Real Coded Clonal Selection Algorithm for Unconstrained Global Optimization using a Hybrid Inversely Proportional Hypermutation Operator Vincenzo Cutello
Giuseppe Nicosia
Department of Mathematics and Computer Science University of Catania V.le A. Doria, 6 95125, Catania, Italy
Department of Mathematics and Computer Science University of Catania V.le A. Doria, 6 95125, Catania, Italy
[email protected]
[email protected]
[email protected]
ABSTRACT
!"#$%&" !'( )* )+ , . /0 )213 45 763 8-9 8:# $ ")";=5vI? "7 )w> $?"T?2( AByxz#CW A"863)2A]$?0>5 1f $ ")"; w 86I"A3 8 >)W 3 > )i" 0 )& A67;wau^ AA 8 \ F )w)2>CWy > wI2-_a2j4jTkhl[m ) )2)2 80 $ @:)2 I . -_)2 #I $ ")";
Categories and Subject Descriptors
a0; x; } @*
.r?
(K:[[?}?r"?
?s[/[yKQ e" )& w >A)";
General Terms
k= 8 >)";
Keywords
kW2 8" ?a9f=n6I)& ")"J!aufik= 8 >)"JZ [n ", k= 8 >)"Jre=6I? fI0 \oh?0 JKk ]oh?2 J?mh$ *oh K;
1.
Mario Pavone
Department of Mathematics and Computer Science University of Catania V.le A. Doria, 6 95125, Catania, Italy
CLONAL SELECTION ALGORITHM
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. SAC’06 April 23-27, 2006, Dijon, France Copyright 2006 ACM 1-59593-108-2/06/0004 ... 5.00.
Z rn" " 5k= 8 >)G = @)2?" . " )2)Gau#
8 >)auk@ iCW> ,>\ f )2 "A\$76] >Z Gn" " c* " f8¡,? }I A"W¢r" !="0> )2)+(i)2 ,>B A I " £ xJ+zQ;Tau^ > )F ?,#CZBI ")2"7# I A ! )2 ^Z ># f} >TJ ?)2A^ ` ¤QJKCW> ,>\ ) )2 "A]$76B >w" [)2 y J? A3)2F W(")&20 A $ I II " ;/gi > )t,C
8 >¥I2-_a2j4jTkhl[mb L,NO yR2STSB}UWVWX+ ,; ¥ AI")i A A )WCW 8 >T> >,
¨ 8 ")5> >@ ")2)h(, H )0 ,JK 72 A" 3$ A ?2 I $ M$76y )w= ]>76I? fI0 y?0 , } A )2" " ^ > I !A©5 I 3I ")";ª2-_a&j4jTkhl/m ")2 A )y86£uCZ¦"7 8 ")"« 7 )dQkI¬ )0 T A®E"" )"; 3 h)& "]°JCZ}> !w f? ]±f² ³´+/)2 "wµ¶, 3ÆËʺ ¸ ¾ ¡,W )f y0 AI|If$?FCW 8 >d ( ÌA )&2 8$I KJ à ÄW ATÂÈ Ä @ >w CZ A3 >@?=$?AI)i >w ["AIAT $ ¼ Á J )2?" !"867;i ")2)w(, " 0>©Ë"" ¼ » Ê]±i¶ }?,0 '" )Z Ã,Ù > y"" !µ7Ö!×w ")r AI" = w 7 "A t? F± ²8Ø ´ +)2 "#µ3ÇTµ7Ö×4·sÜ Ø ;h#$?GKf0 )ZÝÞ ) ! )2"86B ?2 K f$%&" !w(, ] ; ]#$?#@#0 )?,2: "Ad$76d2-_a2j4jTkl/m AI, )2")";6
TAIA©$6^ >B: CW ~7 )" Ë · ² ´ ¾ A ½ ^ · !² ´CW> 3 "I ")2"7 )[ >G#0 @0 ' ! A # ) >G " )2)/(, # Z 5 "A} B ¸ ¾ ¡,_¶ >#} """I Jr ; ;@ >#A ")2 y+ >w: K;F ¼ ÀÁ 132 ·D2&¡ É Æ/ Ç ¼ Á !Å^:Æ5Ç ¼ 4 òÀ ø Ù ñ ¾ CW> ¼ 4 ò"À ø Ù 6 ñ · 5 ¼Á )' }0 A 6,>)2"B $ w AÆHÊH ¸ ¾ ¡ )
f0 AI #$?i$I0 "A3CW 8 >4 8( ¥A )&2 $I K;Za9W ) ?20 7t w> > >+ > GCZ=AwZ)2W 76 AAI ( 5 4"" f >wI $T8; 7=v J >@ $ 5 F )ii)2 AI ACW>"B 5 " # >= ")2)G: ® ; a9)& AJ= >y$?")&]"2 "7B ")2)] y )TAI, )2AT$76B f?, ""70 : 9¶ ¥"" )T \ CZAM © "5 M M >4?. « )]AI 2 A$76 >B 0 îï ¶/g£>ª \º"" i ) îï ÅD¡3A 8= )i0 )2"A3( b >@"I2 "7W? Y >@863,v" >} >E5 1!")W )"'CW>"4",0 B C ? >5)2" " d"0> )2A7")# C >5" © >=`"CW > >=$?)&Z ")2)t: }0ÒÐ ä:Oä:â Ot'" @,v )2 KJ! h" AH" AII )t >i / 8 )Z "7;+k£" A5ª"CW> ,>B)2"")2)&:86BfI0 ")"J! >'>76I? fI0 F> )2JCW > i # i~7 7 h¸I¶auF > ) C' "67JCdy"" )t + !"# F~7 !?2 8u6h =,¢r" !"86 ,v @ >h"0 A)2 ?;Gk=( ,= >w ' >f ##?0 )F >$?")&F)2I I ]D"" )# )2 ", A3( >h? ) ò ± ø ² ³´ A ò ± Ú ² ýÛ þ ÿ´ ;tauB > )'Ci "67J CZ5$0 H >C¹? ©± ²8³ ½ ´ ¾ WµyË"" )"J/:# > v@"0 y°*Å£¡¶ra9G864?µ >Gû«µ4«"" )w> !F)2I I !"AJ >ffÅ? PA@rÒÐÒCBOäQLà\L0NÒ,á2ÏO_LáZ0 AI86)2" ", )iµ É µ?>EDuA F
ò ± ²ýþ0ÿ´ ¶w . $ w!", h X À YJ [¼ » WA )= >}$I%u" !}: T 5 "
A » Ãw·÷QÂ[à Z ¾ Â]à \ ¾ ¶"¶"¶ ¾ Â]à ^r ¾ Âh» È`·ÌQÂhÈ Z ¾ ÂhÈ \ ¾ ¶"¶"¶ ¾ ÂÈ ^ "I ")2"7J[ ")2? !"867J[ > CZf Ay >?F$?AI)w >h . $ ")"J)20>3 > ¼?Á ÊH ÂhÃ8Ä ¾ ÂhÈ Ä !¶ @ $)& ")2)= A]¢r !")2)t2-_a&j4jTkhl/m C'#> !})2A\ C'" 8-_1ICW^$?",>5 1?iuCZ"7u6-Q > "w(, )t. D >I "iA 8¢r / " ")= _Q;+")2]( )f $?wAI A"A3 7 f > "h " ")WCW 8 >4A 8¢r "7W" vI 8 ")" A !( )iI ½ É ba ,JCW> ,># G " !"86} )&6w I "J?$W >FAI ¨"8u6] )2")@ ) >FA ")2 )=t > $ ")F " )2YGf A t: )3I dc É ½Ie ,J*CW> 0> > !f5 76y 7 G 5 IJr " )2"7w >)&}A 8¨" F. )2) G $ ")=:h5 76T ^ 8 >)"Y[#8 A : )tCW> ,>f"70 86#(Cª7" 5 fI I½ f É 3¿ e ,;*a9 )w , ")& 3 3 # > F)2}: )w> !f ~7F: )"g bh3 )5 ^A )2" ))& "d: ¦> ^ ^)2 I fTTY a )} B )&6\: ^ I T ] 8: 86HAI )&2 $I Ah0 A® $tCW 8 > 5 ¸ ¾ ¡,_¶a9w A : ) >#"!, ""#0 )@Iw5 \ 7 )&J[ )@ ># ª )55i
D9> 0jA FHI $T;£oh$ )2867JG:B#8 A : )h >~7 896y+ ># / ")28 )@ )w } ?20 $?" )25 8F l k", )F >5 $ 8u6Hi >A")2 A© 8 > Ò,mâ B Ï,NÒi. ¥ 7 KI 5 ;
3. EXPERIMENTAL RESULTS
H)2 0>º $ 8u6 2-_a&j4jTkhl/m#J=C' > !=)2"A3 F)2WKvI? "7 )ZB >wxzF(, )' "?2 A d _QJK 26I ># 0 ")hn 95 A í ¶? 0 , V 9B " ")2 )F >?,2""70 F)2AB 5A" " )2w >w$?")&W ")2)i: 4 w$I0 A£ « ,>«"0 d 5 "y >y ")2)] £ > 0 y ¸ ¾ ¡,_¶[c* " 26\ ")28#)2>Cß7A?2( 5 "")} I2-_a2j4jTkhl[mD)2 J 9F· r *ovB¶ 7[ ,>@ )&[: ! *¸i A?"AI"7? )rC', G?2( A )2 ª >\: CW £ 0 , )"£µº· ¡"¸¸ ¾ µ7Ö!×b·Þxª A îï ·w¡ *¶Gk'f ,>`0 `CZ3I B >5 d
' >#$?")&w@ A A )@:# *¸3 )w A\ >#)&0 A 0A AI" dCW> 0>ª ))2A ^ AI B >B")2 )& 6d >
8 >T; F?2( 5 "}+I2-_a2j4jTkhl[mº )=" A CW 8 >«)2y >BC'" 8-_")&0 $ )2>A«"! I 26£ 8 >)"J )2,>® ) xwä yhÒ,á2Ò,àOäQÏÐiÍiÎLÐ Ñ OäQL à zfÔ=Ïá,OAä B"Ð Ò @ {GÏá0} æ |/NOä(æf[ä ~ÏP OäLàJ A @ä:æN?ÐÒ=ÍiÎLÐ Ñ OäQLà?Ïá0ãUÐ ;!Lá,ä(uO æ Q; a94 >F: CW J[C'fI ")2"7h ># ")28 )@$I0 Ay$76T2a&j4jTkhl[m#Ji A¦" 3 >" CW 8 >¦ >4)3$0 A¦$76 Y=3
JI ?)2AT H ¤QJCW> 0>T)2")W$ 26])&2 # " ")20 K;`a9`0 $ 4x4C'3 ?2 >]$?")& )28 )$I0 Ad$76 9C'BA 8¢r h1 A)=+a&k=)"Jr)2 >})
##0 T? )";w )h l k" )2-_a&j4jTkhl/m#¬ )w $ 8u6 dÒ,mâ B Ï,Nä(=à ;@( 7 /I 5 I; G $)& ")2)t A#?,2: 5 ")*tI ?)2Aau#'k 8 >TJC'W> !i" AfI2-_a2j4jTkhl[m CW 8 >T >I "w >WC'" 8-_1ICWT"!I 26T 8 >)"J)2
©AI 8¢r "7],v?, 0 I 7"@ ?)2AM Ë Q xwä yÒ,áP Ò,àOäQÏÐÍiÎLÐ Ñ OäLà4pwqG ,JÔ=Ïá0OAä B"Ð& Ò @ {GÏá0 æ |/NOä:æ#[ä ~ÏOäQLà]QcZnow
A«â0ä(æhN?ÐÒ5Í'ÎLÐ Ñ OäQLà?Ïá,ãHUÐ ;!Lá0ä:uO æÌ_nq+k@ ,;+auª AA 8 ¦ >i" )2)2 ?cZno#J >FÏOOá2oÏ BOä:ÎÒ@Ïà Fá Ò9N Ñ Ð â ä:ÎÒ'nÔ @(|dQ 0cZnow C' )G ?)2A 4 QJ7CW> 0> )G wA 8" )20>':'cZno } ! .A5 w 5 i! "";+h(, )"J ?,2: \(5 ,>` )&0 "Tz¸^ AI"?"A"7} )";dj] !,J7 >= I > )+ ")& A > 8 >)G)2 ¡¸¸#AI ")2 ¦ >3h : $ ) K¡ É K¡"zI¶Zauª > ) )2J ð ñGòó Ci )B)2,B *fÇ\¡¸ ¶ !Z 72 A"A} C© f+ 8 > $ )2A5 >W" ?)2" " B " J A5I2-_a2j4jTkhl[m L0NO " ©R9f}U' 8 >B ,J?(w $ t K; 5 f: ")+? >'I2-_a2j4jTkhl[m£ > ' >'( Ci t" F?0 J ! )2"86f ?2 ?>76?, f0 ?0 = AB }?,0 ;GG )2"86F ?2 I>6I? #0 w?2 $)/ ,># A A )2 \ I )2"86TI ?2 K B 8 )= )2)=(, y ; 7/ 867JK > T?0 F" ")@ >fAI")&} A A
)2 I )W( >@"2 =? 4 T0AIW 72 AI" A !, )2 965 A5 } ! .A5 7 r 5 FAI w >="!I 26 )2 ,>TI 7"")2)} r Q; vI? "70 )& A ")[)2>C© > [ >G ?)2AF" )2" "
8 >¥ )= T¢r !w 4 8 >T; 5. ADDITIONAL AUTHORS
khAA 8 ! > )"/mh )2"?th " )2 p@ 2 rjT >"5 )[ AFZ [n" ""J§ ! )2 896h' 0 . J"5 o d(Iojb 3b
6. REFERENCES
8¡,#Z " ¢¡#; Jh ")2 5m#; !D& :hmh $ Kh, " oh J F yÔiá&wL Bd©+L «wuO ¬ Ò @rÒ,ÎÒ,àuO ]R0àªO © ¥ L@ [)2" " 4 " l F; R&ÍGÍ+± Í ¨á&Ïàlâ ©*Là]ÍiÎL[Ð © ¥ LæhYN ©JI!Y +IJzJK;xz r -uox *7¡J _x¸¸x ,; +Fp@F' )&2 5lt;}; J)i h, " Z",>5 1Bct $ ")"; F ¥ L?à ;á Ò,â â}Là]Í'ÎL[Ð © ¥ LæhY N ©·z ¥ Í ¥ ¯l¸IJ!Q;¡J;r¡ r ¸- ¡ r _7Jt_x¸¸¤7 ,; r Fh ")2 5m#; JrZI " V ¡#; Jc* }j^; ! D&k e6I$ A aufk 8 >bCW 8 >TauI: 5 Tm@ ]: > m0 >4Z 5c* $ "¢ FJ+X=Ò,àrÒ,O¹ä BÏà #Í'ÎLÐ Ñ OäQLà?Ïá,ã ¥ LæhN Ñ O_ÏOäLà ¥ Lw à «Ò,á2Ò,à B0Ò,Jrm@qGiio¹x¸¸zIJ?lwin?J!; x _xzIJK;d¡ _7¡- ¡x; o
ºY &»I
ZKo´¼f([d½dIç[I"Ir".? }r\ç/:"?ª ç/è é,DDêBëiìy ú [:/"Iª
+ç/I"I(G ú ( *
(g ç/è é,êy ú /(M["`^/:[d½¦"":[©ç/"Ir""(?¬»Iw/
("BKI?3tII¦?Id?¾j¿:/[çtI/[Irb//o \` · u² lÀs´ \ · ½ ² ´ ç/è é,DDêBëiì ç/,è éDsê3ë'ì =3
Y
j] TZ")& j] 4Z")& j] 4Z")& j] 4G")& _n70ABp@ _n0A5p@" _n0A3ph" _n0A5p@" c 9 ½ ¿  ÁÄÃÃÅ/ÆÇ=¿ÈÉlÊ ¿ [ Á ¿ ¶ x z ^ Ç " ¡ ¸ ¡ ¶ _fÇ^¡"¸ ½ r ¿Á[¿ A x7¶ ¤¤Ç^¡"¸ ½2½ ª ËÁÄÅþ/ÆÇ=¿ ÈÉlÊ zI¶ *fÇ^¡"¸ ½ c ¿Á[¿ ¿Á[¿ ÅYÁĺº ¿ÆÇ=¿ÈÉ #5 )8' ø \ %5 ( ø16Z (5 ,59 ø 5 '$'&% ø \ ! , #$%) ø Z %5 ./ø O '5 ,= øZ ü&ýCþlÿ
ø
ý
") #$&% ý ! #$# ø " # ) ø+! *\ - #./ ø " '$) &% ø ! \ , ' ø ") %$ ) ø ! \ "# ø (-- ("8&% ø 8# , #5 ) $ (5 # ø+* ") 8&$% ø+"*\ ; O # #$(8,=$%./ "; Z * %5 )$) ø Z \ - &,"$./ ø %5 " ø , $($&$%= ø+ *! ! (5 %$ ø , '$ ø 7 5 $9) ø ) > - , #8, ø "O (&,= 3 ø ü&ýCþlÿ
*
(  hd 9\/T}r\ç/:"?^?[4ç/è é,DDêBëiìyÜ$ÝF}TÞ3}TÞ¥/dÝê ?^[ß*"V Å$ [/(/ éê ë8ìjí îïsë8ìjí ì ê(æ ú (£/\[(?Ó=¿ ¿ àoá=â3ãIäªågåæ"çè
Z \ !
* 3
7 & > 6
0 Z2
ZªZ ZA\ Z !
Z ð¬ñMò ô?õQös÷uø ùmú ü&ýCþlÿ ý ý
) ) #5 ( $% ø+* ! %$#5 ) ø '&, #$( D:I@L ? CED51FHG JBZD5BMNCED51FHG ,: #$./-; O ) $= -; \ * DBIDMNCED51FWVX L ? @B@C\D51FWVX ' (8,= \\ \ % "9 ø " #- ) / ø \3\* !Z %5 ($#$%./ ø ø
ôsû ð¬ñ ös÷ø ùsú ü&ýCþmÿ ý ý
,&,"#./ 7> ( (&($./ ø #8,: %&% ø (& %&% DBA:QNCED51FHG ? LDCED51FHG , 8 "; O ) "./ -; \ * ' ("8&% \ª\ \ ' $$'&' ø \CZ - &$% ø * % $# ø \ ø *
ý ý '5 $#./ ZZ22 " $(./ ø D:ZDJ:MC\D5ø 1FU] DB RJNCKD51FWV^ # &&,= ! '$./ ø R+IDM:QBMCEø+D* SUT P DBBLNCED51FU` ) (= ZZ3 3 " #&,/ ø '5 , ) / ø \42 : ø D:ZD-RJMJR JP @RaCED51FU[ ü&ýCþlÿ
Yý ý " &&) ,= -; Z Z 5 ' "; (5 $& -; ! ! 5 %&,= ) "; '5 (- 5) $(.) / \Z ";ø #5 $) "; \ ) ( ' ) #$8,= \ #5 (-Y/ ø \ ) 8'$%._ ø "; O ) " ,$ #$($('$%/ "-; ; ! \ * , "( "; Z ,: ,$% - #&( , 8#./ ZZ 5 ,"'./ ø Z " #$#./ ø Z " #$'./ ø Z (5 $ ø O %5 ,"#./ ø Z ü&ýCþlÿ
ø
Yý
#) ,$$$% /ý -; ! \ (-. "; () %&$($#/-; Z Z ,"'= "; ") - # -; Z Z '&)