Evolution StrategyHow Nature Solves Problems

Ingo Rechenberg

Shanghai Institute for Advanced Studies

CAS-MPG Partner Institute for Computational Biology / 2006-04-11

1 What Evolution Strategy does2 How Evolution Strategy works

1Protoplasm lump in the primordial ocean

What Evolution does

2From this the fish

developed

What Evolution does

3Life peeks out of the water and spreads over the country

What Evolution does

4Our ancestors climb

the treetops

What Evolution does

5Finally we admire

ourselves in the mirror

What Evolution does

History of the Evolution Strategy

Windtunnel

Flexible flow body

to adjust random mutations

Air flow

Gear

DARWIN in the windtunnel

The kink plate for the key experiment with the Evolution Strategy

Number of possible adjustments

515 = 345 025 251

80400

2

4

6

0120 160 200 240 280 320

M u ta t io n s

R e s u lt

Re

sis

tan

ce

The experimentum crucis – Drag minimization of the kink plate

Zigzag after DARWIN

Story in the Magazin

18 th November 1964

Start Result

Six manually adjustable shafts determine the form of a 90°pipe bend

Evolution of a 90° pipe bend

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

35

31

32

33

34

36

37

38

39

40

41

42

43

44

0

45

Evolution of a two phase flow nozzle(Hans-Paul Schwefel)

HistoryEvolution Strategy today

Evolution-Strategy

ES-]),/(,/[

Wright Haldane Fisher ' = Number of offspring populations'= Number of population generations

' = Number of parental populations

= Number of parental individuals

= Number of offspring individuals = Generations of isolation

' = Mixing number for populations

= Mixing number for individuals

Elementary Evolution-Strategic Algorithms

(1 + 1)-ES

DARWINs theory at thelevel of maximum abstraction

(1 , )-ES

Evolution Strategywith more than one offspring

= 6

( , )-ES

Evolution Strategy with more parents and more offspring

= 7 = 2

( , )-ES

Evolution Strategy with mixing of variables

= 8

= 2 = 2

ES]),(,[

12154

New founder populations

The NestedEvolution Strategy

will be an algebraic scheme

The notation

An artificial evolution experiment in the windtunnel

Evolution of a spread wing in the windtunnel

Multiwinglets at a glider designed with the Evolution Strategy

Phot

o: M

icha

el S

tach

e

Darwin was very uncertain whether his theory is correct.

To suppose that the eye, with all its inimitable contrivances for adjusting the focus to different distances, for admitting different amounts of light, and for the correction of spherical and chromatic abberation, could have been formed by natural selection, seems, I freely confess, absurd in the highest possible degree.

He stated in his book „The Origin of Species“:

Fdk

qk

Evolution of an eye lens

Computer simulated evolution of a covergent lens

Flexible glass body

Minimum2kq

Evolution-strategic development of a framework construction

y

x

Weight Minimum

y

x

Weight Minimum

y

x

Weight Minimum

y

x

Weight Minimum

Evolution-strategic optimization of a truss bridge with minimum weight

Arched bridge

Fishbelly bridge

Bridge

designs

Lu Pu Bridge

Dynamic optimization of a truss bridge

Melencolia, engraved in 1514 by Albrecht Dürer

Magic SquareChinese

2 0 0 6

Min)()()()()()()()(

2753

2951

2963

2852

2741

2987

2654

2321

1515151515151515

nnnnnnnnnnnnnnnnnnnnnnnnQ

Objective function for a 3 3-square ?

nn

14

7

2

5

8

3

6

9

nnn

nnn

y

x

The min/max distance problem

DD

min

maxMinimum

ES-Solutions of the min/max-

distance problem

7 Points 12 Points

24 Points 27 Points

9093,2325minmax / DD2minmax / DD

5826,421minmax / DD 8045,4minmax / DD

Maximum distance = 1

Minimum distance

Optimal swarm configuration of 48 individuals

Dmax

Dmin= 6.707

94

94

86

86

103

10377

77

Elements of the optimal structure

Structure of the 48 individual swarm

2 How Evolution Strategy works

Search for a document

(Search)Strategies are of no use in an disordered world

(Search)Strategies need a predictable order of the world

Strategy in military operation

A military strategy is of no use, if the enemy behaves randomly

General

An evolution strategy is of no use, if nature (opponent) behaves randomly

Evolution Strategist

Causality

Weak Causality

Strong Causality

A predictable world order is

Equal cause, equal effect

Similar cause, not similar effect

Similar cause, similar effect !

Billiards-Effect

Example for

weak causality

Strong Causality

Normal behaviour of the world

Weak and strong causality in a graphic view

Weak causality

Strong causality

Experimenter

Plumbing the depth

Search area

The search for the optimum

The search for the optimum

Plumbing the depth

Experimenter

Search area

1. Global deterministic search

3. Local deterministic search

2. Global stochastic search

4. Local stochastic search

4 strategies to localize an optimum

Z

1 m

m

1

1. Global deterministic searchSystematic scanning of the variable space

2)2( mG

nn mG )(

Z

1 m

m

1

2. Global stochastic searchTo find the target with 95% probability

2)2( 99.2 mG

nn mG 99.2)(

1. Global deterministic search

3. Local deterministic search

2. Global stochastic search

4. Local stochastic search

4 strategies to localize an optimum

distance moved uphillnumber of generations

Definition of the rate of progress

Z

x

y

Linearity radius

Progress

3. Local deterministic searchWalking following the steepest ascent

3)2(

grad

1)(

grad nn

Z

x

y

Linearity radius

4. Local stochastic searchRandom drifting along the steepest ascent

1. Offspring

2. OffspringParent

?)2(evo

?)(evon

Plus-offspring

Minus-offspring Center of gravity

Statistical mean of the progress

Determiation of the linear rate of

progress

ParentLinearity radius

2/s

s+

−

Because half of the offspring are failures

rr

rs rs 2

1 rn

ns )(

)(2

121

2 Dim. 3 Dim. n Dim.

s ss

Center of gravity

4

n1

2 n >> 1

rn

s 21

n >> 1

Gradient Strategy contra Evolution Strategy

For n >> 1

nn

2

1)(evon

n )(grad

1/ n

Evolution Strategy

1/n

Gradient Strategy

Local climbing of the Evolution Strategy

linear

Local climbing of the Evolution Strategy

nonlinear

)0(!2

1)0(!1

1)0()0(1 1

2

1ji

n

i

n

j jii

n

i ixxxx

fxxfff

x

TAYLOR series expansionin n dimensions (MACLAURIN series)

1 11

0 ji

n

i

n

jjii

n

ii xxbxaQQ

Transformation to the principle axes

2

110 k

n

ikk

n

kk xdxcQQ

nc

dn

cn

n

k

k 2

2,1

Tabel

1 0

2 0,5642

3 0,8463

4 1,0294

5 1,1630

6 1,2672

7 1,3522

8 1,4236

9 1,4850

10 1,5388

,1c

11 1,5864

12 1,6292

13 1,6680

14 1,7034

15 1,7359

16 1,7660

17 1,7939

18 1,8200

19 1,8445

20 1,8675

,1c

21 1,8892

22 1,9097

23 1,9292

24 1,9477

25 1,9653

26 1.9822

27 1,9983

28 2,0137

29 2,0285

30 2,0428

,1c

35 2,1066

40 2,1608

45 2,2077

50 2,2491

55 2,2860

60 2,3193

65 2,3496

70 2,3774

80 2,4268

90 2,4697

,1c

100 2,5076

200 2,7460

300 2,8778

400 2,9682

500 3,0367

600 3,0917

700 3,1375

800 3,1768

900 3,2111

1000 3,2414

,1c

of the progress coefficients

= zero

=

high

= medium

The complexity

nc

dn

cn

n

k

k 2

2,1

rn2

r

Evolution Window

-5 -3 -1 310

0,2

0,1

0,3

1 01 01 01 010

2,1

c

,1cn

2 Central law of progress

not so

but so

For n >> 1 the white catchment areas of the hills are neglectible small compared with the vaste black space between them

Parent

Evolution Window

-5 -3 -1 310

0,2

0,1

0,3

1 01 01 01 010

How to find the Evolution Window ?

Mutation

Duplicator

DNA

Has made the duplicator

Heredity of the mutabilityCrucial point of the Evolution Strategy

? ? ?

I am the fi rst

Assessment of the climbing style

Climbing alone Climbing in a group

N

Four mountaineers, four climbing styles

Fraidycat

Columbus

Amundsen

Hothead

In a compact notation

Nested Evolution Strategy

Four moutaineers, four climbing styles

On the way to an evolution-strategic algebra

1 +1( ) - ES ,+,

On the way to an evolution-strategic algebra

( ) - ES +,

On the way to an evolution-strategic algebra

/

Example = 2

( ) - ES +,/ 2

Only half of the parental information builds up an offspringMulti-Recombination

( ) - ES +,

On the way to an evolution-strategic algebra

Example:

(1+ 6)4 = (1+ 6) (1+ 6) (1+ 6) (1+ 6)

( ) - ES +,

On the way to an evolution-strategic algebra

+,[ ]

| Family Genus { Species [ Variety ( Individual ) ] } |

Biological equivalent to the strategy nesting

( ) - ES +,

Nested Evolution Strategy

+,[ ]

Adaptationof the objektive

variables xk

Adaptationof the

mutation size

to operate in the Evolution Window!

Reduction of the lateral component of the mutation step using intermediary variable mixing (multi-recombination)

Contour line

Parent

Best of offspring

Recombination of the best of offspring

Reduction of the lateral mutation step

Lateral component

Progress component

The wonder of sexual reproduction

nnc2

,,

nnc2

,,

with multi-recombination

without recombination

42,

max,c

times faster !

I thank you for your attention

www.bionik.tu-berlin.de