Managing Director / CTO NuTech Solutions GmbH / Inc. Martin-Schmeißer-Weg 15 D – 44227 Dortmund
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Transcript of 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
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
Tel.: +31 (0) 71 527 7108Fax: +31 (0) 71 527 6985
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Background I
Daniel Dannett: Biology = Engineering
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Background II
Realistic Scenario ....
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Background III
Phenotypic evolution ... and genotypic
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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 !
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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
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Modeling – Simulation - Optimization
!!!
???
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Simulation
Modeling / Data Mining
Optimization
!!! !!!
!!! ???
??? !!!
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General Aspects
Evaluation
EA-Optimizer
Business Process Model
Simulation
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1
i
iii
i scale
desiredcalculatedweightquality
Function Model from Data
Experiment SubjectiveFunction(s)
...)( yfi
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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
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# 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
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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
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122
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144
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Simulator Calls
P c
om
bin
ed
IKB Results I: Hooke-Jeeves
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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
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Simulator Calls
P c
om
bin
ed
Quality: 7.142 Simulations: 122Quality: 7.142 Simulations: 122
IKB Results II: (1+1)-ES
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Optical Coatings: Design Optimization
Nonlinear mixed-integer problem, variable dimensionality.
Minimize deviation from desired reflection behaviour.
Excellent synthesis method; robust and reliable results.
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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
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Benchmark
Results: Overview of Runs
Factor 2 in quality.
Factor 10 in effort.
Reliable, repeatable results.
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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
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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
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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
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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
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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
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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
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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
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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
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Experimental design optimisation: Optimise efficieny.
Initial design
Final design: 32% improvement in efficieny.
... evolves...
Two-Phase Nozzle Design (Experimental)
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Multipoint Airfoil Optimization (1)
High Lift!
Low Drag!
Start
CruiseClient:
22 design parameters.
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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!
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Evolutionary Algorithms:
Some Algorithmic Details
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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
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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, ...
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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
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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.
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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).
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Crossover
Crossover applied with probability pc.
pc identical for all individuals.
k-point crossover: k points chosen randomly.
Example: 2-point crossover.
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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
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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.
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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.
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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
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Self-Adaptive Mutation
n = 2, n = 1, n = 0
n = 2, n = 2, n = 0
n = 2, n = 2, n = 1
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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
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Self-Adaptation: Dynamic Sphere
Optimum :
Transition time proportionate to n.
Optimum learned by self-adaptation.
n
Rcopt ,
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Selection()
()
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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.
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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.
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Some Theory of EAs
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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
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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)(
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Convergence Velocity Analysis, ES:
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(
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Convergence Velocity Analysis, ES:
Using
one finally gets
One gets:
))(()(: z
dz
dzp
2/~~~2
2,1),1(
2,1),1(
c
ncR
(dimensionless) ln2,1 c
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Convergence Velocity Analysis, ES:
Convergence velocity, ()-ES and ()-ES:
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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
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Convergence Velocity Analysis, GA:
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
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Convergence Velocity Analysis, GA:
p too large:
Exponential
p too small:
Almost constant.
Optimal: O(l ln l) .
p
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Convergence Velocity Analysis, GA:
(1,)-GA (kmin = -fa), (1+)-GA (kmin = 0) :
ikk
ikk
i
fl
kk
ppi
ka
''
1)1(
min,
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Convergence Velocity Analysis, EA:
(1,)-GA, (1+)-GA: (1,)-ES, (1+)-ES:
Conclusion: Unifying, search-space independent theory !?
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Current Drug Targets:
2%2%
11%
5% 7%
45%
28%
receptors enzymeshormones & factors DNAnuclear receptors ion channelsunknown
GPCR
http://www.gpcr.org/
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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)
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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).
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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
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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
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Inverse Design of CAs: 1D
Time evolution diagram:
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Inverse Design of CAs: 1D
Majority problem:
Particle-based rules.
Fitness values:
0.76, 0.75, 0.76, 0.73
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Inverse Design of CAs: 1D
Don‘t care about initial state rules
Block expanding rules
Particle communication based rules
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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%
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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
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Inverse Design of CAs
Learning an AND rule.
Input boxes are defined.
Some evolution plots:
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Inverse Design of CAs
Learning an XOR rule.
Input boxes are defined.
Some evolution plots:
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Inverse Design of CAs
Learning the majority task.
84/169 in a), 85/169 in b).
Fitness value: 0.715
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Inverse Design of CAs
Learning pattern compression tasks.
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Evolution = Computation ?
Yes
Search & Optimization are fundamental problems / tasks in many applications
(learning, engineering, ...).
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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.
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Questions ?
Thank you very much for your time !