Kita, e. 2010: investigation of self-organising map for genetic algorithm


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Kita, e. 2010: investigation of self-organising map for genetic algorithm

  1. 1. ' Adv•ncH 1 EncutfflnJ Sofrw•tt 4 1 (2010) 148- JB n Contents 11111 available at ScieoceOirect Advances in Engineering Software Journal homepage· www Investigation of self-organizing map for genetic algorithm Eisuke Kita ·• Shen Kan. Zha i Fei ARTICLE INFO Ar"dr.tustQI')' lil:«t"l..,fd 29 june 2007 an fCVIK<l. rorm 11 Sc-pft"mbtr 1009 Acctpttd 18 September 2009 A~.-11.-ble onlillt 27 Octo~r 2009 Kr~utr CAIf'UC .~>lpnthm Sdf~Wl'llrn.JP'J ar.,-<:odc'd G.A ABSTRACT This P-iPlf dHcn~s s.tlf..(l'l<fntzang: maps for ~~rwuc •1aonchm (SO'A·GA) 1.11o'hich u tht combln..~hono~~l ,.lgorithm of .o. rf'AI-<odfd &cncht,. .dgorithm (RCGA) And sto1f..Of1"'"'1'f11 rNP (SOM). TOt nlt~orzjnltln, m.1ps olft frilntd wnh 1~ m!orm.JlJOn of the iOOividu•ls •n 1hc popul.atton. Subo-popu1o~uon! .lrt defined by the help of ttt u.,ln~d m~p. Tht RCCA is performed In the sub~popul.ations. The use of 1he sub. pop. ulation se.1rch .ll&Orithm '"'lnovc:-s the local st<11Ch perform.lnct of the RCCA. The sNrch perfom1~nc~ n compared with che l'f'iii·Coded gcnettc algouchm (RCCA) In three cest functions. The results slow that SOM..GA CAn find btUtf ~olut 1 ons 10 ShOI'tt-r CPU liMt than RCC:A. Altll<lugh the computMiOMI COS1 ror rr.umng SOM u txptnsiw, lht' rf'sults sllow tht' convtratnC'e s})Hd ofSOM-GA is .1<'Ctler.Jtf'd ~CC'ord­ mg to tht dt''tloptMnl of SOM tr~mmg.. T"Cn'-n Copyn:,h.t e 2009 PubhshN: by ElSC!V!.er Ltd. All "Jhls rtw"rVrd 1. In troduction An anginal genetic "lgorilhm i.s designed for optim1ut1on probl~ms with binary-coded d~sogn vao ·iables. R eal-<:oded genetic algonthms (RCCA) h>ve been studied widely for oprimiwion problems with real -cod~d dosign variables 11 - 3). The authors will describt th~ us.e of sel•ting Map (4.SJ in order to Jmprove t~ sedfch performance of rhe RCCA. The combonanonalalgorirhm of the evolunonary algorithm and the SO!Iol has been prHOnred by Buc~ er. alj6~ Theu algonrhm os n.amed ...s stlf-orpn•nns: nups for multi-objective evoluuon~ry ••gontluns (SOM·MOE/1~ The SOM·MOE/Iu designed for 1~ mul. tt·ObJtCtive opt1m1 z.tt1on probl~ms. The muJti...ab}«tJve opumtZ~· uon problems do not howe ~ single optimal solut1 00 but P.ueto solut1ons. In the SQM..MOEA. the self-organizing m~p is used for sroring the mformation of the Pareto solutions and 1m proving the convergence spe-ed. In the crossover operation. one par~nt is chosen rAndomly from the population first and then. An· other one is chosen randomly from individu.als related to tM dos· est umt of the best m.atch unit of the first puent. The crossover ~unon lS performed once for uch pair of parents. Nohu th.u two p.irenrs .Jte c~ Without reUuon ro their fitness v.11lue-s. In th< CA. th< average fir~s oft~ populatton should ~ enhancod Keord1ng: to the d~lopment of se.uch. The crosso~r oper.11110n of th< SOM· MOE/1 chooses tht parents randomly and rhtrefore. the average fitness of the population 1s 1m proved very slowly. For overcoming the difficulty. self·org_.anizjng ma,ps for genehc algorirhms (SOM-GA) is presented in rhis paper. SOM·CA i> the • (01-rt"~pondll'¢ ~UihOf f. ,.,.r oddrtss.. br~."-'1'C>Y.Ii u M JP ( E.. Kn~ ~ combin.nione~l .-.tgonthm of tht redl-coded genf'tK <~lsornhm (RCCA) and rhe SOM clustenng. A self·organizing map Is traoned w1th the obJeCtive runct1 and the des1gn vanables oftht mdtvld· on uals in the population. The best m.uch unit for an individual ts chosen and then. • sub·popul.rion is defined by tho ondlvoduals included in the circlt centenng on the best match unl1. RCCA is performed iteranvely m sub-popul.nion .tnd thtn. tht obtamed ~sl mdividuals are added 10 tilt ntxr wholt population. Since the sc.trch 1n e~ch sub·popuJ.-.non is perfonned 1teu· U·tly xcording fitnHs.. the con.-ers:t-nct ~PHd as 2. Background Before explanation of "'lf-or&mozong rmps for ~ntric •110' nrhms ( SOM-GA~ wo woll onrroduce "'lf-org•mzin1 nwps (4.5) .md Stlf-org.lniz•ng mc'IJ)1 for mu l li-o~jeaive evolunon.ary allonrhms (SOM ·MOE/1) (61 2. I. Self-organizing mafiS (SOM} /4.5/ The sclf-<>r~n~>ong map tS • single J•yer feed-forward network where the outpul synu.xe-s ue ~rr•nged in 8J id (F•g. 1). ~ch mpul 096!. 9971 S • we- &ont ruttft' CNWD Cop)'lllht ., 2009 Pub}lshf'd by Ekn'tn LJ4. AU nPt$ rt$lttWC1 dnt 1Clhl l ti_,~WI--.l09 "'0 11 lO lndrv•d~ls' faster than rhat of rh< S01 MOE/I. The SOM..CA performs rltt solu· uon se.uch m uch sub-popul.-.uon. mste.1d of the solulJOn scuch in whole population 1n the RCGA. Therefore. the: SOM...c.A shows beuer local search performance than the origo""'l RCCA. The remaimng or rhe pAper lS organited as follows. In S~C'tiOn 2. the background of the ~tudy is described bnefly. In ~ccion 3, self.. organizing maps and rho SOM·GA algorirhms are explained. In Secuon 4. SQM..CA IS compa.r~ wJth RCCA in thrcl' num<"'riC.ll ex~m.. pitS. Finally. 1he conclus10ni ue summarized in Sect10n 5.
  2. 2. ' 149 L l<lti2 ff ct/AdwmciJ t1t £nttllM1ttgSojtw4ut 41 (1010) 148-ISJ the md1Vidu41ls. ~ mdt-...idu~ls ~rt gentr4lttd 4lccordmg to SOM nttgh.borhood evolutaon .tnd SOM muto~taon .llgonthm1 Un•ti Th< SOM-MOEA algorithm is as (ollows. Now "''e consider the notAtions I and r dtnott the ;md the not.lttons U .1nd V denote the best unu for I and f. rt'Spectivt"ly. Tht" notations .ut ttd in F1g. 2 • .,. 0 • '""'''a)'ef • v' V" is connected to ~111 output nturons. A 'ti,ht veeror with ~~ same dimension.a1iry .u t~ mput 'ectors IS .att.Jched to evtry neuron. The number of 1np1.n damensions as usu~lly ~ lot hagher the output gnd d•menston. SOMs are tnainly used for d•menston.ahry rtducuon r.arher th.Jn expom.Ston. The we1ght vector w at the unn las gwen .as w' ~{w',. w',. Ill',)'. ( l} And the mput vector r! IS as (2) Taking th< Euclid dislanc<l II - w'l as • norm. the best match unit is selected so the norm 1S mmlmtz.ed The besr rnarch unit <(•') can be d<fin<d .s c(lll- argmin ( II (3) w' UJ Once determining C( v ). lh< w<lghl ve<:tor Is updot<d by w'(l + 1) =w'(t) +h.,(tl(ll(tl - h.,(tl :r,<xp ( r, 14) w'(IH wMrt r denotes the dascreuud ume; r hood fur~uon /l.,(t) tS d<fin<d as 0 12 Since the neighbol'hood units howe t he sim1l.u parameters. tht SOM neighborhood evolution tends lO search ne.u the p.11en1 indi· vidu.&ls I and r. Therefore, the SOM mulation is adopted fos improving the glob.1l search performance. In rhe SOM neighborhood <volution orSOM-MOEA. th< p•l'<ntS .are selected without relation to their tirness values .&nd the crossover is performed only once between #.1nd r. Since the mdaviduall' is selected randomly from the individu.&ls rtl.ated to the untt U .the off-springs are often worse than rh< p.renrs. 2.3. S0.1-C.A i S) Apply real-<:od<d gen<ttc algonlhm (RCCA) 10 an tnlltal popularion once to gener.ue .t new popuJ~non. 2. Tr•in self-organizing map by ••Ions the valun or an ob)«ttv< funcnon ~nd des1gn vMi~bles of tlx md1vtdu.tls ...s t~ mput r, ') whtrt r .and rc dtnott tht pos1t10n vecton of tM unu '•nd tM btst TM cotfftc1ent 2, 1S someurnH definai .u R TS (6) wher< atOl =R and <T TS) • 0. The paramtlers TS •nd R denot< rhe number of tr.aining: step .&nd me tnttl~l r~daus. mpectavely. The SOM algorithm is as follows: 1. 2. 3. 4. 5. .1dded to design variables of the individuii!IS. The conceptual dtagram of self-orgilmZm& m.ap5 for gentuc algorithms (SOM-CA) is shown in Ftg. 3. • monotonxally de<:r<as•n& on< In thiS study, howtwr. 11 "d<fin<d ~s rM const~nt W1thm the r~n&e ofO,. 2:, ~ 1 Tht funcrton Ci(t) is d<fin<d •• Glll - C1lt - H 4. Perform SOM mur.nion so that the normal rc1ndom numbers .ue Th< n<ighbor- 2C1'lt ) m.~tch unrt. rtSpK'tlvt'ly 1. C.1Jculate fimess of mdtVKiu.1ls m a popul,uon 2. Train m.&p with the mform,JttOn of the mdividuals. 3. Perform SOM neig!lborhood eYOiuuon .s rot lows: (a) Select .tn individual I randomly from a popul.auon. (b) Search !he b<st match unit U lor the 1ndtvtduall on th< SOM map. (c) Sel«t th< dosesr unit If for the umt U. (d) Select randomly the individuAl T from the ondtvldu•ls rel.tted to the umt fJ. (e) Generate • new individual by applyong the crossov.r operation to the mdivaduals I and 1'. I. d.a.t.a.. 3. Search 1he b<sl march unu U for each tndtVIdual on the map. 4. Define a sub-population by !he ondMdu•ls tnclucl<d m the orcl< antenng on the umt U. S. Apply RCU. ro 1h< sub-popul•rions. 6. Add best ind1viduals m the sub-popul•ltons 10 a populnion. UnotV lniualitt randomly the wetsht vectors Ill. Select the unit c(r!l so>< 10 satiSfY Eq. (l~ UpdAt< the w<tght ve<:tor w occord1 10 Cq. (4). ng Updatt the n<ighbothood function according to Repeat from step 2 tO Slep 4. Eq.(5 ~ 2.2. SOM·M0£11 Self-org•niting maps ror muhl obje<uve evolutionary algorithms (SOM-MOEA). wh1ch was pr<s<nt<d by Buche fl. all61. uses the selr-organizmg maps (SOMs)for k<<ping 1he lnrormotion ofthe tht Partto solutions. After tumma SOMs with thr inrorm..ttion of • lnd1v1dual / n<W
  3. 3. ' 150 ........ ....,.., ............ .... .....,... _ ... - ,. , Q ... It· ; - / I I I I I I I I I I :~: '" -- - ------c:JY- I -~ --- ,__ _:..:N':.:.' ~ ConvergeneeChecl< ~ ... End 7. Rtpellt until .1 desared .soluoon ts obc.ained. (a) Th< SOM netghborhood <v<>lurion or rhe SOM MOEA u ~· fonned only one<. Unless adequate parentS are selected. the SOM neighborhood evolution generates worse off·spnngi than che p.uents. In the SOM-GA. rhe RCGA operation is performed lltratwely among multiple individuals in every sub-popularions .1nd there- fore. It n n find the best or betttr individuals in the subpopulations. (b)<t 1wo indtvtduals from 1he popul.rion P"(c) by roulette selt<t10n. Apply Bl.X l crossover 11.71 to generate new (c) Repeat unul generating M mdtvtdual>. mdividu.als.. 6. Train Map With the values or the objective func1iOn a:nd lhf" design variables of the indiv1duals in the population P (l) according ro Se<rion 2.1. 7, Ceneu.te new individuals u rollows: (a) For txplan.cion or the SOM-GA al&orithm. we will take the fol · P ri i f'ldividu~l P1tl). Perfonn RCGA oper"101' m the sub-populotion SP.I£1 to obt~tninl bt-st tndtV1du~l p, It -- I ). (c) Repeat for <"etY mdtV1dual p, 1 1 ~ 1 2. . M . 8. Terman.ue process 1f convergtnce mtenon t.S uusfitd. tnt (b) low1n1 not.Jtlon.s: p,IIJ M SP,(II Map /In Define sub-populottons on Mop by all individuals within ~ udtus or Rn from t~ best mouch unit for pOpul.uion it Jtneu.uon r .tn tnchvtduo~lltl Jl popul.mon' r) numbn olmdaVJdt.:o~J.s tn Slb-popul.uions xenero~:~ a ~Jo~~non Pir from" popul.1Uon ,.,r) self-orpmzmg maps r<~dius o( su~popul.tt10n in SOM·C.f 9. Connruct ~w p,{I- H •-12 popul~uon .M). P' c + 1, by mdtVIdu.J.Is 10. Update 11me step: 1 - 1 I. I t. Go to step 4. tx~mples. The SOM-GA algorithm (Ftg. 4) is as follows: In the numerical objective functions. I . I 0, 2. Initialize self-organlzmg map Mop. 3. Cener.ue r.mdomly M indivtduals to con~truct Ill' initial pop. 4. Numerical examples ul•uon P"(l ). 4. Ev.atu .u~ fitness of md1v1du.als. S Crner.te the population P (!l as follows: The SOM-GA ts applied to st•rch optimum solutions of test functions: Ra.S(tigin, Rostnbrock. ~nd Cntw~nk funcnons. Tht results ~re com~red w1th RCGA. Tht test funct1ons .are taken .as the test functions are t.tken a:s the
  4. 4. ' t th~ obJKUVt 1S1 KilO tf.t/Adooi'JCt11ll £1tlfltn'f'ttiC So;ltwcvt 41 (10J0J 148-JS.l """ .... .... funct1on.s 4lnd thertfort. tht SOM map 1-' tr~intd with th~ vo~h,;t'S O( tht ttSI funtCfOOS .and tht dtSIJ.T1 :00. V.ln,)blts 1.... 4J Tnt jimcnons 1001 ""' 0 4 I I Rasm&m functton R.lstnpn func11on as drnn<d by FIX)= IOn- tv<- lOcos 2~:><,)) ol -512<. X ,<512J mm!F(x)J = F(O.O .0) • 0 In cue of rwo dtsacn v,an.-.bles. the fun<tton as shown tn Fag. S Since Rastng1n function is 1 mu1u~mocbl function. it h.u one mtnunum .and m.1ny m1n1m.1~ Oestp v•n•blts ue tndependent uch o<h<r .. ~ 120 100 In thiS paper, the num~r Of tht dtSI&n YJrtibltS IS Specified olS ""'··' = 20. eo eo 4.1.2. Rosenbroc~ function Rosenbrock function is deRn•d by 20 0 F(x) = •·I I) tOO(x,., •·I •• x,l'J -xil'•( l (-2 048 < •• < 20481 min(F(x)) • F(l 1.. , I )• 0 (8) In us• or 1wo design varlabl•s. the function Is shown m F1g. 6 Rosenbrock funcuon is .1n sin& funcuon .and there 1s the dependence between des ian vart.ables ThiS funcCion Is one or functtons which nn not be solved by CA t~J1Iy. In thiS ~por. <h• numbtr of th• design varlablts 11 S~J«ified as = 20. N4 • Critw<~nk function Gnowank func11on is d•nn•d by 4I J =1-t~-fl(cos("')) .. , .. a I Fx (-5 12 < .. < 512) m1n F x)) = F!O 0 {9) 0 .o In c.a~ of two cks.lcn v.ari,blH. che runcuon as shown an h&s.. 1 .1nd 8. In Gnew•nk ru.ncuon. ctwre IS the dependency bt-rween de.. Sign v.lnibl~ Thtrt .ue one &1oiNI opumum •nd m•ny loeil opri- so!uuons In th11 ~per. th• numbtr of <h• dts&IJI v•nabl•s 11 S~J«&fied as m<~~l N,.- 20. •• .. 10 Common par.1:me-ters for the SOM~A ""d tht RCCA irt hsttd in hbl• I. Number of th• des1gn vanables 11 20 Popula11on Slzt IS SO. Cros.sover r.ue and the mut•tlon rate .ue t.O •nd 0.002, respte.. t1 vely. Rouleue selee:non .1nd BLX-;r crossover oper.1t1on.s ue .1dopred. In the real-<oded mutcltlon oper•uon. the values or the design v.ariable-s •re ch•nged r•ndomly. 20 10 0 T.tbl~ I lO Common pcr.amttf.'rs for bofh .llgonthm.s. Nutl'lbe-t o( destgn vcn.ablf.'s Popub.1ion sin N4.•l0 ~ltction "· 50 St!lKUOn ROI.II-tUt Crossovtt BLX-2 Crossovtt r.are 1.0 0.002 Muu.hon r.ate Num~r of tri.tls n a. s. R.mriJul furKIIon ul cwo-duntMIOn 10 A 0S
  5. 5. 1S2 Addotoon•l por•~t~s for tht SOM-CA .,. shown on T•ble 2. The n~s&:hborhood r~chus Rn IJ tht r~di us of tM t•rculu rtpon oo the self-orpn~ton, rmp who<h co,;ers • sub-popul•uon. The ~r,llon number 10 lht sub-popul.t~Uon ~ 1s r~ number of tht RCCA oper.uons on toch sub-populouon. use th.n the map stze of 30 )( 30 1s bfi1 from the vttw.-pomts of the ~arch perfonnanct and th< CPU umt. So net tht most •clcqu•t• map soze depends on lht problem to be sol••«!, we t•k• • srmll stze. t.e.• the s1zr of 20 " 20. m the foiiOWtf1 u.amp1H. Ten rri.lls .ue performed tn every uses from d1fferent amtwl 4 J.Z. IIO><nbrock functiOn Convergence htstont>S of the ~~ mdtvtdu.ats ut shown 1n fig. 11. We norice from Ftg. 11 tlul 1he ob)t<tlW runcuon con verges to 0. 1 on 1/CCA but on SOM-CA. it goes to 0.0001 . In 1h1s case. the CPU tmw of the S01-GA wtth SOM lralnlnl time is more th.m tnpled .u long u rtut Without the SOM tr•lnlns rime. This indieartS that the computationo~l cost for SOM rrammg IS more expenstve than the Rastngm function populauons. Mt.t~n v,)Jat-s ut shown m tht numtnuJ rHults.. Moxomum sener•uon steps II, •re shown on n ble l In tht SOM-CA. rht RCCA oper•uon os perfo~ on evel)l sub populo· t1ons. The escim,.uon r1me of fhntss func:t1on 1s ~ual to t~ produo of the> number of m.vumum ~ntr,.uon s:ttp. tht popul.uion s1ze. .lnd the number of the RCGA o~r,uons m e-vt:ry sub-popui..JtJons. In order to equJ!ize the tsflm.atJon tames of fitness funct1ons m RCGA .md SOM-GA. the Jtntrauon seeps .ue spec:"lfied .according 10 4.3.3. Criewank [unction Convergence histories of the ~st indiv1<1u~ l s ue shown tn Fig. 12. We nonce from the results th.u the conver&ence sPf't'(l or where (N1 lJCGA .t~nd (N )su...,GA denote the nuxsmum gent-r.uion step 1 on RCGA •nd SOM·GA. mpectively. • [ •03 - - - - - - c-c~ 1 (.(1 4.3. Numerical rtsults 4.3.1. Rastrlginfunctlon Convergence histories o( the best endlvldu~1s a1 shown m ·e Fo 9. The •bsmsa and the ordinate denote the CPU tome and g. the objtctivt funC'tiOil or tht best individual, respectively. The· RCCA and SOM-C/1 resulis arc label«! with ··seA· •nd ..SOM· CA•• respectovtl~. TI1e convtr,ence speed of RCGA Is faster than the SOM-GA in the tarly Sleps. On tht contmy, the convergence speed of SOM..CA IS slower than the RCCA Ill the .. rly Sleps •nd rhen, ts .acceler.uc.'!d when the ObJ«liVt funcuon vtlut IS smaller than 0.001 . A fina l soluuon Is much bentr than that by RCCA. ~ t t<-e• s. t-o~ ......, ,.. ,... __ ' ,,. ,., ,,. CPU'"_ , ... ... ... The CPU nme oftht SOM.CA Without SOM rr.a~ning tame IS .1lso shown on 11t fisurt. The C1'IJ tome oftht SOM..CA With SOM train· ing rime as rwice ~~ long u 1~1 without tM SOM tr.ammg tune. Thas me-ans that t~ comput4lllon,.1 cost for SOM tr•1mng 15 "-lmost tot~ re~an1ns cost. ·..... ··- Nrxt. Wt would hlce to diSCUSS the ttTKl or thr IJo)p SIZe to the starch perform.~~nu. ~ mo~p Silt v.u1eS from 10 10 ro 80 '( 80. Tht results .ne comp.a.rt'd •n f r&. 10. Tbt ,.I»ClSs.t •nd the ordmatt denore rhe dtmensionlus CPU runt wuh tht nwumum CPU tune . ,.,.. }I' of m.tXJmum mAP Silt ~nd the vo~IJt' of obJf'CliW' funct1on. rtspecnvely and the l• bels denote the rmp soze. Wt nouct that tht CPU rime ancre.u~s .Kcord~n, to the rNIJlltude of the map s1ze. When the m•p size IS 110( sm•ll. i.e.. lrt.ttr than 30 • 10, tht fin.~l solunon conv~r~ to aht same soluuon. ~~fore. we c.1n s.ty m this T.tbk 2 '"- 10. Thf' mult of lt.utn11n functtcm (Uf«t of nwp tlzt). AddltlOn"l ,..r.amrfM• 101 SOM CA .lncl rypc netetlbomood r .td u1 M~p m.t •· • o Tr.unuta ,.,~ o, OS TS'- 1000 .N1'., . Tr.-inin& tlmn ~•zh.borMod ..... Hel'I•JOn o( 20 • 10 r"dms Opernion nu m~t in u!). t £<03 1£.02 S(>,I -(J, ..,._I(ltol, ........ - ! 1 1000 ;;, t £-oOI ~ J 1(•00 J,,.., 1(...01 !E-1!1 T01bk J MMimlm ttMurlon nep ,~>~,. ____ ..,_ , l f -Q.II • · -...... ···-·- IE-ol' - 0 1.000.000 20 20.000.000 200 1.000.000 20 ~ tOO :so .. , ... fir,. 11. 1M rnlll of Ro~nbrocl>' lun(HOn "0 ...
  6. 6. ' IS! l [•)l •nd tht results are compartd wtth tht RCGA. In all US<S. tht ------------ SOM.c.A find bettff soluuons 10 dW)Ittl' t'Omput.11t10N11 11r1~ than rht origin•! RCCA Without SOM EsptO•IIy, tn tht uses of 1 (..;)• ~1(-¢10 Ra.nngm and Gnewank funcnons. t~ computatKMU.I umt of SOM.CA is much shontr than that of RCCA. In tht cost of Rosenbrock function. the tom< ofSO~.CI os much lonc<r ttwn them m the other funcuon.s. Whde the' Rostnbrock funcuon Is .} , (. ' .;_ I (J.I'' t ,,..,. __ ..... .... • smgte. mod.JI function. Rastrigm and Critwank funcuon) .ut muJu .. 000 ... ""'- modal Of')('S. Therefore. lhls result rrwy md1nte 1tut SOM CA IS dft<ti~ for mufn .. modoll funcnon rather th.11n rM sanctt-mocb.l ... one. By the w.ty. rhe numerial results inchute chat the w.uch ptr ~ form..Jnce depends on the s1ze. If a suftioently l.ugt nup 1s used. it un find better $0)urion .and on the conn.ary. the comput<~· tion.11 timt b«omes longer. In the future, we .ue cons1denns the reducuon of the training cos1 of the SOM .alg_orirhm and the exten- the SOM.CA IS much f.,ttr than th., of RCCA. Whilt • RCCA solu· tion converges toO. 1• .t SOM...CA .solul10n re.achO.OOOOt (An exact solullon JS 0 0). So, wt c.n see that tht SOM.CA can find better soluuon the RCCA. We notice that the compu ..tional cost for SOM tuining is almost equal to the rema1ning cost. 5. Conclus ions This p•per descnbed lht self·organlzlna maps for genetic •lgorithms (SOM-CI) fo•· real-v•lued single objeCtive funct ion prob- lems. In the algorithm, the self*organlz1ng 1 naps .trt trained w1th the v.tlut-s o( rhe objecuve funcnon .tnd tht' des1gn vari.tbles of the individuals in a popul.toon .tnd 1he sub- populc1tions c1re defined by tht htlp of lht SOM clust•nn&- Tht rul-<odtd gtnttlc •lgorirhm {RCCA) IS •pphtd 10 tht ondovoduals on sub-popul•uon. Tht prcxe~c:es .111e repe.tted unul c:•usfymg the convergence entenon.. The SOM.CA 1s .tpphed to seuch opumum 501uuons m tt>st functions such u R.tstnf!n. Rosenbrock. o~nd Criewo~nk funct.ons Sion of the .tlgorirhm for reducing rtte convergence speed m the singlt-modol function. References Eslwlm~n LJ. Sch.affH JD.<oded ceneflC .IIJQI'1thn$ .aftd Ultrv•l s<hcnwl• In: Whitley tO, tdltOf. Found.mon of g:rMCIC alsorlrhru,, vol 2 MOrJAn K•ufm~n Pubhutions: 1992. p. 1&1-202. 121 T.tkah.uhi 0. Kita H. K ob.ay.uhi S. A rtolll·co&cl &rntuc al&onthm uslnc cl!suncr dtP-endtnt o~l ttrnation mo<ltl for compltx funclion opumu:•rion In: Whnlcy D.arrtll. VOiclb(rt O~v•<L Cantu~Pal Enck. Sp«tor L«, Parmtot '""· fkoytr Hans Ceorg. editors Pr6t~d•nzs of &r•lfik .and C'VOIlll<lh.IIY cornpuc.-ciontonlrrc-ncoe (GEOC02000), 2000. p, 219- 26, ll) Knot H. A comp.anson study af sel(.,.d..apu!lon in f'""'lhtflnn Sfl•llf'Jif' .and'""' code-d &tnf'tic .-t,gomhm.s. tvolunon.try comput.anon tvolut Comput 2001;9(2):223-42. J Kohoncn T. Sdf-orgamzulg maps..lrd t'd. Spnncrrvnur: 2001 o4J IS) V•n Hullr MM. F~1thful rc-prnt-nutions .tnd loposr.aphK f'IUJ». John W•lt')' 1 SOns; 2000 16) Butbt 0, M•l"no M. Koumoutukos P. ~tf-otf,.llnlllf'J m.t'Pi f« "'l.lllt~blt('ll"1' optun1UI!OI'l. In· 8.trry AM. «lllor GECCO 1002 protn(tlncs of tht btrd of • fe.ther workS.hOpt:. ~~~I( ..and <''OI.ui*OMJY compuc ..tton conJrl't'fCt, New York. MAJ, 2002. p..• S. 11] 111 M, Klt..a H A ~e. opt"'olfOI" un,& ''*Pt~ componrlM .tn.tlys!s Jorre.aJ,odf'd geneoc .a)J:omhms. EYOiut Cof!'J)I.It 2001; 1 64)~g ...