HAL Id: inria-00070802
https://hal.inria.fr/inria-00070802
Submitted on 19 May 2006
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EEDA : A New Robust Estimation of Distribution Algorithms
Michael Wagner, Anne Auger, Marc Schoenauer
To cite this version:
Michael Wagner, Anne Auger, Marc Schoenauer. EEDA : A New Robust Estimation of Distribution Algorithms. [Research Report] RR-5190, INRIA. 2004, pp.16. �inria-00070802�
ISRN INRIA/RR--5190--FR+ENG
a p p o r t
d e r e c h e r c h e
Thème COG
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
EEDA: A New Robust Estimation of Distribution Algorithms
Michael Wagner — Anne Auger — Marc Schoenauer
N° 5190
May 2004
Unité de recherche INRIA Futurs
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nm eqRTW-¡zynpeqWVu2|:egougWÄRD|¾lWeqRTW-§WVuqj®¾lWÀh+egj@u.|lc#egRTWjl}beqnU¦TU«y¿e.eqnUXWÀcVÂ<neh|m·>W-TmDh!W|u
[RoW!egRoWVueqRTWv¡zy·nm%HTWcfegnjlm|lh!ego|la cfj@¾@W~ªegRTWjl}beqnUXn§|:eqnj@m}TuqjlTWVUMjluj@mTawxocfehVjlmd¾lWugk@W~
eqjegRTW§WVuqj¾lWh!egj@uÄ£W|l~TnmTkeqj®|,rs|¢cfW}>j@cgnegn¾lWuqTmD¤+«Q)j½WnUXnmo|egWeqRTn¢c}DjHcgcgnTnnexalÂ)¿WRo|¾@W
UXjb~bnÍoW~&hVjlUXUXjlm¯egWÀcxerTmDh+egnjlmDcÄcfDh2R¨eqRo|:eeqRTW®cgjlbegnjlm&n¢cmTj: |e-|OmTjlmT§WVuqj ¾|loWl«6QRTW
romoh+eqnj@mocegRD|:e[W#cgRo|lWU}oj:aÄnm,jlTuihVjlUX}Tbe2|:egnjlmD|cfego~bnWcz|uqWl¥
ê
f1(x) = 100 10−5+Pn
i=1|yi|, [RTWugW yi = (xi−i+ 1) +yi−1(y1=x1) f2(x) =
Xn
i=1
(xi−i+ 1)2
f3(x) = 1 + Xn
i=1
(xi−i+ 1)2− Yn
i=1
cos(xi−i+ 1)
√i+ 1 .
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f1(x) |lmo~¼UXnmTnUXn§VW
f2(x) |mo~ f3(x)«&QRTWj@}begnU.TU jrÁ|leqRTuqWVWjrªegRTW
|lDj:¾@W rTmoh!egnjlmocªjbhVh!ouqcª|:e
x∗ = (0,1, . . . , n−1)«zQRTW.j@}begnU|K¾:|TWczjr<egRoWcgW#rTmoh!egnjlmocÁ|uqW
f1(x∗) = 107Â f2(x∗) = 0|mo~ f3(x∗) = 0«
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k@W!ei|ugWÀ|ncfegn¢hÁ¾dnW6jrKegRoW}DWufrj@ugU|mDh!WÁjr*¡iyzcz|mo~,|¾lj@n~Ärs|¢cfWRTj@}DW@«
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mojlm$Õjl}beqnU|li¾:|TWl« mo~TWVu| cgnk@R@eÄ}>WVugegougo|egnjlm&jr eqRTW®nmTnpeqn|ln§|egnjlm ±zy
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t¸ x¹ º<«
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cgohh!Wcqc.|lmo~ UXjb~bnÍoW~ eqRTW,nmTnegn¢|~bn¢cxequgnTbeqnj@m« QRTW-}o|luq|lUXW!egWuqcegRo|e¿WVuqWÄmTjleh2Ro|lmTklWÀ~ |uqW
eqRTW.~bnUXWVmocgnjlm
n= 10Âo}>jl}To|egnjlm½cfn§VW
N = 2000ÂTcq|UX}TW.cfn§VW
M = 1000Âo|mD~TmHegu2|mocg|egWÀ~
¾@WVu2cfnjlmDc*jrrTmoh!egnjlmoc
f1(x) f2(x)Âb|mD~ f3(x) £[RTWugW[eqRTWzj@}begnU.oU jbhh!Tu2cv|:eegRTWª§VWugj¾@Wh+eqjlu+¤+Â
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f˜1(x) f˜2(x)Â|mo~ f˜3(x)«Á¬©W¦nmHequgjb~bohVW~½|lm|lcgadUUXWVeguqnh|nmTnegn¢|n§À|:egnjlm
dah2RTjdj@cgnmok
µ0= (2, . . . ,2) |mo~ Σ0 =I, egRTW¦n~bWmHegnexa,U|eguqnp»«zQRTW#egWVuqUXnmD|:egnjlm©h!uqnpeqWVuqn|X|lc cgW!eeqj :oÂ@l-rTmoh+eqnj@m WV¾:|o|:eqnj@moc«Ä¬©W-}TugWÀcfWmHeÁeqRTWÄUXW~Tn|lm |lmo~ cxe2|mo~o|u2~©~bW¾dn|egnjlmOjregRTW
ÍDmo|*¾:|TWXjregRTWÍTeqmTWcqcÁrTmoh!egnjlmOnm |mOjlu2~bWVuqW~®}o|nuÁrjluqUX|eX£sUXW~bn¢|mÂcxe2|mo~T|luq~©~bWV¾dn¢|:eqnj@mD¤+Â
[Ronh2R|uqWhV|¢h!o|egW~-ruqjlU l¦uqTmoc«*tvW|lcgWcfWWQ|lTW@«
n= 10 f˜1(x) f˜2(x) f˜3(x)
±zy ²q³´gµ¶g³ £fl«oÀTÂbT«@
@¤ £xb«l@TÂbT«
@¤ £x d«l
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cfegu2|:eqnmok.rs|lnTuqWl«
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eqRo|:e¦j@mTklWuugTmDc-£[negR uqWcg}>Wh+e.egj®|½¢|uqklWumHoU.>WVu#jlrW¾:|o|:eqnj@moc#|j:WÀ~o¤ªUnklRHe.uqWcgTe.nm
|¨jl}TegnUX|lcfj@beqnj@mª|lmda¯}ougj@j@mTk@|egnjlmjlr#egRTW cfego~bnWcuqWcgTegWÀ~nm6mdTUXWVuqnh| WVuquqjlu2c-~bTW©egj
eqRTWXh!j:¾:|uqn|lmoh!W¦U|:eguqn»½|l}T}Tuqj@|@h2RTnmokegRTWX§WVuqj-U|:equgn»« 5TTugegRoWVuqUj@ugW@Â>egRTWugW|egn¾lW¦UX|lklmTnegD~bWc
jlriegRoWeqWVuqUXnmo|egnjlm¨¾|loWjlriegRTW,ÍTeqmTWcqc¦rTmoh!egnjlm&h!jlUX}o|lugWÀ~ [negR·eqRTW,egWugUXnmo|:eqnj@m¨cfeq|mD~T|u2~
~TWV¾dn|egnjlm·jlr[egRTWÀcfW,¾:|TWc¦}Tuqj:¾dn~bWWV¾dn¢~bWVmohVWÄeqRo|:eegRTW,jl}TegnUX|l[cfj@beqnj@m·[nmTjle¦>WuqW|@h2RTW~«
ZªmoWÄugWÀ|lcgjlmOrjlu#egRTW}TugWU|:egougW-hVjlmd¾lWugk@WVmohVW¦jlri ®±ªy ²q³´qµV¶q³ nc#egRo|e¦ne¦|lc#cgnUX}TaOmTjeXh!j@cgW
WmTjloklR,egj-hVjlmd¾lWugk@Wl¥*nrv ®±zy ²g³´qµV¶q³ cfeq|lufe2cicfhVnWmHega-rs|luz|¿|aÄrugj@UegRoW.cfj@TegnjlmÂDW|lh2R½cfegW}
eqj:¿|luq~egRoWcgjlbegnjlm uqWcgTpe2c#nm·|®~bn¢cxequgnTbeqnj@m [npeqR·Wcqc#¾:|uqn|lmoh!W½£Ýcfnmoh!W-egRTW~bncfeguqnTbegnjlmDc.|:e
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eqRTW§VWugjU|:equgn»o¤|lmo~rTugegRTWu[}Tugj@kluqWcqcn¢c[nUX}DjHcgcgnoW@«
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moj: >jluqugj: rugj@U
vegRoWÄ}o|u2|UXWVegWVu2cªegRo|e#ugWÀcfopeqW~Onm·cgohh!Wcqc|mD~O|lk@|lnmOUXjd~TnpraegRTWÄnmTnegn¢|
~Tncfeguqnobegnjlm«¯¬OW½ugWVeq|nmTWÀ~·eqRTW½~bnUXWmocfnjlm
n = 100Â}Dj@}TT¢|:egnjlm&cgn§W
N = 50Âcq|UX}TW,cgn§W
K = 25Â<|mD~ WÀ|uqmTnmok®uq|egW
α = 0.01«½¬OW,nmHeguqjb~bohVW~ |®UXjb~bnÍoW~·nmTnegn¢|~bn¢cxequgnTbeqnj@m [npeqR UXWÀ|m
µ0 = (25, . . . ,25) |mo~cfeq|lmo~T|luq~&~bW¾Hn¢|:eqnj@moc
σi0 = 1, ∀i«Q)j·>W©|locgjlbegWa¯cfTuqW®jlr eqRTWrs|nTuqW-jlrit¸ x¹ ºM|mo~·WVnUXnmo|:eqW-eqRTW-}>j@cqcfnTnnexa egRD|:e¦¿W~bn~¼mTjeWVe¦eqRTW|kljluqnegRTUugom
rj@u[|cgÄh!nWVmHeqaÄjlmok.eqnUXWlÂoWÁRo|¾lWÁW!»deqWVmo~bWÀ~ÄeqRTW egWugUXnmo|:eqnj@mhVugnegWugn¢|#eqjlÂlloÂ@lÁrTmDh+egnjlm
W¾:|o|:eqnj@mocV«KyikH|nm¿W¿}ougWÀcfWm@e*eqRTW[UXW~bn¢|mX|mo~cfeq|mD~T|u2~¦~TWV¾dn|egnjlmnm|lm¦j@uq~bWugWÀ~#}o|lnu*rj@ugU|e
£sUXW~bn¢|mÂ@cxe2|mo~T|luq~.~bWV¾dn¢|:egnjlm>¤+Â[RTn¢h2R¦|lugW¿hV|lhVT|egWÀ~ruqjlU zugTmDcV«K¬©W[|lcgjz}TuqWcgWVmHe<h!j@U}D|u2|TW
uqWcgTeqcÂdugWVeq|nmTnmTk¦egRTW U|:wxj@ugnexaXjr)eqRTW|>j:¾lWª}o|u2|UXW!eqWVu2c|lmo~jlmTaÄh2Ro|moklnmTk#eqRTW WÀ|uqmTnmok¦u2|:eqW
eqj α = 0.1« 5ojluXeqRTnch|lcgWlÂWj:Wu.eqRTWegWugUXnmo|:eqnj@m¨h!uqnegWVuqn¢|®eqj¼loÂ@l½W¾:|o|:eqnj@mocX|mo~
}ougWÀcfWm@eeqRTW#cxe2|:eqncfegn¢hVc¿rj@uz.uqTmoc«*tvW|lcgW#cgWVWQ|lTWT«
n= 100 f˜1(x) f˜2(x) f˜3(x)
t¸ x¹º £
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¾:|lTWjregRTWXW|lugmTnmTk,uq|egW
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ê
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Level set curve
Linear approximation of the level set curve
= Selected individual
= Unselected individual fitness function = sphere function
Optimum = (0,0)
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