DEPSO: Hybrid Particle Swarm with Differential Evolution Operator

Transcription

1 IEEE Inernaional Conference on Sysems, Man & Cyberneics (SMCC), Washingon D C, USA, 2003: [Cooperaive Group Opimizaion] hp:// DEPSO: Hybri Paricle Swarm wih Differenial Evoluion Operaor Wen-Jun Zhang, Xiao-Feng Xie* Insiue of Microelecronics, Tsinghua Universiy, Beijing 00084, P. R. China * xiexiaofeng@singhua.org.cn Absrac - A hybri paricle swarm wih ifferenial evoluion operaor, erme DEPSO, which provie he bell-shape muaions wih consensus on he populaion iversiy along wih he evoluion, while keeps he selforganize paricle swarm ynamics, is propose. Then i is applie o a se of benchmark funcions, an he experimenal resuls illusrae is efficiency. Keywors: Paricle swarm opimizaion, ifferenial evoluion, numerical opimizaion. Inroucion Paricle swarm opimizaion (PSO) is a novel muliagen opimizaion sysem (MAOS) inspire by social behavior meaphor [2]. Each agen, call paricle, flies in a D-imensional space S accoring o he hisorical experiences of is own an is colleagues. The velociy an locaion for he ih paricle is represene as v i = (v i,,v i,,v id ) an x i = ( x i,..., xi,..., xid), respecively. Is bes previous posiion is recore an represene as p i =(p i,, p i,, p id ), which is also calle pbes. The inex of he bes pbes is represene by he symbol g, an p is calle gbes. A each sep, he paricles are g manipulae accoring o he following equaions [5]: v i =w v i +c ran() ( p i -x i )+c 2 ran() ( p g -x i ) (a) x i = x i + v i (b) where w is ineria weigh, c an c 2 are acceleraion consans, ran() are ranom values beween 0 an. Several researchers have analyze i empirically [,, 20] an heoreically [3, 5], which have shown ha he paricles oscillae in ifferen sinusoial waves an converging quickly, someimes premaurely, especially for PSO wih small w[20] or consricion coefficien[3]. The concep of a more -or-less permanen social opology is funamenal o PSO [0, 2], which means he pbes an gbes shoul no be oo close o make some paricles inacively [8, 9, 20] in cerain sage of evoluion. The analysis can be resrice o a single imension wihou loss of generaliy. From equaions (), v i can become small value, bu if he p i -x i an p g -x i are boh small, i canno back o large value an los exploraion capabiliy in some generaions. Such case can be occure even a he early sage for he paricle o be he gbes, which he p i -x i an p g -x i are zero, an v i will be ampe quickly wih he raio w. Of course, he los of iversiy for p i -p g is ypically occure in he laer sage of evoluion process. To mainain he iversiy, he DPSO version [20] inrouces ranom muaions on he x i of paricles wih small probabiliy c l, which is har o be eermine along wih he evoluion, a leas no be oo large o avoi isorganizaion of he swarm. I can be improve by a bell-shape muaion, such as Gaussian isribuion [8], bu a funcion of consensus on he sep-size along wih he search process is preferable []. A bare bones version [] for saisfying such requiremens is o replace he equaions () by a Gaussian muaion wih he mean (p i +p g )/2 an he sanar eviaion p i -p g, which maybe also be inefficien when p i -p g is very small, an is oo raically since i urns he PSO ino a variay of in evoluion sraegies (ES) [2] /03/$ IEEE

2 This paper escribes a hybri paricle swarm wih ifferenial evoluion (DE) operaor[6], erme DEPSO, which also provie he bell-shape muaions wih consensus on he populaion iversiy, while keeps he paricle swarm ynamics. Then he DEPSO is applie o several benchmark funcions [4, 3, 5], an he resuls illusrae he significan performance improvemen. 2 DEPSO algorihm For propose DEPSO, he muaions are provie by DE operaor [6] on he, wih a rail poin T i =, which for h imension: p i IF (ran()<cr OR ==k) THEN T =p i g + δ 2, (2) where k is a ranom ineger value wihin [, D], which ensures he muaion a leas one imension., CR is a crossover consan, an δ is he case of N=2 for he 2 general ifference vecor δ, which is efine as: N N δ = N (3) N ( ) where he is he essenial ifference vecor [6], means he iffe rence of wo elemens ha ranom chosen from a common poin se, which inclue all he pbes in curren case. N is he number of involve. Then for he h imension: = pa, pb, (4) where p, p are chosen from he pbes se a ranom. A B The experimenal analysis f or δ will be resrice N o a single imension wihou loss of generaliy. Fig. an 2 shows a hisogram of poins ha we re ese in one million ieraions for δ an δ, respecively. Each 2 one imensional elemen p for calculaing is a realvalue ha picke from [-, ] a ranom. I can be seen ha δ is a riangle isribuion an δ is a bell-shape 2 isribuion, which he laer is beer for problemsolving []. Hence he N=2 is selece in equaion (2). The muaion is performe on he pbes insea of x i [8, 20] so as o preven he swarm from isorganizing by unexpece flucuaions, since he replacemen of pbes will follow he seay-sae sraegy, i.e., T i will replace pi only if i is beer han p i. The muaion is also base on p provies he social g learning capabiliy, which migh spee up convergence. p i The learning raio is eermine by CR, which is he counerpar of ineracion probabiliy IP []. If CR=, hen he equaion (2) becomes a bell-shape muaion operaor on p as ES [2]. If CR<, i may reain some g imensions of, which will faciliae he convergence pi for he finess lanscape ha some imensions are irrelevanly. Such capabiliy is also implemene implicily in he some former muaion versions [8, 20] bu is no in he canonical version. Probabiliy Probabiliy FIG. Hisogram of poins ese for δ FIG. 2 Hisogram of poins ese for δ 2 The original PSO operaor an he DE operaor will be performe alernaely, i.e. he equaions () will be performe a he o generaions, an he equaion (2) a he even generaions. The δ 2 will provie a consensus muaion on along wih iversiy of swarm, which pi δ emerge from he naure of he search iself, while ry ing o keep he iversiy of pbes an gbes by changing. 3 Resuls an iscussions δ 2 For DEPSO, c =c 2 =2, w=0.4, he maximum velociy V MAX was se o he half range of he search space on each imension [9, 20]. If wihou eclaraion specially, p i

3 each es case ran for T=2000 generaions an all he cases were run 00 runs each. The consrain-hanling meho is following he crieria [6]: a) any feasible soluion is preferre o any infeasible soluion; b) among wo feasible soluions, he one having beer objecive funcion value is preferre; c) among wo infeasible soluions, he one having smaller consrain violaion is preferre. The bounary consrains are hanle by Perioic moe. For each poin x i, is finess will be calculae by a mapping poin z i = ( z i,..., zi,..., zi D ), which for h imension, i has: IF x i < l THEN z i = u -( l -x i )% s (5a) IF x i > u THEN z i = l +(x i - u )% s (5b) Where % is he moulus operaor, l an u are lower an upper values, an s = u - l is he parameer range of he h imension. 3. Unconsraine funcions The ese unconsraine problems inclue he Rosenbrock funcion, he generalize Rasrigrin an he generalize Griewank funcion [5, 9, 20], which all in 30-D an have same minimum value (F op ) as zero. Table liss he mean finess values of such funcions by FPSO [5], DPSO (w=0.4, c l =0.00) [20], an DEPSO (w=0.4, CR=0) wih ifferen number of agens (m). The CR is se small, since here is lile correlaion beween he parameers for hese funcions. I shows ha DEPSO performs beer han boh ol PSO versions, especially for he Rasrigrin an he Griewank funcions, which wih uncorrelae parameers. TABLE. The mean finess values for he unconsraine funcions f m FPSO[5] DPSO[20] DEPSO 30-D Rosenbrock (F op=0) D Rasrigrin (F op=0) E-9 30-D Griewank (F op=0) Consraine funcions The selece problems inclue eleven funcions ha are propose by Z. Michalewicz e al (G o G ) [3], which inclue eigh funcions wihou equaliy consrains an hree funcions (i.e. G 3, G 5, G ) wih equaliy consrains, an an engineering opimizaion problem: esign of a pressure vessel (Vessel) [4, 9]. The imension of S (D), he opimizaion ype an opimum value (F op ) of each funcion are lis in Table 2. TABLE 2. Summary of consraine funcions f D Type F op G 3 Minimum -5 G 2 20 Maximum G 4 5 Minimum G 6 2 Minimum G 7 0 Minimum G 8 2 Maximum G 9 7 Minimum G 0 8 Minimum G 3 0 Minimum - G 5 4 Minimum G 2 Minimum 0.75 Vessel 4 Minimum Table 3 liss he mean finess values for eigh funcions wihou equaliy consrains by a (30, 200)-ES (T=750) [7], he DE (CR=0.), he canonical PSO (w=0.4), an he DEPSO (w=0.4, CR=0.), where m=70. For G 8, he maximum generaion T=200, an for all he oher cases, T=2000. I shows ha he DEPSO ouperforms eiher he DE or he PSO. By he way, i also provies beer resuls han GA [3] an ES [7] wih much less evaluaion imes. TABLE 3. The F op for he funcions wihou equaliy consrains f ES[7] DE PSO DEPSO G G G G G G G G E-3 E-4 E-5 E-6 Funcion: G PSO (w=0.4) DEPSO(w=0.4, CR =0.) E-7 FIG. 3 Mean relaive performance for G

4 0. 0. E-3 E-5 Funcion: G 8 DEPSO(w=0.4, CR=0.) Funcion: G 2 DEPSO(w=0.4, CR=0.) 0.0 E-7 E-9 E- E FIG. 4 Mean relaive performance for G 2 FIG. 8 Mean relaive performance for G Funcion: G 9 DEPSO(w=0.4, CR=0.) E-4 E-6 Funcion: G 4 E-8 DEPSO (w=0.4, CR=0.) E FIG. 5 Mean relaive performance for G 4 FIG. 9 Mean relaive performance for G Funcion: G 0 PSO (w=0.4) DEPSO(w=0.4, CR=0.) Funcion: G 6 E-4 PSO (w=0.4) DEPSO(w=0.4, CR =0.) E-6 00 FIG. 6 Mean relaive performance for G 6 FIG. 0 Mean relaive performance for G 0 0 Funcion: G 7 DEPSO(w=0.4, CR=0.) FIG. 7 Mean relaive performance for G 7 Figure 3 o 0 show he relaive mean finess value =F mean -F op uring he evoluion generaions () for he eigh funcions, respecively, where F mean is he mean bes finess value in each generaion. I shows ha DEPSO ouperform DE in a leas menione cases. For some problems, such as G 4, G 6, G 8, PSO perform beer han DE an DEPSO wih CR=0., which may ue o he finess lanscape of such problems are episasis, i.e. nee o covary he parameers a he same ime o improve finess. Noes alhough here is no correlaion

5 beween he parameers of G 6, bu is consrains consruc such a S F. This also can explain why PSO is performing worse for such as G, since lack of he capabiliy of varying only few imensions for a poin. For G 3, G 5, G, which have almos 0% feasible space (S F ) ue o he equaliy consrains, are neee o replace he consrain gx ( ) = 0 by an inequaliy consrain gx ( ) < ε for some small ε >0 [3, 4, 7]. Here we choose wo ε values: E-3 [4], an E-4 [7], which he S F of he former is la rger han ha of he laer an will more easily o be solve. Table 4 liss he mean finess values for he hree funcions wih equaliy consrains DE (CR=0.9), he canonical PSO (w=0.4), an DEPSO (w=0.4, CR=0.9), where m=70. In orer o compare wih exising resuls, when ε =E-3, for G 5, T=2500, an for G, T=300; when ε =E-4, for G 3 an G 5, T=5000. For oher cases, T=2000. The values in he brackes gives he number of rails ha are faile in enering S F, an only hose rails ha are succeee in enering S F are coune for he calculaion of mean finess values. DEPSO ouperforms DE an PSO in all cases, an i can fin he opimum soluion in all runs when ε =E-3. Table 5 summarizes he evaluaion imes for exising DE resuls in [4] an DEPSO, which shows DEPSO is also much fas. TABLE 4. The F op for he funcions wih equaliy consrains f ε DE PSO DEPSO G 3 E G 3 E G 5 E (0) G 5 E (2) G E G E TABLE 5. Summary of he evaluaion imes when ε =E-3 f DE[4] DEPSO G G G When ε =E-4, DEPSO also canno always fin he opimum poin, which is beer han normal ES [7] bu is worse han ES wih sochasic ranking [7] for G 3 an G 5 in same evaluaion imes. The objecive of he vessel problem is o minimize he cos of he maerial, forming an weling of a cylinrical vessel [4, 9]. I is a mixe-ineger-iscreeconinuous problem which has four esign variables, wo are ineger an wo are coninuous. Here he closes ineger value will be use o evaluae he finess alhough he algorihms sill works inernally wih coninuous variables. TABLE 6. The mean finess values for he vessel problem Resuls MGA[4] DE PSO DEPSO Bes Mean Wors S.D Table 6 liss he comparison of resuls for he vessel problem by MGA (populaion size is 50, an T=000, which coss evaluaion imes) [4], DE (CR=0.), he canonical PSO (w=0.4), an DEPSO (w=0.4, CR=0.), where for he hree algorihms, m=70, an T=700, which coss evaluaion imes. I can be seen ha he PSO is no sable [9], which may ue o he sep-ype lanscape creae by ineger variables. However, PSO also shows he capabiliy o cach he opimum poin comparing o MGA an DE. DEPSO inheris he meris of PSO an DE, an performs much beer ha MGA wihin almos same evaluaion imes. 4 Conclusions This paper presens a hybri paricle swarm wih ifferenial evoluion operaor calle DEPSO. The hybri sraegy provies he bell-shape muaions wih consensus on he populaion iversiy by DE operaor, while keeps he self-organize paricle swarm ynamics, in orer o make he performance is no very sensiive o he choice of he sraegy parameers as in DE [7]. I is shown o ouperform he PSO an DE for a se of benchmark funcions. However, more comparaive works wih ifferen parameer seings for more problems shoul be performe o provie a full view. The DEPSO seems performing well for he problems wih ineger variables by he help of he bell-shape muaions. However, as eclare for PSO [9], i is als o no very efficienly for hanling hose problems wih exremely small feasible space, such as he problems

6 wih equaliy consrains. Since each agen in DEPSO (an DE, PSO) only refers o few poins (pbes an gbes), i canno employs some sraegies (such as sochasic ranking [7]), which nees a big populaion. Fuure invesigaion may employ exening memory wih a se of poins o saisfy such sraegies. Moreover, accoring o No-Free-Lunch (NFL) heory [8], aken he problem informaion ino accoun will improve he performance of algorihm. For DEPSO, he appropriae CR can be chosen if he correlaion of he parameers is known. Bu for he black-box problems, i is sill a grea challenge o learning such parameers uring run-ime wih efficienly sraegies. Acknowlegemens The work is suppore by he Naional Hi-Tech Research an Developmen Program of China (863 projec) uner Conrac No. 2002AAZ460. The auhors also hank M. Clerc for helpful iscussions. References [] Angeline P J. Evoluionary opimizaion versus paricle swarm opimizaion: philosophy an performance ifference. Annual Conf. on Evoluionary Programming, 998: [2] Beyer H-G, Schwefel H-P. Evoluion sraegies: a comprehensive inroucion. Naural Compuing, 2002, (): 3-52 [3] Clerc M, Kenney J. The paricle swarm - explosion, sabiliy, an convergence in a muliimensional complex space. IEEE Trans. on Evoluionary Compuaion, 2002, 6(): [4] Coello C A C. Theoreical an numerical consrainhanling echniques use wih evoluionary algorihms: a survey of he sae of he ar. Compuer Mehos in Applie Mechanics an Engineering, 2002, 9(-2): [5] Crisian T I. The paricle swarm opimizaion algorihm: convergence analysis an parameer selecion. Informaion Processing Leers, 2003, 85(6): [6] Deb K. An efficien consrain hanling meho for geneic algorihms. Compuer Mehos in Applie Mechanics an Engineering, 2000, 86(2-4): [7] Gäemperle R, Müller S D, Koumousakos P. A parameer suy for ifferenial evoluion. WSEAS In. Conf. on Avances in Inelligen Sysems, Fuzzy Sysems, Evoluionary Compuaion, 2002: [8] Higashi N, Iba H. Paricle swarm opimizaion wih Gaussian muaion. IEEE Swarm Inelligence Symposium, 2003: [9] Hu X H, Eberhar R C, Shi Y H. Engineering opimizaion wih paricle swarm. IEEE Swarm Inelligence Symposium, 2003: [0] Kenney J. The paricle swarm: social aapaion of knowlege. IEEE In. Conf. on Evoluionary Compuaion, 997: [] Kenney J. Bare bones paricle swarms. IEEE Swarm Inelligence Symposium, 2003: [2] Kenney J, Eberhar R C. Paricle swarm opimizaion. IEEE In. Conf. on Neural Neworks, 995: [3] Koziel S, Michalewicz Z. Evoluionary algorihms, homomorphous mappings, an consraine parameer opimizaion. Evoluionary Compuaion, 999, 7(): 9-44 [4] Lampinen J. A consrain hanling approach for he ifferenial evoluion algorihm. Congress on Evoluionary Compuaion, 2002: [5] Shi Y H, Eberhar R C. Fuzzy aapive paricle swarm opimizaion. IEEE In. Conf. on Evoluionary Compuaion, 200: 0-06 [6] Sorn R, Price K. Differenial evoluion - a simple an efficien heurisic for global opimizaion over coninuous spaces. J. Global Opimizaion, 997, : [7] Runarsson T P, Yao X. Sochasic ranking for consraine evoluionary opimizaion. IEEE Trans. on Evoluionary Compuaion, 2000, 4(3): [8] Wolper D H, Macreay W G. No free lunch heorems for opimizaion. IEEE Trans. on Evoluionary Compuaion, 997, (): [9] Xie X F, Zhang W J, Yang Z L. Aapive paricle swarm opimizaion on iniviual level. In. Conf. on Signal Processing, 2002: [20] Xie X F, Zhang W J, Yang Z L. A issipaive paricle swarm opimizaion. Congress on Evoluionary Compuaion, 2002: