what are eas? mathematical formulation & computer...

EVOLUTIONARY ALGORITHMS: What are EAs?

Mathematical Formulation & Computer Implementation Multi-objective Optimization, Constraints Computing cost reduction

Kyriakos

C. GiannakoglouAssociate Professor,

Lab. of Thermal Turbomachines

(LTT),

National Technical University of Athens (NTUA),

GREECE

______________________________________________

Introductory Course to Design Optimization

K.C. Giannakoglou, Associate Professor NTUA, Greece

Outline

From traditional problem solving techniques to EAs.Generalized EA: Basic and advanced operators.Mathematical foundations of EAs.EAs for multi-objective optimization.Distributed Evolutionary Algorithms (DGAs). Hierarchical Evolutionary Algorithms (HGAs).Constraints’ handling.Efficient ways for reducing the computing cost of EAs.Applications in the field of aeronautics, turbomachinery, energy production, logistics.

______________________________________________



Effective/Efficient Problem Solving Techniques

The number of possible solutions in the search space is so large as to forbid exhaustive searchSeeking the best combination of approaches that addresses the purpose to be achievedFinding the solution using the available computing resourcesFinding the solution within the available timeOne or more (contradictory) targetsSolving the problem under a number of (hard/soft) constraints

______________________________________________



Basic Concepts of Problem Solving Techniques

Representation: how to encode alternative candidate solutions for manipulationObjective: describes the purpose to be fulfilledEvaluation function: returns a value that indicates the (numeric of ordinal) quality of any particular solution, given the representation

______________________________________________



Hill-Climbing: A Traditional (Deterministic) PST

Useful Definitions:Neighborhood of a solutionLocal Optimum

x1

x2

Ideas for creating Hill-Climbing Method variantsHow to select the new solution for comparison with the current solution (how to compute the gradient) …To use more than one starting solutions, if necessary …To be “less deterministic” …

Requirements:A starting pointComputation of gradientTermination criteria

______________________________________________



Algorithms Relying on Analogies to Natural Processes

Evolutionary ProgrammingGenetic AlgorithmsEvolution StrategiesSimulated AnnealingClassifier SystemsNeural Networks

Traditional Problem Solving Techniques

Algorithms with analogies

to Natural Processes

______________________________________________



The Subclass we are interested in …

Methods which are based on the principle of evolution (i.e. the survival of the fittest)

Handling populations of candidate solutionsUndergoing unary (mutation-type) operationsUndergoing higher-order (crossover-type) operationsUsing a selection scheme biased towards fitter individuals

Generation

1

Generation

2

Generation

3

Generation

4

______________________________________________



EA –

Schematic Presentation:

if( ( )) end

,

13 2 1

( ),

( ),

( ( ( )))

1

converge

g

g

g g

y F x s S

y s S

S T T T S

g g

g

μφ+

⎡ = ∀ ∈⎢⎢

∀ ∈⎢⎢⎢ =⎢⎢⎢ = +⎢⎢⎢⎣

0, g randomg S S= =

______________________________________________



EA –

Prerequisites:

if( ( ))

13 2 1

( )

( )

( ( ( )))

converge

g g

y F x

y

S T T T S

g

φ

+

⎡ =⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢ =⎢⎢⎢⎢⎢⎣

Parameterization &Representation

Objective Function

FitnessFunction

Evolution Operators

Stopping Criterion

______________________________________________



Encoding the free variables –

Binary Coding:

x

mx Gene

Candidate solution

Binary digits per variable, m=1,Mmb

1,

1

22 1

m

m

bm m i

m m m ibi

U Lx L d−

=

−= +

− ∑

21

0101101.....10111Mx xx

Chromosome:

Lm

, Um

user defined bounds

______________________________________________



A schema

is a similarity template describing a subset of

strings with similarities at certain string positions (Holland, 1968)

Schemata in binary strings:

*10 * 1

01001 01011 11001 11011

Defining length d(S)=5-2=3

Order of schema o(S)=3 (fixed digits)

m=length of string(=5)r=number of *’s (=2)

2m

possible schemata

2r

strings matched by this schema

______________________________________________



The order

of a schema affects its survival probabilities

during mutation

Why dealing with Schemata?

*10 * 1Defining length d(S)=5-2=3

Order of schema o(S)=3 (fixed digits)

The defining length

of a schema affects its survival

probabilities during crossover

(Schema Theorem)

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Encoding the free variable–

Real Coding:

1 2( , ,..., ,..., )m Mx x x xRepresentation:

A step ahead:

Representation including evolution parameters (the concept of ES):

1 2 1 2( , ,..., ,..., , , ,..., ,..., )m M mx x x x σ σ σ σΜ

______________________________________________



The effect of selection:

(mg

) examples of a particular schema (S), generation (g)

F(S)=average fitness of the strings matching schema (S)

Schema Theorem (1/4) :

mg+1

>mg if F(S)>Fmean,g

mg+1

<mg if F(S)<Fmean,g

1,

( )g g

mean g

F Sm m

F+ =Reproductive Schema

Growth Equation

1 (1 )g gm m k+ = +

11 0(1 )ggm m k +

+ = +Long term effect of selection(k=constant)

______________________________________________



The effect of crossover:

Possibility of destructing the schema S during crossover:


( )1des

d Sp

m=

−

****11010*01*1*11*d(S)

m-1

( )1

1main Xover

d Sp p

m= −

−

Possibility of maintaining the schema S:

1,

( ) ( )( ) ( ) (1 )

( ) 1g g Xovermean g

F S d Sm S m S p

F S m+ ≥ −−

Schema Growth Equation (after selection & crossover):

______________________________________________




The effect of mutation:

Possibility of maintaining the schema S after mutation:

( )(1 ) 1 ( )o Ssurv mut mutp p p o S= − ≈ −

****11010*01*1*11*

1,

( ) ( )( ) ( ) (1 ( ) )

( ) 1g g Xover mutmean g

F S d Sm S m S p o S p

F S m+ ≥ − −−

Final Schema Growth Equation:

Short, low-order, above-average

schemata should receive an

(exponentially) increasing number of strings in the next generations

______________________________________________




Lessons Learned:Short, low-order, above-average schemata sould receive an (exponentially) increasing number of strings in the next generations (Schema Theorem).GA explore the search space by short, low-order schemata.GAs seek near-optimal performance through the juxtaposition of short, low-order, high-performance schemata (the so-called building blocks, Building Block Hypothesis).

______________________________________________



Exploration vs. Exploitation :

Exploration: seeking the global optimum in new and unknown areas in the search space.Exploitation: making use the knowledge gained from the previously examined points to guide the search towards new better points in the search space.

Holland 1975: GAs

but:Infinite population sizeFitness function value accurately reflects the utility of a solutionGenes in a chromosome do not interact significantly

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EAs: Binary or Real Coding?

Binary Coding:Creates lengthy binary strings if high accuracy is required.Offers the maximum number of schemata per bit of information, compared to any other coding.Facilitates theoretical analysis and the development of new genetic operators.Smallest alphabet that allows a natural expression of the problem (Goldberg, 1989).

Real Coding:Problem-tailored genetic operators can readily be devised.High-cardinality alphabets contain more schemata (Antonisse, 1989).

______________________________________________



Generalized Evolutionary Algorithm (EA)

λ,gS λ

, 1,mx m M=

μ,gS μ

e,g eS

, 1,kF k K=

( )y F x=

1: Kφ →

eΤ

Τμκ

ρΤrΤm

Offspring Parent Elite

______________________________________________



ΕΑ

–

Schematic Presentation:

( )

( )( )

if( ( )) end

2

,

1, , ,

, , 1,

, , 1,

1, , ,

1, 1, 1,

( ),

( )

( ),

( )

( )

1

converge

g

g e g g ee

g g g e

g g g ee

g g g

g g g em r

y F x s S

S T S S

y s S S S

S T S S

S T S S

S T S S

g g

g

λ

λ

μ λ

λ λ

μ μ λμ

λ μ

φ

+

+

+

+

+ + +

⎡ = ∀ ∈⎢⎢⎢ = ∪⎢⎢ ∀ ∈ ∪ ∪⎢⎢⎢ = ∪⎢⎢⎢ = ∪⎢⎢

= Τ ∪⎢⎢⎢ = +⎢⎢⎢⎢⎣

, , ,0, , , g e g g randomg S S S Sμ λ= = ∅ = ∅ =

______________________________________________



Parent Selection Operator

Τμ

(1/5)

(Neglecting Elitism)

Phase 1:

Phase 2:

( ), , ,,1

prov g gS S Sμ μ λμ= Τ ∪

( )1, ,,2

g provS Sμ μμ

+ = Τ

Life-span κFitness φ

Selective Pressure

(Fitness φ)

(ES)

(GA)

______________________________________________




Τμ

(2/5)

Proportional selection

Linear ranking

Roulette wheel

Probabilistic Tournament selection

Indirect Selection (μ<λ)

( )( )

1, , ,,1

1, ,,2

g g g

g prov

S S S

S S

μ μ λμ

μ μμ

+

+

= Τ ∪

= Τ

______________________________________________




Τμ

(3/5)

Proportional selection:

Select (λρ)

individuals

out of (μ) preselected

ones

( )

( )

1

( )( )

( )

1

1

_ _ int

s

i

ss

i

number of copies

μι

μι

μφ

φ

λρμφ

φ

=

=

≥

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

∑

∑

Premature convergence due to

the presence of a “super-fit”

individual.

______________________________________________




Τμ

(4/5)

Roulette Wheel:

Select (λρ)

individuals

out of (μ) preselected

ones

( )

( )

1

_ 360s

o

i

angle slot μι

φ

φ=

=

∑

Premature convergence due to

the presence of a “super-fit”

individual.

______________________________________________




Τμ

(5/5)

Fitness Ranking:Individuals are sorted in the order of fitness valuesReproductive trials are assigned according to rankLinear RankingExponential Ranking, etc

Overcomes the problem of the presence of an extreme individualFitness ranking performs better than fitness scaling.

______________________________________________



Recombination Operation

Tr

–

Binary Coding

}One-point

Two-pointOne- or two-point per variableDiscrete (variables’ interchanging)Uniform (with parent-depending probability)

01101001110100001100

0110001100

011010011101000011001010001010

0110000010

ρ=2

ρ=3

______________________________________________



Recombination Operation

Tr

– Real Coding

One-point, ρ=2 x1

x2

x3

x4

x5

x6x1

x2

x3

x4

x5

x6 } x1

x2

x3

x4

x5

x6

x4

=x4

+r

(x4

-x4

),

r∈ [0,1]Two-pointM-point

Discrete, ρ=2Discrete Panmictic ρ=Μ (50% selection probability

from each parent {1,2}, {1,3}, ..., {1,Μ} )Generalized Intermediate Panmictic

xm

=xm

+rm

(xm,ρ

-xm

),

m=1,M,

rm

∈ [0,1]

xm

=xm

+rm

(xm

-xm

),

m=1,M,

rm

∈ [0,1]

Blend Xover (BLX-a), c=(1+2a)r-a, a=0.5

______________________________________________



Mutation Operator

Tm

-

Binary Coding

011000110000101011001

1...5m M

mm

Pb

=∑∼

Dynamic Adjustment

of

Pm , depending on

• the number of generations without any improvement

• the number of evaluations without any improvement

______________________________________________



Mutation Operator

Tm –

Real Coding

Dynamic mutation probability per variable

Pm

max

1

1

2max

(1 )

2

( , ), 0.5

( , ), 0.5

( , ) 1

( , ) 1p

m m m

mm m m

p

gg

x D g U x rx

x D g x L r

gD g a a r

g

D g a a r−

⎧ + − <⎪⎪⎪= ⎨⎪ − − ≥⎪⎪⎩⎛ ⎞⎟⎜= ⋅ ⋅ − ⎟⎜ ⎟⎜⎝ ⎠

⎛ ⎞⎟⎜ ⎟= ⋅ −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

ή

…

or, using number of evaluations, instead of number of generations

p~0.2

______________________________________________



Mutation Operator

Tm –

Real Coding

( )exp (0,1) (0,1)

(0,1)

m m m

m m m m

´

x x N

σ σ τ τ

σ

= ⋅ ⋅Ν + ⋅ Ν

= + ⋅

( )( )

1

1

2

' 2

τ

τ

−

−

= Μ

= Μ

______________________________________________



Genetic Algorithms or Evolution Strategies or …

Binary Codingμ=λ, parents=offspringρ=2, two-parent recombinationκ=0, zero life-spanΡr<1, recombination probabilityK=1, one target

[Holland,

1970]

[Goldberg, 1989][Michalewicz, 1994][Fogel]

Real Coding, including the evolution parametersWithout parent Selection Operator (μ<λ)μ<<λκ=0 (μ,0,λ) = (μ,λ) or κ=∞ (μ,∞,λ) = (μ+λ)ρ=2Pr=1Κ=1

[Schwefel, Rechenberg, 1965]

[Bäck, 1996]

GA

ES

______________________________________________



Multi-objective Optimization

Non-Pareto Front

Techniques

Pareto Front Techniques

(K = number of objectives)

______________________________________________



Pareto Front

Front

0 (Pareto) F1

F2

Front

1

Front

2

,g eS

______________________________________________



Pareto Front –

Definition (Minimization Problems):

Dominant Solution:( ) ( ) ( ) ( )

( ) ( )

( ) ( )

{1,..., } :

{1,..., } :

p q p q

p qk k

p qk k

x x s s

k K F F

k K F F

< ⇔ < ⇔

∀ ∈ ≤ ∧

∃ ∈ <

' ' :Mx x x x/⇔ ∃ ∈ >

Pareto Optimal Solution:

______________________________________________



Multi-Objective Optimization

–

Computation of

φ

1

K

k kk

w Fφ=

= ⋅∑

, [1, ]kF k Kφ = ∈

Pareto_Rank( ), k 1,kF Kφ = =

For parent selection, VEGA [Schaffer, 1984]

[Goldberg, 1989]

______________________________________________



Vector Evaluated Genetic Algorithm (VEGA)

K=3

Selection based on φ=F1



Recombination, Mutation

______________________________________________



VEGA: Guess the final solutions…

Front

0 (Pareto) F1

F2

Front

1

Front

2

,g eS

______________________________________________



φ

computation using the Pareto front:

Front Ranking [Goldberg, 1989]

Niched Pareto GA (NPGA) [Horn, Nafpliotis, 1993]

Nondominated Sorting GA (NSGA) [Srinivas, Deb, 1994]

Strength Pareto EA (SPEA) [Zitzler, Thiele, 1998]

Pareto Envelope-based Selection Algorithm (PESA) [Corne, Knowles, Oates, 2000]

NSGA-II [Deb, Agrawal, Pratap, Meyarivan, 2000]

Strength Pareto EA ΙΙ (SPEA II) [Zitzler, Laumanns, Thiele, 2001]

______________________________________________



φ

computation

–

Sorting (NSGA)

F2

F1

______________________________________________



=

1

0, ( , )

( , )1 , ( , )

f shareN

j shareshare

i dummy

d i j

Sh d i jd i j

Sh

σ

σσ

φ φ

=

⎧ ≥⎪⎪⎪⎪= ⎨⎪ − <⎪⎪⎪⎩⋅

∑

d(i,j)i

j

σshare

σshare

φ

computation

–

Niching

(NSGA)

1dummyφ =

F2

F1

_dummy prev frontφ φ ε= +

______________________________________________



φ

computation

–

SPEA

F1

F2

Sg,eDominated SolutionsOther Solutions

( ) ( ) [0,1)

1eFφ

μ λ= ∈

+ +( )( ) 1 ( ) 1iFφ φ= + >∑

______________________________________________



Ways to reduce the number of evaluations:

Improved evolution operators

Hybridization with other optimization methods

Distributed EAs (island model)

Hierarchical EAs (faster solvers)

Use of surrogate models (metamodels, fast approximate model)

______________________________________________



Hybridization with other Optimization Methods

evaluations

Log 1

0(sc

ore)

GA

AM

______________________________________________



Use of Surrogate Evaluation Models

Polynomial InterpolationArtificial Neural Networks

Multilayer PerceptronRadial Basis Function Networks

Kriging

Ways of using the surrogate evaluation model:Decoupled from the exact evaluation tool (+Design of Experiments, DoE)In combination with the exact evaluation tool

Regular Training (depending on the number of new entries in the DB)Dynamical Training (separately, for each new individual)

______________________________________________



Use of Surrogate Models (with Off-Line Training)

Design of Experiments

Exact Evaluations

Surrogate model

Evaluations using the Surrogate

Model

Genetic Operators

“Optimal”

Solution

Inner Loop

Outer Loop

______________________________________________



Use of Surrogate Models (with On-Line Training)

best

Evaluations using the Surrogate Model(s)

Evaluations using the

Exact Model

Genetic Operators

Fitness Value Homogenization

New Population

Build the Surrogate Model(s)

______________________________________________



Inexact Pre-Evaluation (IPE) –

The Concept:

Exact Evaluation of the most promising solutions

λevaluations

Evolution (to the next generation)

Inexact Evaluations

L<λ evaluations

______________________________________________



Local Surrogate Models -

Training Set:

x1

x2min max[ , ]T T T∈

______________________________________________



Surrogate Models –

What else do they tell us???

err

errF

Fitness Function ApproximationConfidence IntervalsHessian Matrix ApproximationSensitivity Derivatives (Importance Factors)

Optimal

______________________________________________



Optimization and Multi-processing

CPUs ≠ λ (ή L )

Loading Distribution:Heterogeneous PlatformsLoading per processorVariable evaluation cost

Synchronization (in each generation)

Master: ΕΑ Module - waiting list for evaluationsSlave: discrete (remote) evaluation process

Parallelization Level

Reduction of the Optimization Wall-Clock Time

CFDCFDEAEA

______________________________________________



Distributed

ΕΑ

Why ?L (<< λ) < CPUs

Persistant diversity in populationsStraightforward parallelization

Additional Parameters:Number of islandsCommunication topologyCommunication frequencyMigration algorithmEA parameters per island

?

______________________________________________



Distributed

ΕΑ

on a Multi-Processor System

Evaluations Server

EA EA EA

Migrations

Islands

Asynchronous Requests for Evaluations

MASTER

SLAVE1 SLAVE2 SLAVE3 SLAVE4

Thread 1

Thread 2

Thread 3 Thread

4 Thread

5

______________________________________________



v1.3

The Evolutionary Algorithms SYstem

Applications

Developed by the National Technical University of Athens, (NTUA), Greece

______________________________________________



Problem: Rastrigin’s

Function

2

1

1

1( ) 20 20 exp( 0.2 )

1exp( cos(2 ))

M

mm

M

mm

F x e xM

xM

π

=

=

= + − −

−

∑

∑

M=30 30 30mx− ≤ ≤

______________________________________________



Results: Rastrigin’s

Function

1

10

100

0 500 1000 1500 2000 2500 3000 3500 4000

Cos

t Fun

ctio

n V

alue

(Ras

trigi

n)

Cost Measured in Exact Evaluations

GADGA

GA-IPE-IFD(GA-IPE-IF)

______________________________________________



Problem: Three-Element Airfoil, Lift Maximization

Initial

Rotation Angle 28.1ο

Δx -0.020

Δy 0.0269

Rotation Angle -37ο

Δx 0.020

Δy 0.0249

0.12, 17.18oM a∞ ∞= =

LF C= −

Initial

Optimal

______________________________________________



Results: Three-Element Airfoil, Lift Maximization

slat

Δy

flap

Initial OptimalRotation Angle 28.1ο 28.02ο

Δx -0.020 -0.03078

Δy 0.0269 0.01982

Rotation Angle -37ο -36.96ο

Δx 0.020 0.02016

Δy 0.0249 0.02469

IFs

Design Variable

Imp

ort

an

ce

______________________________________________



Results: Three-Element Airfoil, Lift Maximization

-5.24

-5.22

-5.2

-5.18

-5.16

-5.14

-5.12

0 500 1000 1500 2000

Cos

t Fun

ctio

n V

alue

(-Li

ft)


GADGA


______________________________________________



Problem: 3D Compressor Blade DesignNURBS

ΑκροπτερύγιοΠόδι

ωref =0.09

Evaluations

Hub Section Tip Section

______________________________________________



Problem: Design of a Compressor Cascade

c=0.07ms=0.68c γ=30ο

Minimum Total Pressure LossesConstraint on the minimum (maximum thickness)Desirable Flow turning

1 1 2F P Pω= ⋅ ⋅

max1 max, maxexp( ), 0.9 , thres

thres ref thresthres

t tP t t t t

t−

= = <

2 1 2exp( max(1, )), 0ref

refP

α αα α α

αΔ −Δ

= − Δ = − >Δ

______________________________________________



Results: Design of a Compressor Cascade

0

0.02

0.04

0.06

0.08

0.1

0.12

0 500 1000 1500 2000

Cos

t Fun

ctio

n V

alue


GADGA


______________________________________________



Problem: Compressor Multi-Point Design

MISES 2.53

c=0.07ms=0.68c γ=30ο

2, 2

1 1

t is t

t

p pp p

ω−

=−

1 2 3 4 5

α1

43o

45o

47o

49o 52o

Μ1 0.6184

0.6182

0.6180

0.6195

0.6214

Re

8.61E5

8.50E5

8.41E5

8.20E5

7.63E5

1/AVDR

1.0909

1.0965

1.1021

1.1027

1.1032

ω

(exp) 0.0232

-

0.0186

-

0.0417

ω

MISES

0.0234

0.0208

0.0189

0.0237

0.0374

α2

(πexp)

20.79

20.80

20.92

21.69

22.74

α2

MISES

20.80

20.80

20.90

21.70

22.701

3 5

ASME 90-GT-140

______________________________________________



Compressor Multi-Point Design (1 OP –

1 target)(20,2,100)

Evaluations

1 2 3 4 5

ω

L=5

0.0244

0.0182

0.0155

0.0275

-

ω

REF

0.0234

0.0208

0.0189

0.0237

0.0374

______________________________________________




2 targets)

1 3 1 2F P Pω= ⋅ ⋅

2 1 5 3 1 2( 2 ) 0F P Pω ω ω= + − ⋅ ⋅ >

Evaluations

______________________________________________




3 targets)

Evaluations

______________________________________________




5 targets)

1 2, 1,..., 5i iF P P iω= ⋅ ⋅ =

Evaluations

______________________________________________



Problem: High-Lift, Low-Drag Optimization

61 : 0.20, 10.8 , Re 5 10oC M a∞ ∞ ∞= = = ×

NS, k-ε

WF 6000

3000

200

62 : 0.77, 1.0 , Re 10 10oC M a∞ ∞ ∞= = = ×

4min

F1

F2

______________________________________________



Results: High-Lift, Low-Drag Optimization

500 evaluations

C1 C2

1500

evaluations 2500 evaluations

______________________________________________



Optimization of Combined Cycle GT Power Plants

ST ST

C T1

2

3

4

5

6

7

1

3

2

7

45

6

HP

LP

G1

G28

9

8

WATER TANK

Natural gas fired, dual-pressure CCGTPP configuration

GT: 260 MWe, 38% efficiency, exhaust gas mass flow 615 kg/sec at 600C.

Design variablesHP steam pressureLP steam pressuresuperheated HP steam temperaturefeedwater temperature at the inlet to the HP evaporator feedwater temperature at the outlet from the first HP economizerfeedwater temperature at the inlet to the LP evaporator superheated LP steam temperaturesteam pressure fed to the water tank exhaust gas mass flow ratio (percentage of mass flowratetraversing the LP economizer) exhaust gas temperature at the HRSG outlet steam extraction pressure from the LP steam turbine exhaust gas temperature at the inlet to the condensate preheater

______________________________________________



Optimization of Combined Cycle GT Power Plants

150

155

160

165

170

175

180

185

0.515 0.52 0.525 0.53 0.535 0.54 0.545 0.55 0.555 0.56 0.565

Ca

pita

l C

ost

(ME

uro

)

Efficiency

Candidate SolutionsOptimal Solution

Pareto front

0

100

200

300

400

500

600

0 50 100 150 200 250 300 350

Tem

pera

ture

(C

elci

us)

Exchanged Heat (MW)

Flue GasHP Water/SteamLP Water/Steam

Preheater

Constrained OptimizationProblem

40

50

60

70

80

90

100

0.515 0.52 0.525 0.53 0.535 0.54 0.545 0.55 0.555 0.56 0.565

Pre

ssur

e V

alue

(H

P, b

ar)

Efficiency

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