Research Article A Swarm Optimization Genetic Algorithm Based on Quantum-Behaved Particle Swarm Optimization

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1 Hidawi Computatioal Itelligece ad Neurosciece Volume 217, Article ID , 15 pages Research Article A Swarm Optimizatio Geetic Algorithm Based o Quatum-Behaved Particle Swarm Optimizatio Tao Su 1,2 ad Mig-hai Xu 1 1 College of Pipelie ad Civil Egieerig, Chia Uiversity of Petroleum, Qigdao 26658, Chia 2 Shegli College, Chia Uiversity of Petroleum, Dogyig, Shadog 257, Chia Correspodece should be addressed to Mig-hai Xu; mighai@upc.edu.c Received 3 Jauary 217; Revised 1 April 217; Accepted 2 April 217; Published 25 May 217 Academic Editor: Ezequiel López-Rubio Copyright 217 Tao Su ad Mig-hai Xu. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Quatum-behaved particle swarm optimizatio (QPSO) algorithm is a variat of the traditioal particle swarm optimizatio (PSO). The QPSO that was origially developed for cotiuous search spaces outperforms the traditioal PSO i search ability. This paper aalyzes the mai factors that impact the search ability of QPSO ad coverts the particle movemet formula to the mutatio coditio by itroducig the rejectio regio, thus proposig a ew biary algorithm, amed swarm optimizatio geetic algorithm (), because it is more like geetic algorithm () tha PSO i form. has crossover ad mutatio operator as but does ot eed to set the crossover ad mutatio probability, so it has fewer parameters to cotrol. The proposed algorithm was tested with several oliear high-dimesio fuctios i the biary search space, ad the results were compared with those from,, ad. The experimetal results show that is distictly superior to the other three algorithms i terms of solutio accuracy ad covergece. 1. Itroductio Particle swarm optimizatio (PSO) algorithm is a populatio-based optimizatio method, which was origially itroduced by Eberhart ad Keedy i 1995 [1]. I PSO, the positio of a particle is represeted by a vector i search space,adthemovemetoftheparticleisdetermiedby a assiged vector called the velocity vector. Each particle updatesthevelocitybasedoitscurretvelocity,thebest previous positio of the particle, ad the global best positio of the populatio. PSO is extesively used for the optimizatio problems because it has simple structures ad is easy to implemet. However, it has some disadvatages, such that it easily falls ito local optima whe solvig the complex ad high-dimesio problems [2, 3]. Hece a umber of variat algorithms have bee proposed to overcome the disadvatages of PSO [4, 5]. The particle swarm algorithm based o the probability covergece is oe of the variat algorithms. This kid of particle swarm algorithm allows the particles to move accordig to probability istead of usig velocity-displacemet particle movemet way. The Bare Boes PSO (B) family is a typical class of probabilistic PSO algorithms [6 8]. The Gaussia distributio was used i the origial versio of B,whichwasproposedbyKeedy[6].theseveral ew B variats used other distributios which seem to geerate better results [7 9]. Ispired by the quatum theory ad the trajectory aalysisofpso[1],suetal.proposedaewprobabilistic algorithm, quatum-behaved particle swarm optimizatio (QPSO) algorithm [11]. I QPSO, each particle has a target poit, which is defied as a liear combiatio of the best previous positio of the particle ad the global best positio. The particle appears aroud the target poit followig a double expoetial distributio. The QPSO algorithm essetially belogs to the B family, ad its update equatio uses a adaptive strategy ad has fewer parameters to be adjusted [12 14]. The QPSO has bee show to perform well i fidig the optimal solutios for cotiuous optimizatio problems ad successfully applied to a wide rage of areas such as multiobjective optimizatio [15, 16], clusterig [17 19], eural etwork traiig [2 22],

2 2 Computatioal Itelligece ad Neurosciece image processig [23, 24], egieerig desig [25], ad dyamic optimizatio [26]. PSO ad QPSO have bee effective tools for solvig global optimizatio problems, but they were origially developed for cotiuous search spaces. Keedy ad Eberhart itroduced a biary versio of PSO for discrete problems amed biary PSO () [27], where the trajectories are defied as chages i the probability that each particle chages its state to 1. Biary PSO has simple structure ad is easy to implemet; hece, it is extesively employed i the optimizatio problems [28 3]. But it also suffers from some disadvatages whe solvig the complex ad high-dimesio problems[28].suetal.proposedbiaryqpso(), i which the target poit is obtaied by usig the crossover operator at the best previous positio of the particle ad the global best positio. Experimet results show that ca fid better solutio geerally tha [31]. I recet years, has bee used successfully i may fields [32 34]. However, although broades the applicatio fields of QPSO, it did ot show the same advatage as i the cotiuous space. QPSO algorithm should have better performace i solvig the problems basedodiscretespace.thispaperaalyzesthemaifactors that impact the search ability of QPSO ad coverts the particle movemet formula to the mutatio coditio by the itroductio of rejectio regio. It the desiged a ew biary codig QPSO, which has crossover ad mutatio operator ad is like geetic algorithm () i form; that is,theproposedalgorithmisaewgeeticalgorithmbut icorporates the core idea of QPSO. So it was amed swarm optimizatio geetic algorithm (). Compared with the, the has o selectio operator, ad each idividual participates i evolutio based o the iformatio of the populatio ad its ow iformatio. At the same time, the mutatio probability of the is ot fixed. I the early stage of the algorithm, the probability of mutatio is large ad the populatio ca keep the diversity, with the iteratio of the algorithm, the mutatio probability teds to zero, ad the algorithm ca fially coverge. The rest of this paper is orgaized as follows. Sectio 2 is a brief itroductio of PSO ad biary PSO; Sectio 3 summarizes QPSO ad biary QPSO; Sectio 4 itroduces the mutatio coditio of biary codig coverted from the particle movemet formula i QPSO; Sectio 5 proposes the ew biary QPSO algorithm,, ad the discusses the differece betwee this algorithm ad QPSO, ; Sectio 6 presets the experimet results from the bechmark fuctios; fially, the paper is cocluded i Sectio Particle Swarm Optimizatio Particle swarm optimizatio (PSO) algorithm is a populatio-based optimizatio techique used i cotiuous spaces. It ca be mathematically described as follows. Assume the size of the populatio is ad the dimesio of the search space is q; the the ith particle of the swarm ca be represeted by a positio vector X i =(x i1,x i2,...,x iq );the velocity of a particle i is deoted by vector V i =(V i1, V i2,..., V iq ); vector P i = (p i1,p i2,...,p iq ) is the best previous positio of particle i, called persoal best positio, ad P g = (p g1,p g2,...,p gq ) is the best positio of the populatio, called global best positio. The velocity of particle i is calculated accordigly: V k+1 ij =ωv k ij +c 1r 1 (p k ij xk ij )+c 2r 2 (p k gj xk ij ), (1) where i = 1,2,...,, j = 1,2,...,q, is populatio size, k istheumberofiteratios,ω is iertia weight, c 1 ad c 2 are acceleratio coefficiets, ad r 1 ad r 1 are radom umbers i the iterval [, 1]. The the ext positio is updated as follows: x k+1 ij =x k ij + Vk+1 ij. (2) The PSO algorithm is applied to solve optimizatio problems i the real search space, but may optimizatio problems are set i discrete space. Keedy ad Eberhart proposed a discrete biary versio of PSO, amed biary PSO (), where the particle positio has two possible values, or 1. The velocity formula i remais uchaged, ad the particle positio is updated as follows: ij = { 1 S(V k+1 ij )>rad { otherwise, { x k+1 where rad is a radom umber i the iterval [, 1] ad the fuctio S(V) is a Sigmoid fuctio as S (V) = 3. Quatum-Behaved Particle Swarm Optimizatio (3) 1 (1+e V ). (4) Ispired by trajectory aalyses of PSO i [1], Su et al. proposed a ovel variat of PSO, amed quatum-behaved particle swarm optimizatio (QPSO), which outperforms the traditioal PSO i search ability. QPSO sets a target poit for each particle; deote G i = (g i1,g i2,...,g iq ) as the target poit for particle i,ofwhichthe coordiates are g ij =φ ij p ij +(1 φ ij )p gj, (5) where φ ij isaradomumberitheiterval[, 1]. the trajectory aalysis i [1] shows that G i is the local attractor of particle i; that is, i PSO, particle i coverges to it. The positio of particle i is updated as follows: x k+1 ij =g k ij ± Lk ij 2 l ( 1 u ), L k ij =2α c j x k ij, where u is a radom umber i the iterval [, 1] ad C= [c 1,c 1,...,c q ] is kow as the mea best positio that is (6)

3 Computatioal Itelligece ad Neurosciece 3 y Figure 1: Probability desity fuctio of double expoetial distributio. defied by the average of the persoal best positio of all particles, accordigly, c j = 1 p tj, j=1,2,...,q, (7) t=1 Parameter α is called Cotractio-Expasio Coefficiet, which ca be tued to cotrol the covergece speed of the algorithms. Because the iteratios of QPSO are differet from those of PSO, the methodology of caot be applied to QPSO. Su et al. itroduced the crossover operator of ito QPSO ad proposed biary QPSO (). I, X i = (x i1,x i2,...,x iq ) still represets the positio of particle i,but it is ecessary to emphasize that X i is a biary strig rather tha a vector, ad x ij is the jth substrig of X i,otthejth bit i the biary strig. Assume the legth of each substrig is l; the the legth of X i is lq. The target poit G i for particle i is geerated through crossover operator; that is, exerts crossover operatio o the persoal best positio P i ad the global best positio P g to geerate two offsprig biary strigs, ad G i is radomly selected from them. Defie p m =α d H (c j,x k ij ) l ( 1 ), u U[, 1], (8) u where k is the umber of iteratios ad d H (c j,x k ij ) is the Hammig distace betwee c j ad x k ij.comparedwiththe two bit strigs, the Hammig distace is the cout of bit differeceithetwostrigs.c j is the jth substrig of the mea best positio, ad the dth bit of c j is determied by the states of the dthbitofallparticles persoalbestpositios.ifmore particles take o 1 at the dth bit, the dth bit of c j is 1; otherwise the bit will be. For each bit of g ij,whep m > rad execute operatios as follows: if the state of the bit is 1, the set its state to ; else set its state to. 4. A Mutatio Coditio Usig i Biary Space The reaso why the QPSO algorithm has better global search capability tha the traditioal PSO algorithm is that it chages the velocity-displacemet model of the traditioal PSO algorithm; i QPSO, the movemet of particle to its target poit has o determied trajectory; it ca appear at aypositioithewholefeasiblesearchspacewithacertai distributio, which is the double expoetial distributio [13, 14]. Such a positio ca be far from the target poit ad may be superior to the curret global best positio of the populatio. This should also be reflected i the costructio of biary QPSO algorithm. The probability desity fuctio of particle i i QPSO is f(x i )= 1 L i e 2 X i G i /L i, (L i >),,2,...,. (9) Set λ i =2/L i,ady i =X i G i ;the(9)caberewritteas f(y i )= λ i 2 e λ i y i, (λ i >). (1) That is, y i obeys the double expoetial distributio, of which the mea ad variace are E(y i )=ad D(y i )=2/λ 2 i.the graph of probability desity fuctio (1) is Figure 1. Sice the domai of y i is (, + ), particle ca appear i ay positio of the search space, but the probability that a particle appears i a positio far away from its target poit is small. Whe λ i +,thevariaced(y i )=2/λ 2 i which meas that X i coverge to G i with probability 1. Whe the positio of a particle uses biary ecodig, it is hard to describe the relative positio of two poits usig the measure of two biary strigs. Similar to set a rejectio regio, we set a threshold value V (V >). Whe the value of y i falls ito the rejectio regio, as show i Figure 2, set y i =;thatis,x i =G i,elsex i = mutatio(g i ).mutatio(g i ) meas mutatio operatio o G i.

4 4 Computatioal Itelligece ad Neurosciece v v Figure 2: Refused domai of probability desity fuctio. For ay u,whichisaradomumberitheiterval[, 1], the coditio that y i does ot fall ito the rejectio regio is P( y i > V) >u. (11) TheleftsideofCoditio(11)cabewritteas P( y i > V) = V λ i 2 eλ it dt + V + + λ =2 i V 2 e λ it dt = e λiv. λ i 2 e λ it dt Thus Coditio (11) meas e λ iv >u, accordigly: (12) λ i V < l ( 1 ). (13) u I order to esure that the algorithm ca coverge, set λ i = 1 d H (C, X i ), (14) where C isthemeabestpositioofthepopulatio. The Coditio (13) is V d(c,x i ) < l ( 1 u ), (15) where d( ) is used to measure the differece of two biary strigs. Hammig distace ca be used here. Assume y=l(1/u);the u=exp ( y), y >. (16) For (16), whe the value of y is small, the fuctio has fast rates of chage as show i Figure 3, so Coditio (15) suffers from the effect of the iitial value of d(c, X i ). So Coditio (15)cabechageditoitsequivaletform: σd (C, X i )>l ( 1 ), (17) u where parameter σ isacostatthatisgreaterthazero u Figure 3: Figure of equatio (16). 5. Swarm Optimizatio Geetic Algorithm Based o the mutatio coditio (17), mutatio operator is itroduced ito. X i still represets the positio of particle i, P i is the persoal best positio of particle i, P g is the global best positio, ad C is the mea best positio which is defied the same as i. Differet from, crossover or mutatio operatio process is applied to the whole biary strig, istead of bits. Because the procedure of the algorithm is similar to, it is amed as swarm optimizatio geetic algorithm (). The process ca be described as follows. (1) Iitialize a populatio of particles X i i biary space; (2) Set persoal best positio P i =X i,adcomputec; (3) Evaluate the fitess of particles f(x i ) ad determie the global best positio P g ; (4) while termiate coditio is ot reached do (5) for each particle i do y

5 Computatioal Itelligece ad Neurosciece 5 (6) Exert crossover operatio o P i ad P g to geerate two offsprig biary strigs, (7) G i is radomly selected from them. (8) if coditio (17) is true, (9) Exert mutatio operatio o G i ; (1) ed if (11) Set X i =G i ; (12) Compute the fitess of particles f(x i ),adupdatep i, (13) ed for i (14) Update P g ad the mea best positio C; (15) ed while p m sigma = 2 sigma = 1 sigma = d Figure 4: Figures of mutatio probability. Compared to the with the same crossover ad mutatio operator, has the followig characteristics: (1) does ot have selectio operator ad crossover probability ad its crossover operator is exerted directly o P i ad P g. Therefore, the form of the fitess fuctio f(x i ) has o effect o the algorithm, ad the target fuctio of the maximizatio problem ca be set as the fitess fuctio. (2) Coditio (17) ca be tured ito u>exp ( σd (C, X i )). (18) Sice σd(c, X i ),therageofexp( σd(c, X i )) is (, 1), adu is a radom umber i the iterval [, 1]; thus Coditio (18) is equivalet to a adaptive mutatio probability: p m =1 exp ( σd (C, X i )), (19) where σ is a costat that is greater tha zero ad d(c, X i ) decreases with the icrease of iteratio times. Therefore, p m is shruk, which causes the algorithm to coverge. (3) σ is the oly parameter of, which ca be tued to cotrol the covergece speed of the algorithms as Cotractio-Expasio Coefficiet α i. Whe the value of σ is.5, 1, ad 2, the curves of mutatio probability p m chagig with d( ) are show i Figure 4. The figure demostrates that the smaller the value of σ,the faster the covergece speed of the algorithm. It also ca be see that the global searchig ability of the algorithm is reduced whe σ is too small. So set σ=1i. 6. Experimetal Results The proposed is compared with,, ad. They are tested o the followig 1 bechmark problems to be miimized [28, 35]: (1)SphereFuctio F 1 (X) = (2)Schwefel sproblem2.22 F 2 (X) = (3)Schwefel sproblem1.2 F 3 (X) = (4)StepFuctio F 4 (X) = x 2 i, x i 1. (2) x i + x i, x i 1. (21) i j=1 ( x j ) 2 ( x i +.5 ) 2, (5)Schwefel sproblem2.21, x i 1. (22) x i 1. (23) F 5 (X) = max { i x i },,2,...,, x i 1. (24) (6) 2 Miima Fuctio F 6 (X) = 1 (x 4 i 16x 2 i +5x i ), x i 5. (25)

6 6 Computatioal Itelligece ad Neurosciece Table 1: Parameters of algorithms applied i the experimets. Algorithm Parameter settigs σ=1 ω =.7, c 1 =c 2 =2, V max =6 α= p c =.9, p m =.1.15 (7)Schwefel sproblem1.2 F 7 (X) = x i si ( x i ), x i 5. (26) (8) Ackley Fuctio F 8 (X) = 2exp (.2 exp ( x 2 i ) cos (2πx i ) )+2+e, (9) Geeralized Pealized Fuctio x i 32. (27) F 9 (X) = π 1 {1 si (πy 1)+ (y i 1) 2 [1 + 1 si 2 (πy i+1 )] + (y 1) 2 }+ u(x i, 1, 1, 4), y i =1+ (x i +1), u(x 4 i,a,k,m)= k(x i a) m { a x i a, { { k( x i a) m x i >a, x i < a. x i 5 (28) (1)GriewakFuctio F 1 (X) = 1 x 2 i 4 cos ( x i )+1, i xi 6. (29) I these fuctios, F 1 F 5 are uimodal ad F 6 F 1 are multimodal. Their optimum values are all zeros except F 6 ad F 7. The miimum values of F 6 ad F 7 are ad , respectively, where is the dimesio of a fuctio. I the experimets, the dimesio of each fuctio is 8, ad the biary code legth of each cotiuous variable is 15, sothelegthofparticleis12foreachfuctio.thesizeof populatio is 5 ad the total umber of iteratios is set to 5. The parameters of algorithms are listed i Table 1, where p c is crossover probability ad p m is mutatio probability i. Four algorithms ra idepedetly 3 times o the bechmark fuctios, ad the best target fuctio value was recorded at each ru. To compare the four algorithms, 3 data sets were aalyzed usig the followig statistic parameters: the mea, the stadard deviatio (STD), the best, the worst, ad the media; these results are reported i Tables 2 ad 3. Moreover, the statistical test is coducted i order to determie whether the average best results are differet with a statistical sigificace. The cofidece level is fixed at.95, ad the tests retur p value which are show i Tables 4 ad5.weusethesasforstatisticaltestig;ithesas system, if the p valueislesstha.1,thesystemdisplays <.1. Thevalueofh i Tables 4 ad 5 shows the result of pairwise compariso; h=1idicates the previous compariso algorithm is sigificatly better tha the latter; h=represets o sigificat differece betwee the two compared algorithms; h = 1 idicates the previous compariso algorithm is sigificatly worse tha the latter. The results of compared with ad arelisteditables2ad4.theresultsshowthat surpasses ad i miimizig the te bechmark fuctios except F 3. Figure 5 illustrates the covergece process of the best target fuctio value of populatio i oe ruig. As show i Figure 5, the coverges faster tha ad. Sicehasalmostthesameformas,thesame crossover ad mutatio operator, sigle-poit crossover ad sigle-poit mutatio, are used i both algorithms. I, the elitist strategy is applied to improve the covergece ad optimizatio results. It should be oted that the ca ot coverge after 5 iteratios for most of the fuctios; for better compariso, the umber of iteratios of the is setto2itable3toesurethatthealgorithmisfully coverget. For high-dimesio fuctios, assume X i =(x i1,x i2,..., x iq ) is the biary strig of the particle (or idividual) i,where q is the umber of dimesios ad x ij is the jth substrig of X i. It is easy for or to exert crossover ad mutatio operatio o each substrig x ij i tur, istead of o the whole X i. For istace, i, Coditio (17) ca be writte as σd (c j,x ij )>l ( 1 ), (3) u

7 Computatioal Itelligece ad Neurosciece Target fuctio Target fuctio Iteratio Iteratio (a) F 1 (b) F Target fuctio Target fuctio Iteratio Iteratio (c) F 3 (d) F Target fuctio Target fuctio Iteratio Iteratio (e) F 5 (f) F 6 Figure 5: Cotiued.

8 8 Computatioal Itelligece ad Neurosciece Target fuctio 2 25 Target fuctio Iteratio Iteratio (g) F 7 (h) F Target fuctio Target fuctio Iteratio Iteratio (i) F 9 (j) F 1 Figure 5: Figures of the covergece processes of,, ad. for each substrig x ij of X i,wherec j are the jth substrig of C. The the process of whe the crossover ad mutatio operatio act o substrig ca be described as follows: the same operatio ca also be used i. (1) Iitialize a populatio of particles X i i biary space; (2) Set persoal best positio P i =X i,adcomputec; (3) Evaluate the fitess of particles f(x i ) ad determie the global best positio P g ; (4) while termiate coditio is ot reached do (5) for each particle i do (6) for each substrig of particle j do (7) Exert crossover operatio o P ij ad P gj to geerate two offsprig biary strigs, (8) G ij is radomly selected from them. (9) if coditio (3) is true, (1) Exert mutatio operatio o G ij ; (11) ed if (12) Set X ij =G ij ; (13) ed for j (14) Compute the fitess of particles f(x i ),adupdatep i, (15) ed for i

9 Computatioal Itelligece ad Neurosciece Target fuctio Target fuctio Iteratio Iteratio (a) F 1 (b) F Target fuctio Target fuctio Iteratio Iteratio (c) F 3 (d) F Target fuctio Target fuctio Iteratio Iteratio (e) F 5 (f) F 6 Figure 6: Cotiued.

10 1 Computatioal Itelligece ad Neurosciece Target fuctio Target fuctio Iteratio Iteratio (g) F 7 (h) F Target fuctio Target fuctio Iteratio Iteratio (i) F 9 (j) F 1 Figure 6: Figures of the covergece processes of ad whe the crossover ad mutatio operatio act o substrig. (16) Update P g ad the mea best positio C; (17) ed while The covergece processes of ad, whe crossover ad mutatio operatio act o substrigs of particles (or idividual), are show i Figure 6; the results of ad are listed i Tables 3, 4, ad 5. The experimetal results show that is obviously superior totheothesolutioaccuracyadthecovergece. For high-dimesio fuctios, it is effective to improve the covergece speed ad optimizatio ability, exertig crossover ad mutatio operatio o substrigs; as show i Tables 3 ad 4, it sigificatly improves the covergece rate of. But its ifluece is ot sigificat for ; accordig to Table 5, it has better performace i miimizig fuctios F 3, F 7, F 8,adF 9, especially for fuctio F Coclusios I this study,, a biary swarm itelligece algorithm, which is based o QPSO ad biary QPSO, is itroduced.

11 Computatioal Itelligece ad Neurosciece 11 Table 2: Miimizatio results for,, ad. Fuctio Algorithm The best Mea SD The worst Media F e e e e 5 F F F F F F F F F It coverts the movemet formula of QPSO to mutatio coditios, thus itroducig the mutatio operator of. has the similar form to but does ot eed to set the crossover ad mutatio probability, so it has fewer parameters to cotrol. itegrates strogpoit of ad PSO. The experimetal results show that is distictly superior to,, ad i terms of solutio accuracy ad covergece. Furthermore, sice has the same crossover ad mutatio operator as, may improvemets o the ca be applied to it; therefore, this algorithm has better applicatios ad research prospects.

12 12 Computatioal Itelligece ad Neurosciece Table 3: Miimizatio results for ad. Fuctio Algorithm The best Mea SD The worst Media 7.451e e e e 5 F e F F F F F F F F F The crossover ad mutatio operatio act o substrig.

13 Computatioal Itelligece ad Neurosciece 13 Table 4: Compariso of with other algorithms ad with. Fuctio Test p value <.1 < F 1 h p value <.1 <.1.1 <.1.34 F 2 h p value.724 <.1 <.1 < F 3 h p value <.1 < F 4 h p value <.1 <.1 <.1 < F 5 h p value <.1 <.1.6 < F 6 h p value.81 < F 7 h p value <.1.2 <.1 < F 8 h p value < <.1 <.1.43 F 9 h p value <.1 < < F 1 h The crossover ad mutatio operatio act o substrig. Table 5: Compariso of with other algorithms. Fuctio Test p value <.1 < F 1 h p value <.1 <.1.2 <.1.38 F 2 h p value < <.1 <.1 <.1 F 3 h p value <.1 < F 4 h p value <.1 <.1 <.1 < F 5 h p value <.1 <.1.2 < F 6 h p value <.1 <.1 <.1 <.1 <.1 F 7 h p value <.1 <.1 <.1 <.1 <.1 F 8 h p value <.1 <.1 <.1 <.1.44 F 9 h p value <.1 < < F 1 h The crossover ad mutatio operatio act o substrig.

14 14 Computatioal Itelligece ad Neurosciece Coflicts of Iterest The authors declare that they have o coflicts of iterest. Ackowledgmets The work was supported by the Natioal Natural Sciece Foudatio of Chia ( ). Refereces [1] R. C. Eberhart ad J. Keedy, A ew optimizer usig particle swarm theory, i Proceedigs of the 6th Iteratioal Symposium o Micromachie ad Huma Sciece, pp , Nagoya, Japa, October [2] F. Va de Bergh ad A. P. Egelbrecht, A ew locally coverget particle swarm optimiser, i Proceedigs of the Iteratioal Coferece o Systems, Ma ad Cyberetics, pp , October 22. [3] F.vadeBerghadA.P.Egelbrecht, Acovergeceprooffor theparticleswarmoptimiser, Fudameta Iformaticae, vol. 15, o. 4, pp , 21. [4] R. Poli, J. Keedy, ad T. Blackwell, Particle swarm optimizatio: a overview, Swarm Itelligece, vol. 1, o. 1, pp , 27. [5] A. A. A. Esmi, R. A. Coelho, ad S. 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