tutorial-slides-coello

Description

A Short Tutorial on EvolutionaryMultiobjective OptimizationCarlos A. Coello CoelloCINVESTAV-IPNDepto. de Ingenier´ıa El´ectricaSecci´ on de Computaci´ onAv. Instituto Polit´ecnico Nacional No. 2508Col. San Pedro ZacatencoM´exico, D. F. 07300, [e-mail] A. Coello Coello, March 2001. Tutorial on Evolutionary Multiobjective OptimizationWhy Multiobjective Optimization?Most optimization problems naturally have several objectives to beachieved (normally conflicting with each other), but in order tosimplify their solution, they are treated as if they had only one (theremaining objectives are normally handled as constraints).EMO’01Carlos A. Coello Coello, March 2001. Tutorial on Evolutionary Multiobjective OptimizationBasic ConceptsThe Multiobjective Optimization Problem (MOP) (alsocalled multicriteria optimization, multiperformance or vectoroptimization problem) can be defined (in words) as the problem offinding (Osyczka, 1985):a vector of decision variables which satisfies constraints andoptimizes a vector function whose elements represent theobjective functions. These functions form a mathematicaldescription of performance criteria which are usually inconflict with each other. Hence, the term “optimize” meansfinding such a solution which would give the values of allthe objective functions acceptable to the decision maker.EMO’01Carlos A. Coello Coello, March 2001. Tutorial on Evolutionary Multiobjective OptimizationBasic ...

Subjects

• Documents
• Ressources professionnelles
• IT systems

Informations

• Published by: Chyok
• Reads: 17
• Language: English

Exrait

A.v

A Short Tutorial on Evolutionary Multiobjective Optimization

-ASMPT .A -PFMMP -PFMMP

-I5V08VATIP5 .FpVP.EFIOgFOIFS´ıA0MF´DVSIDA 8FDDIP´OEF-PNpXVADIP´O IOTVIVXVPPPMIV´FDOIDP5ADIPOAM5P.508 -PM.8AOPFESPZADAVFODP 3F´xIDP,..2.0,3030IX-7 [e-mail]

C5aS]b A, C]CSS] C]CSS]* M5a=O 0../,

OME ./ ’

Tdc]aR5S][vE]SdcR][5afMdScR];jC=cRvCOpcRTRz5cR][

Why Multiobjective Optimization?

Feijefj]c]zTj]edfheUbYcidTjkhTbboTlYiYlYhTbeUYVj]lYijeUY TV]YlYWdehcTbboVedfl]Vj]d[m]jYTVejYh)&Ukj]dehWYhje i]cfb]Zo jY]h iebkj]ed& jYo ThY jhYTjYW Ti ]Z jYo TW edbo edY jY hYcT]d]d[ eUYVj]lYi ThY dehcTbbo TdWbYW Ti VedijhT]dji)(

C5aS]b A, C]CSS] C]CSS]* M5a=O 0../,

EMO’./

Tdc]aR5S][Ev]SdcR][5afMdScR];jC=cRvCpOcRTRz5cR][

Basic Concepts

KY3XMVIPCjFDVIvF 7pVINIzAVIPO PSPCMFNFHP) Tbie VTbbYWckbj]Vh]jYh]Tefj]c]zTj]ed&ckbj]fYhZehcTdVYehlYVjeh efj]c]zTj]ed fheUbYc) VTd UY WYfidYW ]d mehWi) Ti jY fheUbYc eZ fidW]d[HioVzaT&,430)5

T lYVjeh eZ WYV]i]ed lTh]TUbYi m]V iTj]ifiYi VedijhT]dji TdW efj]c]zYiTlYVjehZkdVj]edmeiYYbYcYdjihYfhYiYdjjY eUYVj]lY ZkdVj]edi( KYiY ZkdVj]edi Zehc T cTjYcTj]VTb WYiVh]fj]ed eZ fYhZehcTdVY Vh]jYh]T m]V ThY kikTbbo ]d Vedfl]Vjm]jYTVejYh(AYdVY&jYjYhc“efj]c]zY”cYTdi fidW]d[ ikV T iebkj]ed m]V mekbW []lY jY lTbkYi eZ Tbb jY eUYVj]lY ZkdVj]edi TVVYfjTUbY je jY WYV]i]ed cTaYh(

C5aS]b A, C]CSS] C]CSS]* M5a=O 0../, Tdc]aR5S ][ vE]SdcR][5af MdScR];jC=cRvC pOcRTRz5cR][

’ / EMO .

Basic Concepts

KY [YdYhTb Fkbj]eUYVj]lY Hfj]c]zTj]ed PheUbYc FH)P VTd UY ZehcTbbo WYfidYW Ti5 ;]dW jY lYVjehx~ ∗ = Px / ∗ , x 0 ∗ , . . . , x ∗ S S T m]V m]bb iTj]iZo jYO ]dYgkTb]jo VedijhT]dji5

F F x~)≥+M= ,,-, . . . , O

jYpYgkTb]jo VedijhT]dji

L F x~) = +M= ,,-, . . . ,

TdWm]bbefj]c]zYjYlYVjehZkdVj]ed

p

~ x E~) = PE / x~), E 0 x~), . . . , E k x~)S

T

,) 

-) 

 .)

C5aS]b A, C]CSS] C]CSS]* M5a=O 0../,

OME’./

Tdc]aR5S][Ev]SdcR][5afMdScR];jC=cRvCpOcRTRz5cR][

Basic Concepts

ATl]d[ iYlYhTb eUYVj]lY ZkdVj]edi& jY dej]ed eZ “efj]ckc” VTd[Yi& UYVTkiY ]d FHPi& mY ThY hYTbbo jho]d[ je fidW [eeW Vecfhec]iYi eh “jhTWYeffi”) hTjYh jTd T i]d[bY iebkj]ed Ti ]d [beUTbefj]c]zTj]ed(KYdej]edeZ“efj]ckc”jTj]iceij Veccedbo TWefjYW ]i jTj eh][]dTbbo fhefeiYW Uo ;hTdV]i Ni]Whe :W[Ymehj ]d ,33,(

C5aS]b A, C]CSS] C]CSS]* M5a=O 0../,

MOE’./

Tdc]aR5S][Ev]SdcR][5afMdScR];jC=cRvCOpcRTRz5cR][

Basic Concepts

K]idej]edmTibTjYh[YdYhTb]zYWUoM]bZhYWePThYje]d,341)( 6bjek[ iecY Tkjehi VTbbEdgeworth-Pareto optimumje j]i dej]ed& mY m]bb kiY jY ceij Veccedbo TVVYfjYW jYhc5Pareto optimum(

C5aS]b A, C]CSS] C]CSS]* M5a=O 0../,

MEO’./

Tdc]aR5S][vE]SdcR][5afMdScR];jC=cRvCOpcRTRz5cR][

Basic Concepts

WYiTojTjTlYVjeheZWYV]i]edlTh]TUbYix~ ∗ ∈ F]iPareto optimal ]Z jYhY WeYi dej Yn]ij TdejYhx~∈ FikV jTjE F x~)≤Ex~ ∗ ) Zeh F TbbM= ,, . . . , kTdWE L x~) E L x~ ∗ ) Zeh Tj bYTij edYN(

C5aS]b A, C]CSS] C]CSS]* M5a=O 0../,

OME’./

Tdc]aR5S][vE]SdcR][5afMdScR];jC=cRvCOpcRTRz5cR][

Basic Concepts

~ ∗ Bd mehWi& j]i WYfid]j]ed iToi jTjx]i PThYje efj]cTb ]Z jYhY Yn]iji de ZYTi]UbY lYVjeh eZ WYV]i]ed lTh]TUbYix~∈ Fm]V mekbW WYVhYTiY iecY Vh]jYh]ed m]jekj VTki]d[ T i]ckbjTdYeki ]dVhYTiY ]d Tj bYTij edY ejYh Vh]jYh]ed( LdZehjkdTjYbo& j]i VedVYfj Tbceij TbmToi []lYi dej T i]d[bY iebkj]ed& Ukj hTjYh T iYj eZ iebkj]edi VTbbYW jYPareto optimal set( KY lYVjehix~ ∗ VehhYifeW]d[ je jY iebkj]edi ]dVbkWYW ]d jY PThYje efj]cTb iYj ThY VTbbYW nondominated( KY fbej eZ jY eUYVj]lY ZkdVj]edi meiY dedWec]dTjYW lYVjehi ThY ]d jY PThYje efj]cTb iYj ]i VTbbYW jY Pareto front(

C5aS]b A, C]CSS] C]CSS]* M5a=O 0../,

MOE’./

L

Tdc]aR5S][Ev]SdcR][5afMdScR];jC=cRvCpOcRTRz5cR][

4

L

An Example

3

1

2F

2

F

L

;][khT ,5 6 ZekhUTh fbTdY jhkii(

F

C5aS]b A, C]CSS] C]CSS]* M5a=O 0../, Tdc]aR5S ][ Ev]SdcR][5af MdScR];jC=cRvC OpcRTRz5cR][

OME’./

Example

F]d]c]zY    E E 0 /   x~ x~ ) =) = F  E   -x  x / 0 1 % % √ 0 x - √ x 2 0 0 − % 0 x √ √ 3 x 1 0 % % x x 0 4 2  

ikV jTj5

)

F /σ)≤x / ≤.F /σ √ -F /σ)≤x 0 ≤.F /σ √ -F /σ)≤x 1 ≤.F /σ    F /σ)≤x 2 ≤.F /σ

) ) )

mYhYF= ,+ aG&= -×,+ 5 aG)Vc 0 &= -++ Vc&σ= ,+ aG)Vc 0 (

/)

0)

C5aS]b A, C]CSS] C]CSS]* M5a=O 0../,

Tdc]aR5S][Ev]SdcR][5afMdScR];jC=cRvCpOcRTRz5cR][ Example

+(+/ +(+.0 +(+. +(+-0 E 0 +(+- +(+,0 +(+, +(++0 ✸ + /+1++,-++-/++-1++-3++ ,-++ , + , 3++ -+++ - E /

OEM ./ ’

;][khT -5 KhkY PThYje Zhedj eZ jY ZekhUTh fbTdY jhkii fheUbYc(