evolution strategy
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Evolution StrategyHow Nature Solves Problems
Ingo Rechenberg
Shanghai Institute for Advanced Studies
CAS-MPG Partner Institute for Computational Biology / 2006-04-11
80400
2
4
6
0120 160 200 240 280 320
M u ta tio n s
R e s u lt
Re
sis
tan
ce
The experimentum crucis – Drag minimization of the kink plate
Start Result
Six manually adjustable shafts determine the form of a 90°pipe bend
Evolution of a 90° pipe bend
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0
45
Evolution of a two phase flow nozzle
(Hans-Paul Schwefel)
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
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
Min)()(
)()()(
)()()(
2753
2951
2963
2852
2741
2987
2654
2321
1515
151515
151515
nnnnnn
nnnnnnnnn
nnnnnnnnnQ
Objective function for a 3 3-
square ?
nn
1
4
7
2
5
8
3
6
9
nnn
nnn
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
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
Causality
Weak Causality
Strong Causality
A predictable world order is
Equal cause, equal effect
Similar cause, not similar effect
Similar cause, similar effect !
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
2. Global stochastic search
To 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
Z
x
y
Linearity radius
Progress
3. Local deterministic search
Walking following the steepest ascent
3)2(
grad
1)(
grad n
n
Z
x
y
Linearity radius
4. Local stochastic search
Random 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
21 r
n
ns
)()(
21
21
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
21)(
evonn )(
grad
1/ n
Evolution Strategy
1/n
Gradient Strategy
)0(!2
1)0(!1
1)0()0(1 1
2
1
ji
n
i
n
j jii
n
i ixx
xxfx
xfff
x
TAYLOR series expansionin n dimensions (MACLAURIN series)
1 11
0 ji
n
i
n
j
jii
n
i
i xxbxaQQ
Transformation to the principle axes
2
11
0 k
n
i
kk
n
k
k xdxcQQ
nc
d
nc
n
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
For n >> 1 the white catchment areas of the
hills are neglectible small compared with the
vaste black space between them
Parent
Mutation
Duplicator
DNA
Has made the duplicator
Heredity of the mutability
Crucial point of the Evolution Strategy
( ) - 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
+,[ ]
Adaptation
of the objektive variables xk
Adaptation
of 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
4
2,
max,
c times faster !
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