 1.
Since we are not finding better solutions, we have reached the top of the hill. 
Rechenberg’s 1/5 rule reduces the standard deviation σ in the case that the system was
not very successful in finding better solutions.
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Another historical example:
the jet nozzle experiment
Task: to optimize the shape of a jet nozzle
Approach: random mutations to shape + selection
Initial shape
Final shape
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Another historical example:
the jet nozzle experiment cont’d
In order to be able to vary the length of the nozzle and the
position of its throat, gene duplication and gene deletion was
mimicked to evolve even the number of variables, i.e., the nozzle
diameters at fixed distances. The perhaps optimal, at least
unexpectedly good and so far best-known shape of the nozzle
was counter-intuitively strange, and it took a while, until the
one-component two-phase supersonic flow phenomena far from
thermodynamic equilibrium, involved in achieving such good
result, were understood.
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
The disadvantages of (1+1)-ES
• Fragile nature of the search point by point based on the 1/5
successful rule may lead to stagnation in a local minimum point.
• Dispersion (step size) is the same for each dimension
(coordinate) within search space.
• Does not use recombination; it is not using a real population
• There is no mechanism to allow individual adjustment of
stride for each coordinate axis of the search space. The lack of
such a mechanism is that the procedure will move slowly to the
optimum point.
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
(+), (,) - (multi membered
Evolution Strategies)
 parents give birth to  offspring
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
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)
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
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
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
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
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
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
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Mutants with equal likelihood
Circle: mutants having the same chance to be created
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
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 parmeters:
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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
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Mutants with equal likelihood
Ellipse: mutants having the same chance to be created
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
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
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Correlated mutations cont’d
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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x stands for the vector  x1,…,xn 
C’ is the covariance matrix C after mutation of the  values
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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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A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Mutants with equal likelihood
Ellipse: mutants having the same chance to be created
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Recombination
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Creates one child
Acts per variable / position by either
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Averaging parental values, or
Selecting one of the parental values
From two or more parents by either:
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Using two selected parents to make a child
Selecting two parents for each position anew
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Names of recombinations
Two parents
Two fixed parents
selected for each i
zi = (xi + yi)/2
Local
intermediary
Global
intermediary
zi is xi or yi
chosen randomly
Local
discrete
Global
discrete
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
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)
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
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
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Survivor selection cont’d
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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
Selective pressure in ES is very high (  7 •  is the
common setting)
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Self-adaptation illustrated
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Given a dynamically changing fitness
landscape (optimum location shifted every 200
generations)
Self-adaptive ES is able to
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follow the optimum and
adjust the mutation step size after every shift !
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Self-adaptation illustrated cont’d
Changes in the fitness values (left) and the mutation step sizes (right)
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Prerequisites for self-adaptation
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 > 1 to carry different strategies
 >  to generate offspring surplus
Not “too” strong selection, e.g.,   7 • 
(,)-selection to get rid of misadapted ‘s
Mixing strategy parameters by (intermediary)
recombination on them
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
ES Applications:
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Lens shape optimization required to Light
refraction
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Distribution of fluid in a blood network
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Brachystochrone curve
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Solving the Rubik's Cube
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Example application:
the Ackley function (Bäck et al ’93)
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The Ackley function (here used with n =30):
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1 n 2
f ( x )  20  exp   0.2
  xi
n i 1

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
1 n

  exp   cos( 2xi )   20  e

 n i 1
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Evolution strategy:
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Representation:
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-30 < xi < 30 (coincidence of 30’s!)
30 step sizes
(30,200) selection
Termination : after 200000 fitness evaluations
Results: average best solution is 7.48 • 10 –8 (very good)