managing director / cto nutech solutions gmbh / inc. martin-schmeißer-weg 15 d – 44227 dortmund

Managing Director / CTONuTech Solutions GmbH / Inc.

Martin-Schmeißer-Weg 15D – 44227 Dortmund

[email protected]

Tel.: +49 (0) 231 / 72 54 63-10Fax: +49 (0) 231 / 72 54 63-29

Thomas Bäck

October 11, 2004

Evolution Strategies: A Different Type of EC and its Applications

Natural ComputingLeiden Institute for Advanced Computer Science (LIACS)Niels Bohrweg 1NL-2333 CA Leiden

[email protected]

Tel.: +31 (0) 71 527 7108Fax: +31 (0) 71 527 6985

2

Algorithms TheoryExamplesOverview Other

Background I

Daniel Dannett: Biology = Engineering

3


Background II

Realistic Scenario ....

4


Background III

Phenotypic evolution ... and genotypic

5


Optimization

max)(,: xfMf

E.g. costs (min), quality (max), error (min), stability (max), profit (max), ...

Difficulties:

• High-dimensional

• Nonlinear, non-quadratic: Multimodal

• Noisy, dynamic, discontinuous

• Evolutionary landscapes are like that !

6


Overview Evolutionary Algorithm Applications: Examples

Evolutionary Algorithms: Some Algorithmic Details Genetic Algorithms

Evolution Strategies

Some Theory of EAs Convergence Velocity Issues

Other Examples Drug Design

Inverse Design of Cas

Summary

7


Modeling – Simulation - Optimization

!!!

???

!!!

Simulation

Modeling / Data Mining

Optimization

!!! !!!

!!! ???

??? !!!

8


General Aspects

Evaluation

EA-Optimizer

Business Process Model

Simulation

215

1

i

iii

i scale

desiredcalculatedweightquality

Function Model from Data

Experiment SubjectiveFunction(s)

...)( yfi

9


Examples I: Inflatable Knee Bolster Optimization

Support plateFEM #4

Initial position of knee bag model deployed knee bag (unit only)

Volume of 14L

Load distribution plate

Tether

Support plate

Vent hole

MAZDA Picture

Load distribution plate FEM #3

Tether FEM #5

Knee bagFEM #2

Straps are defined in knee bag(FEM #2)

Low Cost ES: 0.677GA (Ford): 0.72Hooke Jeeves DoE: 0.88

Low Cost ES: 0.677GA (Ford): 0.72Hooke Jeeves DoE: 0.88

10


# Variables Characteristics HIC CG Left foot load Right foot load P Combined

4 Unconstrained 576,324 44,880 4935 3504 12,3935 Unconstrained 384,389 41,460 4707 4704 8,7589 Unconstrained 292,354 38,298 5573 5498 6,951

10 Constrained 305,900 39,042 6815 6850 7,289

IKB: Previous Designs

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Design Variable Description Base Design 1 Base Design 2 GA (Yan Fu)dx IKB center offset x 0 0 0,01dz IKB center offset y 0 0 -0,01rcdex KB venting area ratio 1 1 2massrat KB mass inflow ratio 1 1 1,5rcdexd DB venting area ratio 1 1 2,5Dmassratf DB high output mass inflow ratio 1 1 1,1Dmassratl DB low output mass inflow ratio 1 1 1dbfire DB firing time 0 0 -0,003dstraprat DB strap length ratio 1 1 1,5emr Load of load limiter (N) 3000 3000 2000Performance Response DescriptionNCAP_HIC_50 HIC 590 555.711 305,9NCAP_CG_50 CG 47 47.133 39,04NCAP_FMLL_50 Left foot load 760 6079 6815NCAP_FMRL_50 Right foot load 900 5766 6850P combined (Quality) 13.693 13.276 7,289

Objective: Min Ptotal Subject to: Left Femur load <= 7000 Right Femur load <= 7000

IKB: Problem Statement

12


Quality: 8.888 Simulations: 160Quality: 8.888 Simulations: 160

0,08

0,085

0,09

0,095

0,1

0,105

1 12 23 34 45 56 67 78 89 100

111

122

133

144

155

Simulator Calls

P c

om

bin

ed

IKB Results I: Hooke-Jeeves

13


0,06

0,065

0,07

0,075

0,08

0,085

0,09

0,095

0,1

0,105

0,11

1 11 21 31 41 51 61 71 81 91 101

111

121

131

141

Simulator Calls

P c

om

bin

ed

Quality: 7.142 Simulations: 122Quality: 7.142 Simulations: 122

IKB Results II: (1+1)-ES

14


Optical Coatings: Design Optimization

Nonlinear mixed-integer problem, variable dimensionality.

Minimize deviation from desired reflection behaviour.

Excellent synthesis method; robust and reliable results.

15


Dielectric filter design.

n=40 layers assumed.

Layer thicknesses xi in [0.01, 10.0].

Quality function: Sum of quadratic penalty terms.

Penalty terms = 0 iff constraints satisfied.

min215

1

i

iii

i scale

desiredcalculatedweightquality

Client:

Corning, Inc., Corning, NY

Dielectric Filter Design Problem

16


Benchmark

Results: Overview of Runs

Factor 2 in quality.

Factor 10 in effort.

Reliable, repeatable results.

17


Grid evaluation for 2 variables.

Close to the optimum (from vector of quality 0.0199).

Global view (left), vs. Local view (right).

Problem Topology Analysis: An Attempt

18


18 Speed Variables (continuous) for Casting Schedule

TurbineBladeafter Casting

FE mesh of 1/3 geometry: 98.610 nodes, 357.300 tetrahedrons, 92.830 radiation surfaces

large problem:

run time varies: 16 h 30 min to 32 h (SGI, Origin, R12000, 400 MHz)

at each run: 38,3 GB of view factors (49 positions) are treated!

Examples II: Bridgman Casting Process

19


Quality Comparison of the Initial and Optimized Configurations

Initial (DoE)GCM(Commercial Gradient Based

Method)Evolution Strategy

Global Quality

Turbine Bladeafter Casting

Examples II: Bridgman Casting Process

20


Generates green times for next switching schedule.

Minimization of total delay / number of stops.

Better results (3 – 5%) / higher flexibility than with traditional controllers.

Dynamic optimization, depending on actual traffic (measured by control loops).

Client:Dutch Ministry of TrafficRotterdam, NL

Examples IV: Traffic Light Control

21


Minimization of passenger waiting times.

Better results (3 – 5%) / higher flexibility than with traditional controllers.

Dynamic optimization, depending on actual traffic.

Client: Fujitec Co. Ltd., Osaka, Japan

Examples V: Elevator Control

22


Minimization of defects in the produced parts.

Optimization on geometric parameters and forces.

Fast algorithm; finds very good results.

Client: AutoForm Engineering GmbH,Dortmund

Examples VI: Metal Stamping Process

23


Minimization of end-to-end-blockings under service constraints.

Optimization of routing tables for existing, hard-wired networks.

10%-1000% improvement.

Client: SIEMENS AG, München

Examples VII: Network Routing

24


Minimization of total costs.

Creates new fuel assembly reload patterns.

Clear improvements (1%-5%) of existing expert solutions.

Huge cost saving.

Client: SIEMENS AG, München

Examples VIII: Nuclear Reactor Refueling

25


Experimental design optimisation: Optimise efficieny.

Initial design

Final design: 32% improvement in efficieny.

... evolves...

Two-Phase Nozzle Design (Experimental)

26


Multipoint Airfoil Optimization (1)

High Lift!

Low Drag!

Start

CruiseClient:

22 design parameters.

27


Multipoint Airfoil Optimization (2)

Pareto set after 1000 Simulations Three compromise wing designs

Find pressure profiles that are a compromise between two given target pressure distributions under two given flow conditions!

28


Evolutionary Algorithms:

Some Algorithmic Details

29


Unifying Evolutionary Algorithm

t := 0;

initialize(P(t));

evaluate(P(t));

while not terminate do

P‘(t) := mating_selection(P(t));

P‘‘(t) := variation(P‘(t));

evaluate(P‘‘(t));

P(t+1) := environmental_selection(P‘‘(t) u Q);

t := t+1;

od

30


Evolutionary Algorithm Taxonomy

Evolution StrategiesEvolution Strategies

Genetic AlgorithmsGenetic Algorithms

Genetic ProgrammingGenetic ProgrammingEvolutionary ProgrammingEvolutionary Programming

Classifier SystemsClassifier Systems

Many mixed forms; agent-based systems, swarm systems, A-life systems, ...

31


Real-valued representation

Normally distributed mutations

Fixed recombination rate (= 1)

Deterministic selection

Creation of offspring surplus

Self-adaptation of strategy

parameters:

Variance(s), Covariances

Binary representation

Fixed mutation rate pm (= 1/n)

Fixed crossover rate pc

Probabilistic selection

Identical population size

No self-adaptation

Genetic AlgorithmGenetic Algorithm Evolution StrategiesEvolution Strategies

Genetic Algorithms vs. Evolution Strategies

32


Genetic Algorithms Often binary representation.

Mutation by bit inversion with probability pm.

Various types of crossover, with probability pc.

k-point crossover.

Uniform crossover.

Probabilistic selection operators.

Proportional selection.

Tournament selection.

Parent and offspring population size identical.

Constant strategy parameters.

33


Mutation

0 1 1 1 0 1 0 1 0 0 0 0 0 01

0 1 1 1 0 0 0 1 0 1 0 0 0 01

Mutation by bit inversion with probability pm.

pm identical for all bits.

pm small (e.g., pm = 1/l).

34


Crossover

Crossover applied with probability pc.

pc identical for all individuals.

k-point crossover: k points chosen randomly.

Example: 2-point crossover.

35


Selection Fitness proportional:

f fitness

population size

Tournament selection: Randomly select q << individuals.

Copy best of these q into next generation.

Repeat times.

q is the tournament size (often: q = 2).

1

)(

)(

jj

ii

af

afp

36


Evolution Strategies Real-valued representation.

Normally distributed mutations.

Various types of recombination.

Discrete (exchange of variables).

Intermediate (averaging).

Involving two or more parents.

Deterministic selection, offspring surplus .

Elitist: ()

Non-elitist: ()

Self-Adaptation of strategy parameters.

37


Mutation

-adaptation by means of

– 1/5-success rule.

– Self-adaptation.

)1,0(iii Nxx

Creation of a new solution:

Convergence speed:

Ca. 10 n down to 5 n is possible.

More complex / powerful strategies:

– Individual step sizes i.

– Covariances.

38


Self-Adaptation

Learning while searching: Intelligent Method.

Different algorithmic approaches, e.g:

• Pure self-adaptation:

• Mutational step size control MSC:

• Derandomized step size adaptation

• Covariance adaptation

))1,0()1,0(exp( iii NN )1,0(iiii Nxx

2/1)1,0( if , /

2/1)1,0( if ,

Uu

Uu

)1,0(iiii Nxx

39


Self-Adaptive Mutation

n = 2, n = 1, n = 0

n = 2, n = 2, n = 0

n = 2, n = 2, n = 1

40


Self-Adaptation: Motivation: General search algorithm

Geometric convergence: Arbitrarily slow, if s wrongly controlled !

No deterministic / adaptive scheme for arbitrary functions exists.

Self-adaptation: On-line evolution of strategy parameters.

Various schemes: Schwefel one , n , covariances; Rechenberg MSA.

Ostermeier, Hansen: Derandomized, Covariance Matrix Adaptation.

EP variants (meta EP, Rmeta EP).

Bäck: Application to p in GAs.

tttt vsxx 1

Step size

Direction

41


Self-Adaptation: Dynamic Sphere

Optimum :

Transition time proportionate to n.

Optimum learned by self-adaptation.

n

Rcopt ,

42


Selection()

()

43


Possible Selection Operators

(1+1)-strategy: one parent, one offspring.

(1,)-strategies: one parent, offspring.

• Example: (1,10)-strategy.

• Derandomized / self-adaptive / mutative step size control.

(,)-strategies: >1 parents,> offspring

• Example: (2,15)-strategy.

• Includes recombination.

• Can overcome local optima.

(+)-strategies: elitist strategies.

44


Advantages of Evolution Strategies Self-Adaptation of strategy parameters.

Direct, global optimizers !

Faster than GAs !

Extremely good in solution quality.

Very small number of function evaluations.

Dynamical optimization problems.

Design optimization problems.

Discrete or mixed-integer problems.

Experimental design optimisation.

Combination with Meta-Modeling techniques.

45


Some Theory of EAs

46


Robust vs. Fast:

Global convergence with probability one:

General, but for practical purposes useless.

Convergence velocity:

Local analysis only, specific functions only.

1))(Pr( *lim

tPxt

)))(())1((( maxmax tPftPfE

47


Convergence Velocity Analysis, ES:

A convex function („sphere model“).

Simplest case: ()-ES

Illustration: (1,4)-ES

)( 2:1

2),1( rRE

n

iixxf

1

2)(

48



Order statistics:

p(z) denotes the p.d.f. of Z

Idea: Best offspring has smallest r / largest z‘.

The following holds from geometric considerations:

One gets: :1

222: '2 RZRlr

::2:1 ... ZZZ

dzzpndzzpzR

dzzpnzR

nRZE

)()(2

)()2(

)'2(

:2

:

:2

2:),1(

49



Using

one finally gets

One gets:

))(()(: z

dz

dzp

2/~~~2

2,1),1(

2,1),1(

c

ncR

(dimensionless) ln2,1 c

50



Convergence velocity, ()-ES and ()-ES:

51


Convergence Velocity Analysis, GA:

(1+1)-GA, (1,)-GA, (1+)-GA.

For counting ones function:

Convergence velocity:

Mutation rate p, q = 1 – p, kmax = l – fa.

l

iiaaf

1

)(

jfljfl

ij

aifif

i

aa

a

a

k

k

a

a

a

a

qpj

flqp

i

fkp

kafamfkp

kpk

10

0)11(

)(

))())((Pr()(

)(max

52



Optimum mutation rate ?

Absorption times from transition matrix

in block form, using where

llafp

1

)1)((2

1*

QR

IP

0

Tj

iji ntE )(

1)()( QInN ij

53



p too large:

Exponential

p too small:

Almost constant.

Optimal: O(l ln l) .

p

54



(1,)-GA (kmin = -fa), (1+)-GA (kmin = 0) :

ikk

ikk

i

fl

kk

ppi

ka

''

1)1(

min,

55


Convergence Velocity Analysis, EA:

(1,)-GA, (1+)-GA: (1,)-ES, (1+)-ES:

Conclusion: Unifying, search-space independent theory !?

56


Current Drug Targets:

2%2%

11%

5% 7%

45%

28%

receptors enzymeshormones & factors DNAnuclear receptors ion channelsunknown

GPCR

http://www.gpcr.org/

57


Goals (in Cooperation with LACDR): CI Methods:

Automatic knowledge extraction from biological databases – fuzzy rules.

Automatic optimisation of structures – evolution strategies.

Exploration for Drug Discovery,

De novo Drug Design.

N N

NH

NH2

Charge distribution on VdW surface of CGS15943

“Fingerprint”

New derivative with goodreceptor affinity.

Initialisation

Final (optimized)

58


Evolutionary DNA-Computing (with IMB): DNA-Molecule = Solution candidate !

Potential Advantage: > 1012 candidate solutions in parallel.

Biological operators: Cutting, Splicing.

Ligating.

Amplification.

Mutation.

Current approaches very limited.

Our approach: Suitable NP-complete problem.

Modern technology.

Scalability (n > 30).

59


UP of CAs (= Inverse Design of CAs)

1D CAs: Earlier work by Mitchell et al., Koza, ...

Transition rule: Assigns each neighborhood configuration a new state.

One rule can be expressed by bits.

There are rules for a binary 1D CA.

1 0 0 0 0 1 1 0 1 0 1 0 1 0 0

Neighborhood(radius r = 2)

1,01,0: 12 r

122 r

1222r

60


UP of CAs (rule encoding)

Assume r=1: Rule length is 8 bits

Corresponding neighborhoods

1 0 0 0 0 1 1 0

000 001 010 011 100 101 110 111

61


Inverse Design of CAs: 1D

Time evolution diagram:

62



Majority problem:

Particle-based rules.

Fitness values:

0.76, 0.75, 0.76, 0.73

63



Don‘t care about initial state rules

Block expanding rules

Particle communication based rules

64


Inverse Design of CAs: 1D Majority Records

Gacs, Kurdyumov, Levin 1978 (hand-written): 81.6%

Davis 1995 (hand-written): 81.8%

Das 1995 (hand-written): 82.178%

David, Forrest, Koza 1996 (GP): 82.326%

65


Inverse Design of Cas: 2D

Generalization to 2D (nD) CAs ?

Von Neumann vs. Moore neighborhood (r = 1)

Generalization to r > 1 possible (straightforward)

Search space size for a GA: vs.

10

0

1

1 10

0

1

1

0

0

1

1

52 92

522922

66


Inverse Design of CAs

Learning an AND rule.

Input boxes are defined.

Some evolution plots:

67



Learning an XOR rule.

Input boxes are defined.

Some evolution plots:

68



Learning the majority task.

84/169 in a), 85/169 in b).

Fitness value: 0.715

69



Learning pattern compression tasks.

70


Evolution = Computation ?

Yes

Search & Optimization are fundamental problems / tasks in many applications

(learning, engineering, ...).

71


Summary Explicative models based on Fuzzy Rules. Descriptive models based on e.g. Kriging method. Few data points necessary, high modeling accuracy. Used in product design, quality control, management decision

support, prediction and optimization. Optimization based on Evolution Strategies

(and traditional methods). Few function evaluations necessary. Robust, widely usable, excellent solution

quality. Self-adaptivity (easy to use !). Patents: US no. 5,826,251; Germany no. 43 08 083,

44 16 465, 196 40 635.

72


Questions ?

Thank you very much for your time !

managing director / cto nutech solutions gmbh / inc. martin-schmeißer-weg 15 d – 44227 dortmund

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