Lecture 3

Evolutionary Algorithms

Invited Guest Professorship

Université Lois Pasteur, StrasbourgUniversité Lois Pasteur, Strasbourg

Prof. Dr. Gisbert Schneider

Goethe-University, Frankfurt

25 November 2008, (c) G. Schneider

Global optimum

Fitness

Adaptive Walk

in a “fitness landscape”

Start

Xn

Optimization Tasks

Local optimum

Fitness

X1X2

End

X1

X2

Xn

2

Fitn

ess

Types of fitness landscapesF

itn

ess

The Dream The Nightmare

3

Start

q C

BA

Local Neighborhood Search

Neighborhood N Local

optimumx

• fixed N

• variable (“adaptive”) N

4

Local (neighborhood) search

Any-ascent/Stochastichill-climbing

Steepest/First-ascenthill-climbing

Population-based

Local Neighborbood Strategies

Threshold-acceptingSimulatedannealing

Tabu search

Geneticalgorithms

Evolutionstrategies

Evolutionaryprogramming

5

Elements of artificial adaptive systems (Koza)

• Structures that undergo adaptation

• Initial structures (starting solutions)

• Fitness measure that evaluates the structures

• Operations to modify the structures• Operations to modify the structures

• State (memory) of the system at each stage

• Method for designating a result

• Method for terminating the process

• Parameters that control the proces

6

• Genetic algorithms

• Evolution strategies

• Particle Swarm Optimization (PSO)

Learning from Nature ...

• Particle Swarm Optimization (PSO)

• Ant Colony Optimization (ACO)

7

Which search strategy should be applied?

Global & local search techniques guided by

• Random parameter variation

• Stepwise systematic parameter variation

• Gradients in the fitness landscape ...

There is no one best method.There is no one best method.

Evolutionary algorithms are

• easily implemented

• robust

• able to cope with high-dimensional optimization tasks

8

Searching in real and artificial fitness landscapes

Quality

(experimental)

“Fitness”

(calculated)

Idealized path

Neural network model

of peptide antibody recognition

9

a) Griewangk Funktion b) Rastrigin Funktion c) De Jong‘s Sphere Funktion

globales Minimum globales Minimum

1cos4000

)(11

2

+

−= ∏∑

==

n

i

in

i

i

i

xxxf

∑=

=n

i

ixxf1

2)(

( )∑=

⋅⋅⋅−+⋅=n

i

ii xxnxf1

2)2cos(1010)( π

d) Schaffer F6 Funktion e) Rosenbrock Funktion

globales Minimum

( ) ( )21

1

22

1 1100)( ∑−

=+ −+−⋅=

n

i

iii xxxxf( )222

222

)(001.01

5.0sin5.0)(

yx

yxxf

+⋅+

−++=

10

Operators of Evolutionary Optimization

• Mutation

• Crossover

• Mutation

• CrossoverVariation

Adds new information to a population

Exploits information within a population• Crossover

• Selection

• Crossover

• Selection

Exploits information within a population

11

clogPcMWO+NH-donors Surf [Å2]

S

NH

O

O

5.19 100 110404 011 0105 100 1102 001 00159.9 010 011

Real Binary GrayDescriptors

Genetic Algorithms: Chromosome Representation

Surfpolar [Å2]

NH

O

59.9 010 011

chromosome

genes

100011100001010

110010110001011

12

Principle of Evolutionary Searching

Step 1: Random search

Step 2 and following:

“Local hill climbing”

Best value

Optimum

Generate a diverse set of parametervalues and hope for a hit

Generate localized distributionsof parameter valuesand improve steadily

Best value

à Adaptive neighborhood 13

(Rechenberg, 1973)The (µ,λλλλ) Evolution Strategy

14

Algorithm of the (1,λλλλ) Evolution Strategy

Initialize parent (ξξξξP,σσσσP,QP);

For each generation:

Generate λλλλ variants (ξξξξV,σσσσV,QV) around the parent:

Quality (“Fitness”)

Stepsize

Variables

σσσσV = abs(σσσσP + G);

ξξξξV = ξξξξP + σσσσV • G;

calculate fitness QV;

Select best variant by QV ;

Set (ξξξξP,σσσσP,QP) ≡≡≡≡ (ξξξξV,σσσσV,QV)best ;

Box-Muller: )2sin()ln(2),G( jiji π⋅−=

i,j: pseudo-random numbers in ]0,1]

in [-∞,∞]15

Evolutionary Optimization: Protein backbone folding

Variables: Coordinates of “Beads on a string”

Fitness function: Empirical potential

„Folding“ of the ribonucleasese A backbone

X-ray structure (1rtb)

16

Multiobjective Optimization (MO)

f(property) = w1 * property1 + w2 * property2 + …

Disadvantages:

One possibility: weighted Fitness Function

Disadvantages:

• The setting of the weights is non-intuitive

• Regions of the search space may be excluded due to

the chosen weight setting

• Objectives may be correlated

• Only a single solution is found17

MOGA – Multiobjective Genetic Algorithm

• Most optimization algorithms can only cope with a single

objective

• Evolutionary Algorithms use a population of solutions

MOGA (Fonseca and Fleming 1993):

• Maps out the hypersurface in the search space where all

solutions are tradeoffs between the different objectives

• Searches for a set of non-dominated solutions

• The ranking of solutions is based on dominance instead

of a fitness function (Pareto ranking)18

Multiobjective Optimization (MO)

• Many practical optimization applications are

characterized by more than one objective

• Optimal performance in one objective often correlates • Optimal performance in one objective often correlates

with low performance in another one

• Strong need for a compromise

• Search for solutions that offer acceptable performance

in all objectives

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MOGA – Multiobjective Genetic Algorithm

0

0

24

f2

number of times the

Potential solutions in a two-objective (f1 and f2) minimization problem:

0

0

0

0

1

f1

number of times thesolution is dominated

non-dominated solution(Pareto solution)

dominated solution

20

Particle Swarm Optimization (PSO)

• Biological motivated optimization paradigm

• Individuals move through a fitness landscape

• Particles can change their movement direction and velocities

• “Social” interactions between individuals of a population• “Social” interactions between individuals of a population

• Each particle knows about its own best position and all

particles know about the overall best position

21

Particle Swarm Optimization (PSO) – cont.

T. Krink, University of Aarhus

22

PSO Algorithm

begin

initialize particle Positions and Memories

evaluate Fitness

while (stop criterion ≠ true)while (stop criterion ≠ true)

compute new Particle Velocities

compute new Particle Positions

evaluate Fitness

update Memories

end

end

23

PSO: Update Rule 1 (Standard Type)

))((2))((2)(v)1(v 21ii txprtxprtt ibii −⋅⋅+−⋅⋅+=+

Individual

Memory

Social

Memory

v: Velocity vector of a particle

x: Position of a particle

i: Dimension

t: Epoch (“time”)

r1 and r2: pseudo-random numbers between 0 and 1

pi and pb: best Positions stored in the individual and social Memories

24

PSO: Update Rule 2 (Inertia Weight Type)

))(())(()(v)1(v 2211ii txprntxprntwt ibii −⋅⋅+−⋅⋅+⋅=+

Inertia

Weight

Individual

Memory

Social

Memory

EpochsMaxEpochs

wwww endstart

start ⋅−

−=

25

v: Velocity vector of a particle

x: Position of a particle

w: inertia weight

i: Dimension

t: Epoch (“time”)

n1 and n2: constants for the individual and social terms

r1 and r2: pseudo-random numbers between 0 and 1

pi and pb: best Positions stored in the individual and social Memories

global

minimum

Convergence of PSO for the minimization of theGriewangk-Function after 100 epochs

PSO: Individual and Social Memory

n1 = 1n2 = 2

n1 = 2n2 = 1

Higher weight on the social memory

26

PSO: Update Rule 3 (Constriction Type)

)))(())(()(v()1(v2221ii

txprntxprntKt ibii −⋅⋅+−⋅⋅+⋅=+

Constriction

ConstantIndividual

Memory

Social

Memory

|42|

2

2 ϕϕϕ −−−=K

4,21

>+= ϕϕ nn

27

v: Velocity vector of a particle

x: Position of a particle

K: constriction constant

i: Dimension

t: Epoch (“time”)

n1 and n2: constants for the individual and social terms

r1 and r2: pseudo-random numbers between 0 and 1

pi and pb: best Positions stored in the individual and social Memories

The Constriction constant regulates swarm behavior in two ways (Kennedy & Eberhart, 2001):

• It leads to swarm convergence after a certain period of time.

• If the swarm has found a local optimum, the value of K determines its ability to escape the local optimum.

Rosenbrock function

Global minimum

Global minimum

Constriction-Type

Rosenbrock function

(after 100 epochs)

Standard-Type

28

Particle Trajectories

Global minimum

• The particle visited several local optima bevor the global optimum was found

29

Exercise

1) Go to the modlab website: www.modlab.de à software à PSOViz

• make yourself familiar with the PSOViz software options

2) Start PsoViz from your local computer (or directly from the website)

Choose a fitness function and change the various PSO parameter settings.

• Do you see an influence on the convergence behavior?

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• Do you see an influence on the convergence behavior?

• How do you explain the differences?

• Do you observe „typical problems“ with PSO?