Particle Swarm Optimization Algorithm with Reverse-Learning and Local-Learning Behavior

Transcription

1 35 JOURNAL OF SOFTWARE, VOL. 9, NO. 2, FEBRUARY 214 Parcle Swarm Opmzaon Algorhm wh Reverse-Learnng and Local-Learnng Behavor Xuewen Xa Naonal Engneerng Research Cener for Saelle Posonng Sysem, Wuhan Unversy, Chna Hube Engneerng Unversy, Xaogan, Chna Emal: Jngnan Lu Naonal Engneerng Research Cener for Saelle Posonng Sysem, Wuhan Unversy, Chna Yuanxang L School of Compuer, Wuhan Unversy, Hube Wuhan 4379, Chna Absrac In order o resolve conflc beween convergence speed and populaon dversy of parcle swarm opmzaon (PSO) algorhm, an mproved PSO, called reverse-learnng and local-learnng PSO () algorhm, s presened n whch a reverse-learnng behavor mplemened by some parcles whle local-learnng behavor adoped by ele parcles n each generaon. urng he reverse-learnng process, some nferor parcles of nal populaon and each parcle s hsorcal wors poson are reserved o arac a parcle o leap ou of local opmums. Furhermore, he Hammng dsance beween he nferor parcles s se o no less han a defaul rejecon dsance he am of whch s o manan he dversy of populaon and mprove s exploraon ably. In each generaon, he dfference beween he bes parcle and he second-bes parcle s used o gude he bes one o carry ou local search whch s crucal for mprovng s exploaon ably. The resuls of expermens show ha has a good global searchng ably and convergence speed especally n hgh dmenson funcon. Index Terms parcle swarm opmzaon, reverse learnng, local learnng, premaure convergence I. INTROUCTION Parcle swarm opmzaon (PSO) algorhm s a global opmzaon mehod ha orgnally developed by Kennedy and Eberhar n 1995[1]. Unlke oher evoluonary compuaon algorhms, n whch some genec operaons such as selecon, crossover and muaon operaors are adoped o manpulae a new ndvdual, PSO searches for an opmum by each parcle s flyng hrough he problem space. Ths search mechansm enables PSO has hgh operang effcency, fas convergence speed, and mplemenaon smplcy. Owng o s ousandng characers, PSO has been an effecve ool for solvng global opmzaon problems durng he pas wo decades, such as non-lnear funcon opmzaon, sgnal processng[2], mage reconsrucon Correspondng auhor: Xuewen Xa, emal: laughkd@163.com, xwxa@whu.edu.cn [3], engneerng opmzaon [4], ec. Alhough PSO has obaned some promsng resuls n many research felds, many expermens also ndcae ha he radonal PSO algorhm easly falls no local opma whle solvng complex mulmodal problems. Amng a he ssue, many researchers have made grea effor durng he pas decade whch wll be dscussed n Secon II. Based on prevous works, a novel algorhm, called reverse-learnng and local-learnng parcle swarm opmzer (), s proposed n hs paper. In, some new feaures are proposed. Frs, much useful nformaon whn n some weak parcles as well as n he bes parcle s aken no accoun o preven populaon from premaury. Second, a smple dfferenal operaor on some ele parcles s adoped o enhance he exploaon ably of PSO. The res of hs paper s organzed as follows. Secon II descrbes he framework of radonal PSO and revews some mproved PSOs. The deal of algorhm s presened n Secon III. Secon IV descrbes he expermenal sudy on and comparson of 3 varous mproved PSO algorhm. Fnally, conclusons are gven n Secon V. II. RELATE WORKS A. PSO Smlar o oher swarm nellgence heursc algorhm, PSO s a populaon-based sochasc opmzaon echnque. In PSO, a connuous process of opmzaon s descrbed as each parcle s flyng whle s descrbed as populaon s evoluon n genec algorhm (GA). When searchng n a -dmensonal hyperspace, he h parcle n PSO has a velocy vecor V = [v 1, v 2,..., v ] and a poson vecor X = [x 1, x 2,..., x ] o ndcae s curren sae, where s a posve neger ndexng he parcle n he swarm and s he dmensons of he problem under sudy. The poson vecor X s regarded as a canddae soluon of he problem whle he velocy vecor V s reaed as a parcle s search drecon and sep. urng he process of opmzaon (or flyng), each 214 ACAEMY PUBLISHER do:1.434/jsw

2 JOURNAL OF SOFTWARE, VOL. 9, NO. 2, FEBRUARY parcle decdes s rajecory accordng o s personal hsorcal bes poson vecor pbes = [pb 1, pb 2,, pb ] (.e., pbes locaon) and he global bes-so-far poson vecor gbes = [gb 1, gb 2,,gb ] (.e., gbes locaon). The updae rules of a parcle s velocy and poson are very smple, whch are defned as ( ) ( ) v + = ωυ + c r pb x + c r gb x (1) 1 j j 1 1 j j 2 2 j j x + = x + v (2) 1 j j j where ω s he nera wegh; c1 and c2 are he acceleraon coeffcens ha deermne he relave learnng wegh for pbes and gbes, whch called self-cognve and socal nfluence, respecvely; r1 and r2 are wo random numbers ha are unformly dsrbued over [, 1]; represens he curren parcle; and j represens he curren dmenson. B. Some Modfed PSO Alhough PSO has been successfully appled n varous felds snce was frs nroduced, especally n some complex, large scale, nonlnear and non-dfferenable opmzaon problems, here are sll many defcences n PSO algorhm, such as low accuracy of resoluon, premaure convergence and curse of dmensonaly, ec. In order o solve above-menoned problems, several modfcaons have been nroduced o mprove he performance of he radonal PSO durng he pas decade. From he aforemenoned parameers n PSO we know ha he nera weghω and he acceleraon consans c play mporan roles n opmzaon process. I s a smple and effecve sraegy o mprove he effcency of PSO ha selecng a proper se of parameers or nroducng a useful paramerc adapaon mechansm. For example, n order o regulae swarm s convergence speed and global searchng ably, Sh and Eberhar adoped a mehod of lnearly deceasng ω wh he eraon for PSO n [5] and hen nroduced a fuzzy adapve ω mehod for PSO n [6]. In [7], he adjusmen of ω no only depends on he curren eraons of evoluon bu also reles on he curren populaon dversy. Ranaweera e al. [8] developed a self-organzng H-PSO wh me-varyng acceleraon coeffcens. Afer makng sably and asrngency analyss for radonal PSO, a se of opmzed parameers was proposed by Trelea and M. Jang n [9][1], respecvely. The populaon opology has a sgnfcan effec on he performance of PSO. There are several common opologes, such as rng, wheels, sars and Von Neumann neghbor srucure, have been wdely used n many varans of PSO [11]. Afer ha, some opology unng sraeges have been proposed basng on he common opologes. In [12], a fully nformed PSO, called FIPS, was proposed by Mendes e al., n whch a parcle uses a sochasc average of pbes s from all of s neghbors nsead of usng s own pbes and he gbes n he updae equaon. In [13], Eucldan dsance beween parcles s deemed as a creron o selec proper pbes and gbes for a specfc parcle. In addon, many dynamc opologes, n whch he sac opologes have been replaced by some dynamcally adjused neghbor models, were nroduced n [14][15] o avod defcences of fxed neghborhoods. These modfcaons on opology mprove he local searchng ably and overcome he premaure phenomena of PSO n deferen degree. The choce of whch learnng paern o use s very mporan for PSO o acheve a good resuls. Lang e al. [16] developed a comprehensve learnng PSO (CLPSO) for mulmodal problems. In CLPSO, a parcle uses dfferen parcles hsorcal bes nformaon o updae s velocy, and for each dmenson, a parcle can poenally learn from a dfferen exemplar. Changhe L e al.[17] presened a self-learnng parcle swarm opmzer (SLPSO), n whch each parcles has four canddaes learnng-paerns o cope wh dfferen suaons n he search space. In [18], a common sngle gbes s replaced by a group of ele parcles. And a rejecon sraegy s adoped o enable he eles wdely dsrbued n he search space. In [19], a novel Baldwn effecs based learnng sraegy s adoped o mprove he performance of PSO, n whch he hsorcal benefcal nformaon s ulzes o ncrease he poenal search range and reans dversy of he parcle populaon. Alhough he modfcaons of PSO can generally be caegorzed no he aforemenoned ypes, should be poned ou ha some complcaed PSO varans may adop mulple sraeges above-menoned smulaneously. The mprovemen sraegy adoped n hs paper nvolves parameer adjusmen and learnng paern adjusmen. III. ALGORITHM As nroduced above, he man ams of hese sraeges ha adoped by PSO are o manan populaon dversy and o mprove global search ably wh a hgh convergence speed. In hs paper, a reverse learnng sraegy s seleced o manan populaon dversy whle a local research sraegy s adoped by gbes o mprove he convergence speed. The deals of he sraeges are descrbed as follow, respecvely. A. Reverse Learnng of Moderae Parcles So far, mos PSO algorhms use a smlar learnng paern for all parcles, n whch each parcle n a swarm learns from s own bes hsorcal experence and s neghborhood s bes hsorcal experence. Ths monoonc learnng paern may cause he swarm lack of nellgence o deal wh dfferen complex suaons. The monoonc learnng paern s wdely appled n mos PSO because a hypohess ha here s lle useful nformaon n an nferor parcle. However, s jus as rue ha an nferor parcle may have good values on some dmensons of he soluon vecor hough he parcle has he wors values on oher dmensons. The reason why he performance of ele parcles s beer han he performance of nferor parcles s ha he ele parcles have good values on more dmensons han he 214 ACAEMY PUBLISHER

3 352 JOURNAL OF SOFTWARE, VOL. 9, NO. 2, FEBRUARY 214 nferor parcles, or he soluon s vulnerable o he dmensons on whch he ele parcles acqure good values. So s an arbrary decson o regard ha here s lle useful nformaon embedded n an nferor parcle. In addon, here s usually a long dsance beween an nferor parcle and a subopmal parcle. Ths ra of he nferor parcles may be employed o drag some parcles o jump ou of local opmal. Hence, how o dscover more useful nformaon embedded n he nferor parcles and hus how o ulze he nformaon o mprove PSO s global search ably are mporan for us. In hs paper, when he opmum nformaon of he swarm s sagnan durng he search process n a -dmensonal hyperspace, n parcles wll parcpae n a reverse learnng. An nferor poson n he nal populaon W ( k = wk1, wk2,... wkj,..., wk) (1 k m ) and he hsorcal nferor poson of he h parcle, denoed as W = ( w1, w2,..., wj,..., w) (1 n), are used as wo new exemplars o gude parcle flyng by changng s speed and drecon( wheher can be effecvely appled n he adjusmen of parcle s poson). Ths behavor s called reverse-learnng process n hs paper, whch s descrbed as follows: ( ) ( ) υ + = ω υ + c r x w + c r x w (3) 1 j j 3 3 j j 4 3 j kj where c3 and c4 are he acceleraon coeffcens ha deermne he relave wegh of self-cognve and socal nfluence from nferor posons; rand r3 and rand r4 are wo random numbers ha are unformly dsrbued over [, 1]; w kj represens he k h nferor poson n he j h dmenson; wj represens personal hsorcal wors poson of parcle n he j h dmenson a me. The meanng of oher parameers can refer o equaons (1) and (2). Afer reverse-learnng process, some parcles could jump ou of local opmzaon and redsrbue over he search space. To make hese parcles have a more wdespread dsrbuon condon, a longer rejecon radus R beween he nferor posonsw s necessary. So when choosng he nferor parcles poson n he nal populaon, we should accoun for no only he fness of hem bu also he rejecon radus beween hem. The algorhm o nalze a subpopulaon conssng of m nferor posons ha sasfy a predefned rejecon radus s descrbed as Algorhm 1. B. Local Research by Hsorcal Opmal Poson To oban a accurae soluon, each ele n a populaon should no only ake responsbly for gudng he evoluonary search bu also make self-dreced learnng. In hs paper, he global hsorcal bes poson gbes makes local research based on he dfferenal beween self and he second global hsorcal bes poson sgbes, whch s defned as (4). k ( ) * gbes gbes r d gbes sgbes = + (4) where r s a random numbers ha unformly dsrbued over [, 1], by whch gbes could adjus drecon of he local research; d s a weghng facor of he dfferenal value a he h eraon. In general, he dsance beween gbes and he global opmal soluon wll become shorer and shorer along wh evoluon of populaon. So he radus of local research should have smlar varaon rend durng he evoluon. In hs paper, d decreases lnearly durng he run me, whch s updaed as follow: ( ) d = 1 d 1 + / T (5) where T s a maxmum number of search eraons and s he curren search eraon. Afer he local research, greedy-choce sraegy s appled o P g1 and P ' g1, whch s defned as (6). gbes = gbes, ( ) ( ) * * gbes, f f gbes > f gbes oher where f(x) s he fness of poson x. C. Adjusable Maxmum Flyng Velocy V max As nroduced n Secon II, we know ha a well-desgned maxmum flyng velocy Vmax could make a parcle search he problem space wh a reasonable sep lengh. Many expermens demonsraed ha V max s se n he nerval of 1% ~2% of search space could oban a preferable resul. However, we hnk s preferable ha dfferen V max should be adoped by dfferen parcles a dfferen sages n he opmzaon. For example, a parcle ha responsble for exploraon requres a hgh flyng velocy o mprove probably of global search. On he conrary, and he end of opmzaon, a parcle wh sronger velocy resran enables self o oban a more accurae resoluon. When he opmum nformaon of a swarm s sagnan durng he search process, wha we wan s o help some parcles o jump ou of local opmal va he reverse learnng process n whch a larger V max could enable hese parcles o escape from he local opmal wh a hgh velocy. Fnshng he reverse learnng, however, he parcles need o adjus her velocy o a smaller value n order o mee he requremen of local search. Togeher wh he aforemenoned componens, he mplemenaon of he algorhm s summarzed n Algorhm 2. IV. BENCHMARK TESTS AN ISCUSSION A. Tes Funcons The performance of he developed s evaluaed hrough nne well-known funcons ha ofen used as es sue n numercal expermens. The deals of hese funcons are descrbed as Table I. The opmal objecve values of hese funcons for mnmzaon are all zero. (6) 214 ACAEMY PUBLISHER

4 JOURNAL OF SOFTWARE, VOL. 9, NO. 2, FEBRUARY Algorhm 1. Consrucon of nferor subpopulaon INPUT: Inal populaon X { } = X, X,... X, Rejecon radus R, Inal nferor populaon W = { φ} 1 2 N OUTPUT: Populaon W = { W, W,..., W 1 2 m } Begn Sep 1. Evaluae X and rearrange n ascendng order sor by fness. Afer rearrange, whou loss of generaly, he populaon s sll X, X,..., X, f f,, j [1, N]& & < j recorded as : { 1 2 } N X X j W = X, coun=1, W W { W } 1 1 coun Sep 2. FOR =2 TO N IF coun m Break; ELSEIF ( ) j coun++; =, =1; j W X > R,1 j coun W = X ; W W coun { W coun } EN IF ENFOR Sep 3. WHILE coun < m Random generae a new pacles nd; End EN IF EN WHILE IF ( ) j = ; j nd W > R,1 j coun coun++; W coun = nd ; W = W { W coun } ; Algorhm 2. Algorhm INPUT: -dmensonal opmzed funcon; N (Populaon sze); m (Inferor subpopulaon sze); Coeffcens c 1,c 2,c 3 and c 4 ; V ( maxmum flyng velocy); L mes (eraons of reverse learnng); n(he number of parcles ha ω (maxmum and mnmum of nera wegh); mn max parcpae n a reverse learnng); d (weghng facor); OUTPUT: Populaon hsorcal bes poson P g1 ; Begn X X, X,... X 1 2 N Sep 1. Generae nal populaon = { } and velocy V ( V1, V2,..., V ) N Sep 2. Evaluae populaon X ; Se P g1 =he bes parcle, P g 2 =he second bes parcle; Sep 3. Consruc a nferor populaon W accordng Algorhm 1; Sep 4. WHILE (No mee he sop condon) Sep 5. Updae V and X accordng o formulas (1) and (2), respecvely; = ;Se P = B = X,=; Sep 6. Evaluae populaon X ; Updae P, B, P g1 and P g 2 ; Sep 7. Pg 1 performs local learnng procedure accordng o formula (3); Updae d accordng formula (4); Sep 8. IF Mee he reverse learnng condon Sep 9. Adjus V ; max Sep 1. Updae V 1 ~ V and X ~ 1 Sep 11. Updae ~ V V and Q+ 1 N X Sep 12. EN IF Sep 13. =+1; Sep 14. EN WHILE End Q ~ Q+ 1 B. Expermenal Seng Three varan PSO algorhms are seleced for comparsons. The frs one s EPSO algorhm [21] n whch E and PSO are adoped as wo basc opmzaon mehods. Accordng o he curren success raes of E and PSO, EPSO selec a beer one o gude subsequen opmzaon process. The second one s whch nroduced n [18]. In, a common sngle gbes s replaced by a group of ele parcles. To X accordng o formula (5)and (2), respecvely; Q X N accordng o formula (1)and (2), respecvely; ωmax keep he dversy of populaon, a rejecon sraegy s adoped by he ele parcles. Accordng o Secon II, he modfcaon of EPSO and can be caegorzed no Hybrdzaon sraeges and Learnng-paerns adjusmen, respecvely. The las peer algorhm s a recen sandard PSO algorhm named PSO27 avalable on he Parcle Swarm Cenral: hp:// For more deals abou hese peer algorhm, he reader s referred o correspondng leraures. and 214 ACAEMY PUBLISHER

5 354 JOURNAL OF SOFTWARE, VOL. 9, NO. 2, FEBRUARY 214 In hs expermenal, we chose wo dfferen dmensons, ncludng =1 and =1, for hese es funcon nroduced n Table I o es he scalably of algorhms on low-dmenson and hgh-dmenson funcon opmzaon. Accordng o he dfferen dmensons, wo populaon szes are se as N=2 and N=3 respecvely. The confguraon of each peer algorhm aken from he leraure s exacly he same as ha used n he orgnal paper. Noe ha all opmzers nvolved n he expermen were mplemened on a PC wh 2.93GHz Penum CPU, 4.GB memory, Wndows 7 operaor sysem and all algorhms were mplemened n MATLAB 29a. C. Resuls and scusson In hs secon, each algorhm was execued 1 ndependen runs over he 9 es funcons n wo cases, whch are 1 and 1 dmensons, respecvely, wh he correspondng populaon sze seng as 2 and 3, respecvely. The expermenal resuls n he form of he mean value (Mean), mnmum value (Mn), sandard devaon value (S) and mean run-me (RT) of fnally dscovered bes objec values are presened n Table II and Table III, where he bes resul of Mean and Mn on each problem among all algorhms s shown n bold. From he expermenal resuls n Table II, can be seen ha RSPSO obans he bes mean resuls on F 2, F 6 and F 9 where acheves he second-bes resuls ha slghly worse han he bes resuls. Snce F 6 and F 9 are wo unmodal funcons whch seleced o examne he exploaon ably of an algorhm, we can draw a concluson ha rejecon mechansm n RSPSO and local learnng mechansm n are benef o mprove he exploaon ably and ncrease he soluon accuracy. Meanwhle, obans he bes resuls on F 1, F 3, F 7 and F 8 whle SPS27 acheves he bes resuls on F 4 and F 5. Alhough EPSO and RSPSO are much more nferor o and on funcon F 1, should be poned ou ha hese four algorhms have obaned almos dencal soluon accuracy n mos cases excep EPSO rappng n local opmum 23 ou of 1 runs whle SPS27 rappng n local opmum 3 ou of 1 runs. From he expermenal resuls n Table III, we can see ha, wh he ncreasng of funconal dmenson from 1 o 1, obans he bes resuls on mos funcons excep on F 2 and F 6. I also should be poned ou ha n spe of he mean value obaned by on F 2 and F 6 are no he bes one, he number of mnmum values on F 2 ha less han 1-14 s 79 ou of 1 runs whle he number of mnmum values on F6 ha less han 1-38 s 85 ou of 1 runs. So we can say ha s he bes one among he four peer algorhms especally for solvng hgh-dmensonal funcon when success rae and soluon accuracy are consdered ogeher. From he expermenal resuls shown n Table II and Table III we can see ha me-consumng of PSO27 s he lowes han oher algorhms benef of s smple operaors whle he me-consumng of RSPSO s he longes due o s consrucon of ele subpopulaon. For example, f he ele subpopulaon sze s M n RSPSO, we need o make a leas M*(M-1)/2-1 Eucldean dsance calculaons beween parcles n each generaon n order o manan he ele subpopulaon dversy. In addon, he rejecon radus also needs o be calculaed n each generaon. EPSO s more me-consumng han PSO27 no only because of EPSO need o make sascal analyss bu also because of dfferenal evoluon algorhm s more me consumng he radonal PSO. In, wo novel learnng mechansm called reverse learnng and local learnng are adoped. Noe ha here s no exra me-consumng durng he reverse learnng model for here s no addonal operaors excep a reverse flyng drecon of some parcles. Alhough an nferor subpopulaon s needed n, unlke he consrucon of ele subpopulaon n RSPSO, he nferor subpopulaon only consruced one me afer populaon nalzaon. urng he local learnng process, parcle P g1 makes local research n some dfferen fly drecons whch cause he me-consumng of s more han of PSO27. As he dmensons s ncrease from 1 o 1, Pg1 wll ake more me o execue local research. So s less me-consumng han EPSO n 1 whle s more me-consumng han EPSO n 1. In fac ha f we execue he local research of P g1 and he fly process of oher parcles n parallel, can oban he same me-consumng as PSO27. Asde from he aforemenoned arges presened n Table 4 and Table 5, convergence speed s also an mporan assessmen creron for PSO algorhm. The plos n Fg 1 show he convergence progress of he mean soluon values of he 5 rals durng he run for funcons F 4, F 7, F 8 and F 9. In he four funcons, F 8 and F 9 are unmodal funcons seleced o es convergence speed and solvng accuracy of algorhm whle F 4 and F 7 are mulmodal funcons adoped o examne global search ably of algorhm. I can be observed from he fgure ha PSO27 has faser convergence speed n solvng F 8 and F 9 wh low dmensons han oher peer algorhms n he nal sage of evoluon. However, EPSO, RSPSO and have faser convergence speed han PSO27 a he laer evoluon process. A he same me, we see ha has he fases convergence speed and mos accurae resuls n solvng hese unmodal funcons when he funconal dmenson ncreases from 1 o 1. The same concluson can be obaned from he opmzaon of funcon F 7. urng he process of opmzaon of funcon F 4 wh 1 dmensons, obans he bes resul and he fases convergence speed. However, once he dmenson s 1, only provdes he second-bes performance for funcon F 4. In fac, obans he bes performance for all ral funcons excep funcon F 4. ue o lmed space, he convergence progresses of oher funcons lsed n Table3 are no presened n hs paper. So could be an undersaemen o say ha s more suable for hgh-dmensonal funcon opmzaon. 214 ACAEMY PUBLISHER

6 JOURNAL OF SOFTWARE, VOL. 9, NO. 2, FEBRUARY TABLE I. BENCHMARK PROBLEMS Funcon name efnon Search space Global f mn (x*) x* F ( ) 2 2exp.2 exp cos 2 = + 1 π = = 1 [ 32,32 ]. Acley ( ) Alpne F2 ( X) = x sn( ).1 1 x + x = [ ] 2 Rasrgn F3 ( X) = ( x 1cos( 2 ) 1) 1 π x = + [ ] Schwefel P1.2 F4 ( X) ( xsn ( x = )) Grewank = [ ] 2 5 = 1 = 1 [ ] = [ 1,1] = F ( X ) = ( x / 4) cos( x / ) + 1 Schwefel P2.22 F 6 ( X ) = x 1 + x 1 ( ) Rosenbrock F7( X) = 1( x 1 ) ( 1 ) 1 + x + x = [ ] Sphere Sum of dfferen power 2 8( ) x = 1 F X 1, , , ,6.. 3,3. = [ 1,1]. 9 = 1 ( + 1) F ( X) = x [ 1,1]. TABLE II. COMPARISON RESULTS OF MEANS AN VARIANCES IN 1 IMENSIONS (=1, N=2) F 1 F 2 F 3 Mean Mn S RT Mean Mn S RT Mean Mn S RT EPSO 2.17e e e e e e e+ 4.46e e e e e e e e+ 9.95e e+.93 SPS e e e e-6 1.7e e e+ 9.95e-1 2.2e e e e e e e e-1.7 F 4 F 5 F 6 Mean Mn S RT Mean Mn S RT Mean Mn S RT EPSO 8.51e e e e e e e e e e e e-2 7.4e-3 3.2e e e e SPS e e e e-2 2.e e e e e e e e e e e-45 2.e F 7 F 8 F 9 Mean Mn S RT Mean Mn S RT Mean Mn S RT EPSO 6.39e e-3 1.6e e e e e e e e+ 1.19e e e e e e-17 4.e e SPS e+ 4.82e e e e e e e e e+ 1.15e e e e e e e e TABLE III. COMPARISON RESULTS OF MEANS AN VARIANCES IN 1 IMENSIONS (=1, N=3) F 1 F 2 F 3 Mean Mn S RT Mean Mn S RT Mean Mn S RT EPSO 2.47e+ 1.25e+ 3.9e e+ 5.69e e e e e e+ 1.63e+ 3.96e e+ 1.72e e e e e SPS27 2.2e+ 1.22e+ 4.5e e+ 2.62e-1 3.7e e+2 1.5e e e e e e+ 3.9e e e e+ 2.44e F 4 F 5 F 6 Mean Mn S RT Mean Mn S RT Mean Mn S RT EPSO 1.95e e e e e e e+ 1.17e e e e e e+ 1.2e e e e e SPS e e e e e e e e e e e e e+ 5.89e e F 7 F 8 F 9 Mean Mn S RT Mean Mn S RT Mean Mn S RT EPSO 8.5e e e e+ 6.91e-1 2.9e e-25 1,53e-3 2.1e e e e e e e e-3 3.8e e SPS e+2 3.1e e e e e e e e e e e e-2 1.8e e e e e ACAEMY PUBLISHER

7 356 JOURNAL OF SOFTWARE, VOL. 9, NO. 2, FEBRUARY Fness EPSO SPSO27 Fness EPSO SP S Generaon Generaon Fness (a) EPSO SP SO27 Fness (b) EPSO SPSO Generaon Generaon (c) (d) Fness EPSO SPSO Generaon (e) Fness EPSO SPSO Generaon (f) Fness 1-1 EPSO SPSO Generaon (g) Fness EPSO SPSO Generaon Fgure 1. Comparson of convergence performance on 4 es funcons n wo dfferen dmensons. (a)f 4 (1). (b)f 4 (1). (c)f 7 (1). (d)f 7 (1). (e)f 8 (1). (f)f 8 (1). (g)f 9 (1). (h)f 9 (1). (h) 214 ACAEMY PUBLISHER

8 JOURNAL OF SOFTWARE, VOL. 9, NO. 2, FEBRUARY V. CONCLUSIONS In hs paper, we presened a modfed PSO called algorhm n whch a reverse-learnng process and a local-learnng process are adoped o make he search effecve and effcen. In each generaon, a local-learnng by he bes parcle s execued afer sandard flyng process. Whle he populaon raps n local opmum, he reverse-learnng process s carred ou o help some parcles jump ou of he local opmum. urng he reverse-learnng process, some nferor parcles n nal populaon and he hsorcal nferor posons of parcles are used as new exemplars o gude parcles flyng. Whle selecng he nferor parcles n he nal populaon, we se a rejec radus o make he nferor parcles wdely dsrbued n search space whch can mprove he dversy of populaon whle afer reverse-learnng process. Furhermore, he dfference of he bes parcle and he second-bes parcle n each generaon s used o gude he bes one o carry ou local search. Smulaon resuls reveal ha has a good global searchng ably and convergence speed especally n hgh dmenson funcon. ACKNOWLEGMENT The research work was jonly suppored by he Naonal Key Projec for Basc Research of Chna (Grans No ), he Naonal Scence Fund (Grans No. 6179) and Hube Educaon eparmen Scence Fund (Grans No. X212394) REFERENCES [1] J. Kennedy and R. C. Eberhar, Parcle swarm opmzaon, Proceedngs of IEEE Inernaonal Conference on Neural Nework, Perh, Ausrala, vol.4, pp ,1995. [2] LI Ja-sheng WANG Yng-de, TIAN wang-lan, Exracng communcaon sgnals based on parcle-swarm-opmzaon and adapve-neural-fuzzy-conrol, Inernaonal Journal of Advancemens n Compung Technology, vol.5, no.3, pp , 213. [3] Baojan Zhang, Fugu Chen and Lnfeng Ba, An mprove parcle swarm opmzaon algorhm, Journal of Compuer, vol. 6, no.11, pp , 211 [4] Sbn Zhu, Guxan L and Junwe Han, An mprove PSO algorhm wh objec-orened performance daabase for flgh rajecory opmzaon, Journal of Compuer, vol. 7, no.7, pp , 212 [5] Y. Sh and R. C. Eberhar, A modfed parcle swarm opmzer, Proceedngs of IEEE Inernaonal Conference on Evoluonary Compuaon, Anchorage, AK, pp.69 73, [6] Y. Sh and R. Eberhar, Fuzzy adapve parcle swarm opmzaon, Proceedngs of IEEE Conference on Evoluonary Compuaon, Soul, Korea, pp.11 16, 21 [7] FAN Hu-Lan, ZHONG Yuan-Chang, Two-subpopulaon parcle swarm opmzaon based on pheromone dffuson, Journal of Sysem Smulaon, vol.23, no.1, pp , 211. [8] A. Ranaweera, S. K. Halgamuge, and H. C. Wason, Self-organzng herarchcal parcle swarm opmzer wh me-varyng acceleraon coeffcens, IEEE Transacon on Evoluonary Compuaon, vol.8, no.3, pp , 24. [9] Ioan Crsan Trelea, The parcle swarm opmzaon algorhm: convergence analyss and parameer selecon. Informaon Processng Leers, vol.85, no.6, pp , 23. [1] M Jang, Y P Luo, S Y Yang, Sochasc convergence analyss and parameer selecon of he sandard parcle swarm opmzaon algorhm, Informaon Processng Leers, vol.12, no.1, pp.8-16, 27. [11] J. Kennedy and R. Mendes, Populaon srucure and parcle swarm performance, Proceedngs of Congress on Evoluonary Compuaon, Honolulu, HI, pp , 22. [12] R. Mendes, J. Kennedy, and J. Neves, The fully nformed parcle swarm: Smpler, maybe beer, IEEE Transacon on Evoluonary Compuaon, vol.8, no.3, pp , 24. [13] Suganhan P.N., Parcle swarm opmser wh neghborhood operaor, Proceedngs of IEEE Congress on Evoluonary Compuaon, Washngon,.C., USA, pp , [14] P. N. Suganhan, Parcle swarm opmzer wh neghborhood operaor, Proceedngs of IEEE Congress on Evoluonary Compuaon, pp , [15] S. Janson and M. Mddendorf, A herarchcal parcle swarm opmzer and s adapve varan, IEEE Transacon on Sysems, Man, and Cybernecs Par B, Cybernecs, vol.35, no.6, pp , 25. [16] J. J. Lang, A. K. Qn, P. N. Suganhan, and S. Baska, Comprehensve learnng parcle swarm opmzer for global opmzaon of mulmodal funcons, IEEE Transacon on Evoluonary Compuaon, vol.1, no.3, pp , 26. [17] Changhe L, Shengxang Yang, A Self-learnng parcle swarm opmzer for global opmzaon problems, IEEE Transacon on Sysems, Man, and Cybernecs Par B, Cybernecs, vol.42, no.3, pp , 212. [18] TAN Yang, TANG e-quan, QUAN Hu-Yun, Rejecon of compeon parcle swarm opmzaon, Journal of Sysem Smulaon, vol.23, no.2, pp , 211. [19] J Qang Zha, Ke Q Wang, Baldwn effec based parcle swarm opmzer for mulmodal opmzaon, Journal of Compuer, vol. 7, no.9, pp , 212 Xuewen Xa, born n 1974, Ph.., asssan professor. He receved Ph from Wuhan Unversy n 29. Hs research neress nclude compuaon nellgence and applcaon. Jngnan Lu, born n 1943, professor, Ph.. supervsor, academcan of Chnese Academy of Engneerng. Hs research neress nclude GPS heores, mehod and daa processng. Yuanxang L, born n 1962, Ph.., professor, Ph.. supervsor. He receved Ph from Wuhan Unversy n Hs research neress nclude parallel compuaon, compuaon nellgence and applcaon. 214 ACAEMY PUBLISHER