Particle Swarm Optimization with Inertia Weight and Constriction Factor

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1 ICSI : Inrnaonal confrnc on sarm nllgnc d- Parcl Sarm Opmzaon h Inra Wgh and Consrcon Facor Mauro Sbasán Innocn Johann Snz Cl and Compuaonal Engnrng Cnr STUTE projc Dgal Tchnum Buldng Collg of Engnrng Sansa Unrsy Snglon Park S 8PP Und Kngdom M.S.Innocn@sansa.ac.uk J.Snz@sansa.ac.uk bsrac In h orgnal Parcl Sarm Opmzaon PSO formulaon conrgnc of a parcl oards s aracors s no guarand. locy consran s succssful n conrollng h xploson bu no n mprong h fn-gran sarch. Clrc and Knndy sudd hs sysm and proposd consrcon mhodologs o nsur conrgnc and o fn un h sarch. Thus hy dlopd dffrn consrcon mhods accordng o h corrlaons among som coffcns ncorporad o h sysm. Typ consrcon bcam ry popular bcaus h basc upda quaons rmand rually unmodfd and h orgnal and nu maphor ald. Th man drabacks of hs consrcon yp ar ha consrcon bcoms oo srong ry quckly as h acclraon s ncrasd; ha h spd of conrgnc canno b asly conrolld; and ha hr s no flxbly o s a dsrd form of conrgnc. nohr problm s ha h mhod can b found n h lraur formulad so as o consrc a PSO sysm hch alrady ncluds h nra gh for hch h calculaon of a consrcon facor usng h formula prodd by Clrc and Knndy dos no guaran conrgnc. Ths papr analyzs Typ consrcon n dal for hch Typ consrcon s also rlan. Th formula for Typ and Typ consrcon facors suabl for a PSO algorhm ncludng h nra gh ar prodd. Kyords Consrcon facor guarand conrgnc nra gh parcl sarm. Inroducon Parcl Sarm Opmzaon PSO s a Sarm Inllgnc SI-basd mhod nsprd by h coopra bhaour obsrd n socal anmals n naur. Th Orgnal PSO OPSO algorhm s unsabl and parcls nd o drg from h aracor h so-calld xploson. Th ncorporaon of h locy consran prns h xploson bu dos no mpro h poor fn-gran sarch. In Classcal PSO CPSO h ncorporaon of h nra gh allos for coffcns sngs ha nsur conrgnc and conrol h dsrd balanc bn xplora and xploa bhaour rfr o [ 5]. Smlarly h Consrcd Orgnal PSO COPSO ncorporas a consrcon facor o h OPSO hch nsurs conrgnc and mpros h fn-unng of h sarch rfr o [6]. s an nndd gnralzaon formulaons of h PSO algorhm h boh nra gh and consrcon facor can b found n h lraur.g. [7 ] hr hr no anon s pad o conrgnc ssus or h formula prsnd o calcula a consrcon facor ha supposdly ould guaran conrgnc s planly rong. Ths s bcaus h formula proposd by Clrc and Knndy [6] hch s only ald for h OPSO = s appld o h Consrcd Classcal PSO CCPSO. Th lar s a PSO algorhm h nra gh and consrcon facor. Th rmandr of hs papr s organzd as follos: Th PSO mhod s rd n scon hr h OPSO s prsnd n.; h classcal formulaon CPSO s prsnd n scon.; and Crgy Franc Jun -5

2 d- ICSI : Inrnaonal confrnc on sarm nllgnc h COPSO s dscussd n scon.3 ncludng Typ and Typ consrcon classs; a formal sudy of h CCPSO.. PSO h nra gh and consrcon facor s offrd n scon 3; and concludng rmarks ar prodd n scon. Ths sudy addrsss msundrsandngs msformulaons and/or msuss of h consrcon facor; hlps o undrsand s manng as nsond by Clrc and Knndy [6] and ho orks n conjuncon h h nra gh. Fnally prods h radr h h appropra formula o calcula a consrcon/scalng facor χ hch guarans conrgnc for any sng of h nra gh and of h acclraon coffcn ϕ n Consrcd Classcal PSO CCPSO. Parcl Sarm Opmzaon Th PSO mhod as proposd by Knndy and Ebrhar [3] nsprd by brd flock smulaons amd a sudyng socal bhaour. Th mhod as also nfluncd by ohr smulaons xprmns and hors n socal psychology rfr o [ 5]. Compuaonal Inllgnc CI chnqus such as Eoluonary lgorhms Es also nfluncd s dlopmn onc h mhod as hough of as a SI-basd opmzr. Hnc h PSO paradgm also has s o rfcal Inllgnc I and mahmacal opmzaon.. Orgnal Parcl Sarm Opmzaon Th formulaon of h OPSO algorhm [3] s as follos: j j xj xj U j pb x s U gb x j j j j hr j pb j x j : onn/coordna j of h locy and poson rspcly of parcl a msp ; gb j : coordna j of h bs xprnc of parcl and of h bs xprnc n h sarm rspcly by m-sp ; = s = : nddualy gh and socaly gh rspcly boh orgnally qual o [3]; U : random numbr gnrad form a unform dsrbuon n h rang [] rsampld an ry m s rfrncd. Ths sysm s unsabl and parcls nd o drg from h aracor. Th frs sragy usd o conrol hs xploson conssd of lmng h sz of ach onn of a parcl s locy. Ths hlps prn h xploson bu dos no hlp h h conrgnc or fn-unng of h sarch. Knndy [6] prformd a dald analyss of h rajcory of a drmnsc parcl pulld by saonary aracors on a on-dmnsonal spac. H consdrd boh aracors dncal p and carrd ou an obsraonal analyss of h rajcory of a parcl h nal poson x = p. Snc hr s a sngl aracor hr s also a sngl acclraon coffcn ϕ hch s assumd o b consan. Knndy [6] nocd ha h parcl dsplays psudo cyclc bhaour for < ϕ < and drgs for ϕ = and ϕ and carrd ou a dald analyss of h mrgng parns. Ozcan and Mohan [7 8] prsnd h frs horcal analyss of h dynamc bhaour of h OPSO sudyng h sam smplfd sysm as Knndy [6]. Thy sudd h rajcory of h sam drmnsc parcl for h sam parcular cas of x = p. Undr hs assumpon hy analyzd h rajcory for a f parcular alus of ϕ clamd o rs boundary cass for dffrn parcl s bhaours prsnng h parcl s rajcory quaon and sang ha h rajcory undr hs condons s a snusodal a hr h sngs of ϕ drmn h amplud and frquncy of h Crgy Franc Jun -5

3 ICSI : Inrnaonal confrnc on sarm nllgnc d-3 a. Thus hy argu ha h parcl dos no rally fly or h sarch-spac bu surfs on sn as. Th parcl s hn aracd by h ghd arag of h o bs xprncs mong n sp szs randomly oband from a snusodal a. Th yp of a caugh ould b drmnd by h random ghs hl h consran hlps h parcl jump ono anohr a.. Classcal Parcl Sarm Opmzaon In CPSO h sarm s bs xprnc s rplacd by h bs xprnc n h h parcl s nghbourhood. In addon h nra gh nally proposd n [9] s ncorporad hch allos conrollng h xploson and h conrgn bhaour. Gn ha hs s h mos dsprad formulaon s rfrrd o hr as h CPSO. Th currn rnd s o prsn h sngl ssnal upda quaon rahr han h radonal spara ons as shon n : hr x x U pb x s U lb x x x j j j j j j j j lb j : coordna j of h bs xprnc n h nghbourhood of parcl up o m-sp ; : nra gh of parcl a m-sp ; s : nddualy gh and socaly gh rspcly of parcl a m-sp. To h bs of our knoldg h frs horcal sudy n CPSO as prsnd by an dn Brgh [] ho also basd hs suds on a drmnsc parcl. H prsnd h rajcory quaon for lnarly ndpndn gnalus. H also prodd a conrgnc rgon n h ϕ plan alhough runcad only > and hou formal dfnon of s boundars. Trla [] sudd h CPSO n rms of a drmnsc parcl as ll usng rsuls from h Dynamc Sysm Thory. H prsnd a conrgnc rangl hou furhr dals. Innocn and Snz [5] nassd and xndd h horcal suds of a drmnsc parcl. Thy prsnd all hr rajcory quaons for h hr yps of soluons of h rcurrnc rlaons n rms of h o ssnal nal condons; hy formally drd and fully bound h lx and conrgnc rgons n h ϕ plan; hy sudd h rajcors of h parcl on h hr boundars of h conrgnc rgon and h spd and form of conrgnc nsd add by h rajcory quaons and by rajcory plos; and fnally proposd a mans of sng h coffcns ϕ and so as o oban a dsrd bhaour..3 Consrcd Orgnal Parcl Sarm Opmzaon Clrc and Knndy [6] analyzd h rajcory of a drmnsc parcl for h OPSO algorhm and dlopd consrcon facors amng o nsur conrgnc. Thy bul a sysm of o rcurrnc rlaons of frs ordr ha dscrbs h smplfd sysm by usng h chang of arabls n 3. Thus h sysm of o lnar frs ordr rcurrnc rlaons s as shon n. y p x 3 y y y Th sysm can b xprssd n marx noaon as n 5 and 6 hl h gnalus of h sysm marx M ar offrd n 7 hr = for h pos sgn. P y M P M P M 6 5 Crgy Franc Jun -5

4 d- ICSI : Inrnaonal confrnc on sarm nllgnc Crgy Franc Jun -5 ; 7 By dagonalzng M Clrc and Knndy [6] shod ha h poson of a parcl dpnds on h nal condons and on s gnalus rasd o h por of h m-sp. If a las on gnalu s no smallr han on h sysm dos no conrg. In hs cass hy proposd o buld a surroga sysm hos gnalus ' and ' ar boh smallr han on. For hs purpos hy addd f coffcns o h sysm hos alus can b chosn so as o nsur conrgnc. Such a sysm s shon n 8 hl M s as n 9. No ha γ n 8 and 9 s dffrn from γ n 7. y y y 8 M 9 Thn f h sysm n dos no ly h h conrgnc condon of boh gnalus bng smallr han on consrcon coffcns ar appld as n hr h gnalus of h surroga sysm ar forcd o ha magnuds smallr han on. ' ccordng o ho h addd coffcns α β γ δ η ar corrlad Clrc and Knndy [6] sudd dffrn consrcon yps; namly Typ and s draons Typ and Typ and Typ. For dals on hs yps rfr o hr orgnal ork [6]. Th only consrcon consdrd hr s Typ bcaus s h only on ha manans h orgnal nu concp of h locy as h dffrnc bn o conscu posons. Indrcly Typ consrcon s also rlan hr..3. Typ consrcon Th coffcns ncorporad ar corrlad as n. Clrc and Knndy [6] suggs ha sng h corrlaons as n nsurs ral coffcns. Thrfor h sysm marx M s as n 3. M 3 Ths consrcon yp s parcularly nrsng bcaus h gnalus of M n 3 ar as n. Hnc h consrcon facor ha nsurs conrgnc can b calculad as n 5. ' ; 5 For ϕ h gnalus of h orgnal sysm ar ral-alud and >. Thrfor ; ' 6 For < ϕ < h gnalus of h orgnal sysm ar lx conjugas h = =. Hnc χ s as n 7: ; 7

5 ICSI : Inrnaonal confrnc on sarm nllgnc d-5 To summarz h Typ consrcon facor ha nsurs conrgnc s calculad as n 8: f ; 8 f If h gnalus of h orgnal sysm ar lx conjugas or ral-alud so ar hos of h surroga sysm. Ths consrcon facor s smply h rao bn h corrspondng gnalus of boh sysms. Hor h classcal PSO formulaon s nocably dsrupd bcomng as n 9: x p.3. Typ consrcon p x p x p x p x For hs consrcon yp h addd coffcns ar corrlad as n. For smplcy Clrc and Knndy [6] also suggs sng. Thrfor h χ nsad of α h sysm marx s as n and h gnalus as n 3. ' 9 M Δ In Typ consrcon f h gnalus of h orgnal sysm ar lx conjugas or ralalud so ar hos of h consrcd on. Ths s no h cas n Typ consrcon. Th alus of ϕ dpndng on χ for hch Δ n 3 quals zro ar as n : mn 3 If h gnalus of h surroga sysm ar lx conjugas hr magnuds ar gn by h squar roo of χ. Hnc conrgnc s nsurd smply by χ <. Thrfor nforcng boh condons nsurs conrgnc. Clrc and Knndy [6] proposd usng h sam consrcon facor as for h Typ consrcon s 8. If ϕ < Δ < for ϕ mn < ϕ < ϕ. In hs cas solng s asy snc χ = κ. I s also asy o s from ha ϕ mn < for ϕ < and /9 < κ < as sad n [6] and shod n 5: mn f 5 9 If ϕ = χ sll quals κ s 8 and hrfor hr s connuy n h curs ϕ Δ. If ϕ > Δ < for < ϕ < ϕ mn h ϕ mn as n. Hor h calculaon of ϕ mn s no sraghforard for ϕ > s and 8 h ϕ mn ncrasng as κ ncrass. Clrc and Knndy [6] ud h alus of ϕ mn for κ =. ϕ mn = 8.7 and κ =.99 ϕ mn = Thy rfrrd o hs alus as ϕ nsad bcaus hy rs h uppr bound for Δ <. W rfr o hm as ϕ mn bcaus hy ar calculad h h nga sgn n. Th dsrabl faur of hs consrcon s ha h classcal formulaon of PSO s margnally alrd mananng h nu noon of h locy as h dffrnc bn o conscu posons. Th PSO upda quaons for hs consrcon yp rsul as shon n 6. No ha h upda quaons of h OPSO h Typ COPSO and h Typ COPSO concd for χ =. On Crgy Franc Jun -5

6 Consrcon Facor χ Consrcd Inra Wgh c d-6 ICSI : Inrnaonal confrnc on sarm nllgnc undsrabl faur s ha h calculaon of χ s no sraghforard as no longr rprsns h rao bn h magnuds of h corrspondng gnalus of h orgnal and h surroga sysms. x x x x p x 6 Snc ϕ s a random arabl s common pracc o rplac ϕ by ϕ n h full PSO sysm o b on h saf sd. Thus conrgnc s nsurd for all randomly gnrad ϕ. Ths mpls ha ry gnrad ϕ < ϕ ll b consrcd mor srongly han ncssary. Gnralzng for coffcns ha may dffr for dffrn parcls and chang or m h upda quaons of Typ COPSO ar: x x U pb x s U lb x x x 7 j j j j j j j j h f ohrs Th Typ COPSO can b asly rducd o h CPSO usng h consrcon facor as h consrcd nra gh c and h consrcd acclraon coffcn ϕ c nsad of h orgnal on ϕ as shon n 9: c 9 c Hnc hs consrcon bascally orks as a scalng-don facor for h orgnal coffcns of h sysm = and ϕ. Thrfor conrgnc s nsurd f h par of consrcd coffcns ϕ c c falls hn h conrgnc rangl dfnd n [5]. Each ln n Fgur rprsns all h pars ϕ c χ or ϕ c c for a gn alu of ϕ n h orgnal sysm for all alus of κ. Th op-rgh nd of h lns corrspond o κ =. Rcall ha χ = κ for ϕ ϕ =. -. ϕ = ϕ =. -. ϕ = ϕ = ϕ = c = ϕc+ *SQRTϕc Consrcd cclraon Coffcn ϕ c Fgur : Consrcon facor χ rsus consrcd acclraon coffcn ϕ c for dffrn alus of h acclraon coffcn ϕ n h orgnal sysm. Each ln rprsns all pars ϕ c χ or ϕ c c for all alus of κ. Th op-rgh nd of h lns corrspond o κ =. Th shadd rangl s h conrgnc rgon hl h lx conjuga gnalus ar hn h dod parabola s [5]. Th conrgnc rgon h < as ll as som pars of h conrgnc rgon h canno b cord by all possbl sngs of h COPSO. Th rangular rgon nclosd by h op and lf boundars of h conrgnc rgon and by h ln for ϕ = n Fgur can b cord by Crgy Franc Jun -5

7 Consrcon Facor χ Consrcd Inra Wgh c ICSI : Inrnaonal confrnc on sarm nllgnc d-7 scalng h ln corrspondng o any < ϕ < h < χ = κ <. Hor h consrcon bcoms oo srong for ϕ > and h rgons nar h rgh-boom boundary of h conrgnc rgon canno b cord by Typ consrcon usng 8. In fac no n h hol lx rgon can b cord. Th par of h conrgnc rgon ha can b cord by h COPSO s gn by h nlop of h op-rgh nds of h lns corrspondng o all alus of ϕ n Fgur as shon n Fgur Consrcd cclraon Coffcn ϕ c Fgur : Th gray shadd rangl shos h conrgnc rgon hras h nnr black loop ncloss h rgon n h ϕ plan hch can b cord by Typ consrcon usng 8. In addon h mpac of dffrn ϕκ sngs or h parcl s rajcory s no as sraghforard as ha of dffrn ϕ sngs hn h conrgnc rangl n CPSO hr dffrn bhaours for dffrn sub-rgons can b dnfd rfr o [5]. 3 Consrcd Classcal Parcl Sarm Opmzaon I s no nfrqun o fnd n h lraur h formulaon of h locy upda quaon ncludng boh nra gh and consrcon facor χ. Hor prhaps du o h fac ha follong h xclln ork prsnd by Clrc and Knndy [6] s no sraghforard hr consrcon facor s ofn msnrprd and hrfor mscalculad and/or msusd. In som cass h formulaon s ncorrc bu nr usd hras n ohrs hr s no calculaon of h consrcon facor bu smply som fxd or adap sng dsrgardng s conrgnc proprs. Zhang al. [8] formula h CCPSO h a Typ consrcon facor as n 8 h κ = hch s only ald for =. In CCPSO h rang of alus of ϕ for hch h gnalus ar lx conjugas s dos no only dpnd on χ bu also on. In addon h gnalus of h surroga sysm bng lx conjugas h χ < no longr guarans ha hr magnud b smallr han on bcaus hy ar no longr gn by bu by. Thrfor h calculaon of χ as n COPSO dos no apply o CCPSO. Hor lar n hr xprmns hy smply us χ = and random nra ghs. Chn al. [9 p. 38] also formula h CCPSO h a Typ consrcon facor as n 8 h κ =. Bar ha hy calld h consrcon facor κ and gnord h coffcn κ n 8 aloghr. In hr xprmns hy do no us ha formulaon and smply mplmn an adap consrcon facor hn xprmnally oband lms [..] h lnarly dcrasng n h rang [.9.] and ϕ [..] dsrgardng conrgnc ssus. Mahor al. [ p. 37] also posd h CCPSO h a Typ consrcon facor as n 8 h κ = h lnarly dcrasng. Hgash and Iba [7 p.7] s =.9 ϕ = and χ as random numbrs from Crgy Franc Jun -5

8 d-8 ICSI : Inrnaonal confrnc on sarm nllgnc.9 o.. Bu al. [ p. 336] usd a CCPSO h an adap χ h drmnsc dcay ruls =.79 ϕ =.. Blsn al. [] formula h CCPSO h a Typ consrcon facor bu us χ = n hr xprmns. Th formula prodd hrafr nsur conrgnc of a drmnsc parcl for any alu of ϕ. For h full PSO sysm suffcs o rplac ϕ by ϕ o nsur conrgnc. 3. Typ CCPSO Ths consrcon facor s such ha 3 s sasfd and ' <. Hor h gnalus of h orgnal sysm ar no longr calculad as n 7 bu as n 3. Thrfor h calculaon of hs consrcon yp rmans as n 5 smply modfyng h calculaon of h gnalus o accoun for h nra gh. Rcall ha for hs consrcon yp h PSO upda quaon s as n 9. ' 3 3. Typ CCPSO ; Consrcng h CPSO h Typ consrcon s no as sraghforard snc 3 s no ald. lhough Clrc and Knndy [6] usd h sam consrcon as for Typ consrcon hs only nsurs conrgnc for h OPSO. Typ consrcon may b d as scalng = and ϕ by a facor of χ. L us call and ϕ h alus of and ϕ nally s.. of h orgnal sysm. Thr consrcd alus c and ϕ c ould b on h ln dfnd as n 3: 3 Th pars ϕ on hs ln hch nsur conrgnc ar hn h sgmn dfnd bn h orgn and h nrscon of 3 h hr h op or h rgh-boom lns boundng h conrgnc rgon n Fgur. Thus h upprmos pon dfnng h sgmn of alus ϕ c c hch nsur conrgnc s gn by 33 hras h lormos pon s =. f con con 33 ohrs Th pars ϕ on 3 hch nsur conrgn lx gnalus ar hn h sgmn dfnd bn h nrscons of 3 h h parabola n Fgur for h quaon of h parabola rfr o [ 5]. Th upprmos pon ' dfnng h sgmn s gn by 3 and 35 hras h lormos pon ' s gn by 36. f ' 3 par par ohrs par par 35 3 Crgy Franc Jun -5

9 ICSI : Inrnaonal confrnc on sarm nllgnc d-9 Crgy Franc Jun -5 mn mn ' par par par par 36 L us call h orgnal sngs as n 37: 37 To nsur conrgnc by consrcng a CPSO sng a consrcon facor ha ls h 38 suffcs: con con 38 If h usr ans o nsur ha h consrcd coffcns fall hn h conrgn lx rgon sng a consrcon facor ha ls h 39 suffcs: mn mn ' ' 39 Rfr o Fgur 3 for a sualzaon of h locaons of pons ' and ' n h ϕ plan for hr dffrn nally s coffcns = ϕ n CCPSO. 3.. Ensurng conrgnc summary procdur ha nsurs conrgnc can b rducd o and : ohrs f con con No ha χ s grar han f h orgnally s ϕ par s nsd h conrgnc rgon. For χ approachng h uppr lm conrgnc spd dcrass bu h oppos s no ru for χ approachng. 3.. Ensurng lx gnalus summary procdur ha nsurs lx gnalus for h surroga sysm can b rducd o o : ohrs f mn 3 mn No ha χ s grar han f h orgnally s ϕ par s alrady nsd h lx rgon. Hr conrgnc spd ndd ncrass for dcrasng alus of χ hn h hol rang dfnd n. s opposd o h orgnal Typ COPSO n [6] h calculaon of h rang of ϕ hch rsuls n lx gnalus of h surroga sysm s smpl and cors h hol lx rgon. In addon an nral s dfnd for χ from hch any alu slcd lads o conrgnc

10 Inra Wgh d- ICSI : Inrnaonal confrnc on sarm nllgnc and h lor h alu h fasr h conrgnc. Hnc h manng of choosng alus closr o h lms of h nral s obous = ϕ.. ' '..8.6 = ϕ.. ' ' '. ' cclraon Coffcn ϕ Fgur 3: Thr dffrn nally s coffcns ϕ and h corrspondng pons ' and ' along hr corrspondng scalng ln n CCPSO. Concludng rmarks Th Typ and Typ COPSO r analyzd n som dal o clarfy h undrlyng concps. Th COPSO as proposd by Clrc and Knndy [6] as a mans o nsur conrgnc gn an nally s acclraon coffcn ϕ and no nra gh. Whl Typ consrcon appls o any ϕ > Typ s sad o b ald only for ϕ ϕ mn ϕ as urns ou s also ald for any ϕ >. Common msaks hn smulanously applyng consrcon facor χ and nra gh n CCPSO r pond ou and dscussd. Th conrgnc suds carrd ou for h CPSO n [5] nabld us o prod quaons for h calculaon of Typ and Typ consrcon facors hn hr s also nra gh n h sysm. In addon h quaons prodd allo for any nally s ϕ and for br conrol or h conrgnc spd: f lx conrgn gnalus ar nsurd an nral for χ s calculad hr h closr o h lor lm h fasr h conrgnc and h closr o h uppr lm h slor h conrgnc. Snc χ can b grar han on h proposd consrcon facor s n fac a scalng facor. Thus conrgnc may b dlayd y sll nsurd f h nally s ϕ par rsuls n oo fas a conrgnc. No ha hs ork s lmd o h drmnsc parcl and h proposd quaons ar oband usng quaons and rsuls from [5]. Th xnson o h full PSO sysm s smpl: rplacng ϕ by ϕ nsurs conrgnc. Th sudy of h ffcs of allong pars of h ϕ rang o la h conrgnc rgon s byond h scop of hs papr. Rfrncs [] VN DEN BERGH F. n nalyss of Parcl Sarm Opmzrs PhD hss Unrsy of Prora. [] TRELE I.C. Th parcl sarm opmzaon algorhm: conrgnc analyss and paramr slcon. Informaon Procssng Lrs 85 pp [3] VN DEN BERGH F. ENGELBRECHT.P. sudy of parcl sarm opmzaon parcl rajcors. Informaon Scncs 76 pp [] INNOCENTE M.S. Dlopmn and sng of a Parcl Sarm Opmzr o handl hard Crgy Franc Jun -5

11 ICSI : Inrnaonal confrnc on sarm nllgnc d- unconsrand and consrand problms PhD hss Sansa Unrsy. [5] INNOCENTE M.S. SIENZ J. Drmnsc Parcl h Saonary racors: Toards Coffcns Sngs Gudlns n Parcl Sarm Opmzaon. Submd o IEEE Transacons on Eoluonary Compuaon. [6] CLERC M. KENNEDY J. Th Parcl Sarm Exploson Sably and Conrgnc n a Muldmnsonal Complx Spac. IEEE Transacons on Eoluonary Compuaon Vol. 6 No. pp [7] HIGSHI N. IB H. Parcl Sarm Opmzaon h Gaussan Muaon. Procdngs of h IEEE Sarm Inllgnc Symposum SIS 3 pp [8] ZHNG L. YU H. HU S. N pproach o Impro Parcl Sarm Opmzaon. Procdngs of h Gnc and Eoluonary Compuaon Confrnc GECCO 3 pp [9] CHEN X.-H. LEE W.-P. LIO C.-Y. DI J.-T. dap Consrcon Facor for Locaonrlad Parcl Sarm. Procdngs of h 8h WSES Inrnaonal Confrnc on Eoluonary Compung pp [] MHOR. PRSD V. RNGNEKR S. Economc dspach usng parcl sarm opmzaon: r. Rnabl and Susanabl Enrgy Rs 3 pp [] BEIELSTEIN T. PRSOPOULOS K.E. VRHTIS M.N. Tunng PSO Paramrs Through Snsy nalyss. Unrsy of Dormund Dparmn of Compur Scnc Rh Compuaonal Inllgnc CI / Tchncal Rpor. [] BUI L.T. SOLIMN O. BSS H.S. Modfd Sragy for h Consrcon Facor n Parcl Sarm Opmzaon. In Randall M. bass H.S. Wls J. Eds. Lcur Nos n rfcal Inllgnc 88 pp [3] KENNEDY J. EBERHRT R.C. Parcl Sarm Opmzaon. Procdngs of h IEEE Inrnaonal Confrnc on Nural Norks pp [] KENNEDY J. EBERHRT R.C. Sarm Inllgnc. Morgan Kaufmann Publshrs. [5] KENNEDY J. Ho orks: Collabora Tral and Error. Inrnaonal Journal of Compuaonal Inllgnc Rsarch Vol. pp [6] KENNEDY J. Th bhaor of parcls. In Poro V.W. Saraanan N. Waagn D. Ebn.E. Eoluonary Programmng VII: Procdngs of h 7h nnual Confrnc on Eoluonary Programmng San Dgo pp [7] OZCN E. MOHN C.K. nalyss of a smpl parcl sarm opmzaon sysm. Inllgn Engnrng Sysms Through rfcal Nural Norks Vol. 8 SME books pp [8] OZCN E. MOHN C.K. Parcl Sarm Opmzaon: Surfng h Was. Procdngs of h IEEE Congrss on Eoluonary Compuaon Washngon DC pp [9] SHI Y. EBERHRT R.C. modfd parcl sarm opmzr. Procdngs of h IEEE Inrnaonal Confrnc on Eoluonary Compuaon pp Crgy Franc Jun -5

Consider a system of 2 simultaneous first order linear equations

Consider a system of 2 simultaneous first order linear equations Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm