Particle Swarm Optimization with Adaptive Mutation in Local Best of Particles

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1 1 Internatonal Congress on Informatcs, Envronment, Energy and Applcatons-IEEA 1 IPCSIT vol.38 (1) (1) IACSIT Press, Sngapore Partcle Swarm Optmzaton wth Adaptve Mutaton n Local Best of Partcles Nanda ulal Jana a*, Tapas S a and Jaya Sl b a epartment of Informaton Technology, Natonal Insttute of Technology, urgapur, Bengal, Inda b epartment of Computer Scence and Technology, BESU, West Bengal, Inda Abstract. Partcle Swarm Optmzaton (PSO) has shown ts good search ablty n many optmzaton problems. But PSO easly gets trapped nto local optma whle dealng wth complex problems due to lacks n dversty. In ths work, we proposed an mproved PSO, namely PSO-APMLB, n whch adaptve polynomal mutaton strategy s employed n local best of partcles to ntroduce dversty n the swarm space. In the frst verson of ths method (PSO-APMLB1), each local best s perturbed n the current search space nstead of entre search space. In the second verson of ths method (PSO-APMLB), each local best s perturbed n terms of the entre search space. We also proposed another local best mutaton method, namely, PSO-AMLB, n whch mutaton sze s controlled dynamcally n terms of current search space. In ths work, we carred out our experments on 8 well-known benchmark functons. Fnally the results are compared wth PSO. From the expermental results, t s found that the proposed algorthms performed better than PSO. Keywords: Partcle swarm optmzaton; adaptve polynomal mutaton;adaptve mutaton 1. Introducton Partcle swarm optmzaton [1] s a populaton based global search technque havng a stochastc nature. It has shown ts good search ablty n many optmzaton problems wth faster convergence speed. However, due to lack of dversty n populaton, PSO easly trapped nto local optma whle dealng wth complex problems. fferent mutaton strateges lke Cauchy Mutaton [3], Gaussan Mutaton [], Power Mutaton [7], Adaptve Mutaton [6] s ntroduced nto PSO for solvng local optma problem. Changhe L et al. [4] ntroduced fast partcle swarm optmzaton wth cauchy mutaton and natural selecton strategy. Andrew Stacey et al. [5] used mutaton n PSO wth probablty 1/d, where d s the dmenson of partcles. JIAOWe et al. [9] proposed elte partcle swarm optmzaton wth mutaton. Xaolng Wu et al.[7] ntroduced power mutaton nto PSO (PMPSO). Coello [13] presented a hybrd PSO algorthm that ncorporates a non-unform mutaton operator smlar to the one used n evolutonary algorthms. Pant [1] used an adaptve mutaton operator n PSO. Hgash [] proposed a PSO algorthm wth Gaussan mutaton (PSO-GM). A new adaptve mutaton by dynamcally adustng the mutaton sze n terms of current search space s proposed n [6]. A.J. Nebro et al. [1] appled a polynomal mutaton to the 15 percentage of the partcles. Tapas S et al. [17] ntroduced adaptve polynomal mutaton n global best partcles n PSO (PSO-APM). In ths work, our obectve s to use adaptve polynomal mutaton and adaptve mutaton n local best soluton n PSO to solve local optma problem and to analyss the performance and effectveness of adaptve polynomal mutaton. A comparatve study s also made wth PSO havng lnearly decreasng nerta weght.. Partcle Swarm Optmzaton(PSO) PSO s an optmzaton algorthm by smulatng the behavour of fsh schoolng and brd s flockng. PSO algorthms use a populaton of ndvdual called partcles. Each partcle has ts own poston and velocty to move around the search space. Partcles have memory and each partcle keep track of prevous best poston and correspondng ftness. The prevous best value s called as pbest. Thus pbest s related only to a

2 partcular partcle. It also has another value called gbest, whch s the best value of all the partcles pbest n the swarm. The basc concept of PSO technque les n acceleratng each partcle towards ts pbest and the locatons at each tme step. pbest gbest ( ) ( ) S ( t+ 1) = w v ( t) + c r x ( t) x ( t) + x ( t) x ( t) (1) 1 1 x( t+ 1) = v( t) + x( t) () c 1 and c are personal and socal cognzance of a partcles respectvely and r 1 and r are two unformly dstrbuted random numbers n the nterval (,1).The nerta weght w n (7) was ntroduced by Sh and Eberhart [14]. They proposed a w lnearly decreasng wth the teratve generaton as w= wmax ( wmax wmn ) g (3) G where g s the generaton ndex representng the current number of evolutonary generaton and G s the predefned maxmum number of generatons. 3. Proposed Algorthms Polynomal mutaton s based on polynomal probablty dstrbuton. u X = ( t+ 1) = x( t) + ( x x) δ (4) u l Where x s the upper bound and x s the lower bound of x. The parameter δϵ [-1, 1] s calculated from the polynomal probablty dstrbuton. η p( δ) =.5 ( η 1)(1 ) m m + δ (5) η m s the polynomal dstrbuton ndex. δ = { 1/( ηm+ 1) ( r) 1, r.5 1/( ηm+ 1) 1 [(1 r) ], r.5 η = 1 + t (7) m In the proposed algorthm, adaptve polynomal mutaton s done n local best of the partcles wth the mutaton probablty: 1 t 1 pm = + 1 (8) d tmax d where d s the dmenson of the problem,t s the current teraton number and s the maxmum teraton number. pbest pbest mx = x + ( b( t) a ( t)) δ (9) Where δ s calculated usng Eq.(6) and η m s calculated usng Eq.(7) a() t = mn( x()) t b() t = max( x()) t (1) In ths work, mutaton s employed n PSO wth decreasng nerta weght. The man feature of ths method s that each local best of partcles s muted wth the ncreasng probablty Pm by dynamc mutaton sze n terms of current search space and decreasng polynomal ndex wth ncreasng teraton PSO-APMLB1 Algorthm 1. Intalze the partcle wth unform dstrbuted random numbers.. Update velocty and poston vectors usng Eq.(1 ) and Eq.(). 3. Calculate ftness value, f ftness() 4. f ( ftness() <= Pbest()) then set Pbest() = ftness(). f (Pbest () <= gbest) then set gbest = Pbest (). 5. f (U(,1)<= Pm) then Apply mutaton to Pbest usng Eq.(8) 6. Evaluate the ftness value, Pftness() (6)

3 7. f ( ftness() <= Pbest()) then set Pbest() = ftness(). 8. f (Pbest () <= gbest) then set gbest = Pbest(). 9. Repeat the loop untl maxmum teraton or maxmum number of functon evaluaton s reached. In PSO-APMLB algorthm, mutaton s done by usng the followng equaton mx pbest pbest ( u = x + x x ) δ (11) In PSO-AMLB algorthm, mutaton s done by usng the followng equaton pbest pbest mx = x + ( b ( t) a ( t)) rand (1) 4. Expermental Studes There are 8 dfferent global optmzaton problems, ncludng 4 un-modal functons (f1- f4) and 4 mult-modals functons (f5- f8), are chosen n our expermental studes. These functons were used n an early study by X. Yao et al. [18]. All functons are used n ths work to be mnmzed. The descrpton of these benchmark functons and ther global optma are gven n Table 1. In ths experment, the parameters are set as followng: populaton sze s set to, =3,1 5 number of functon evaluatons are allowed for each problem. Intal dstrbuton ndex for polynomal mutaton = 1. c 1 = c = , w max =.9 and w mn =.4 The obtaned results are presented n Tables & 3. Mean Best s the mean of best solutons and standard devaton of the best soluton of 3 runs for each test problem are reported n Table & 3.In Fg. 1, convergence graph of PSO-APLB1 for functon f5 s gven. Form the Table & 3,t can be easly sad that our proposed algorthms PSO-APMLB and PSO-AMLB performed better then Adaptve PSO.PSO-AMLB performed better for functons f 1,f 3, f 5 and f 7.PSO-APMLB performed better for functon f. PSO-APMLB1 performed better for functons f 4,f 6 and f8.for a comparson of overall performance, PSO-APMLB1 perform better for all functons except the hghly mult-modal functon f 5 comparatve to PSO-AMLB. Table 1: The 8 benchmark functons used n our experments, where s the dmenson of the functons, fmn s the mnmum values of the functons, and S R n the search space Test Functon S f mn f 1( x) = x = 1 f ( x ) = x = 1 = n 1 3 ( ) = ( 1 ) = 1 f x x 5. Concluson 1 4( ) = [1( + 1 ) + ( 1) ] = 1 5( ) = * sn( ) = 1 f x x x x f x x x x x f 6( x) = cos( ) + 1 = 1 4 f ( x ) = * exp(. * x ) exp( cos( π x )) + + e f 8 x x π x = 1 = 1 = 1 ( ) = [ 1 cos( )) + 1] [-1,1] [-1,1] [-1,1] [-1,1] [-5,5] [-6,6] [-3,3] [-5.1,5.1] In ths work, we proposed adaptve polynomal mutaton and adaptve mutaton n local best of partcles n partcle swarm op-best of partcles to ntroduce dversty n the populaton and to solve local optma problem. PSO wth adaptve polynomal mutaton and adaptve mutaton n local best of partcles produces the better results than PSO. But PSO-APMLB produces poor performance for mult-modal functon f 5 and f 8

4 wth respect to the global optmums of the aforementoned functons. Our future works wll be drected towards solvng local mnma problem n complex mult-modal functon optmzaton. Fg. 1: Convergence graph of PSO for f5 Table : Functon values acheved by PSO and PSO-APMLB1 PSO wth decreasng ω PSO-APMLB1 Problem Mean Std. ev. Mean Std. ev. f 1.58e e-1 1.5e e-13 f f e e-1.16e-9 f f f e-.41e- 1.54e- 1.5e- f f Table 3: Functon values acheved by PSO-APMLB and PSO-APML 6. References Problem PSO-APMLB PSO-AMLB Mean Std. ev. Mean Std. ev. f 1.64E E-1 3.7e e-14 f f e e e e-14 f f f 6.54e-.5e- 1.76e- 1.77e- f f [1] Kennedy J and Eberhart R. Partcle Swarm Optmzaton, In Proc. IEEE Internatonal Conference on Neural Networks, Perth, Australa, [] Hgash N, lba H, Partcle Swarm Optmzaton wth Gaussan Mutaton, In Proc. IEEE Swarm Intellgence Symposum, Indanapols, 3, pp [3] Wang H, Lu Y, L C. H., Zeng S. Y. A hybrd partcle swarm algorthm wth Cauchy mutaton, In Proc. of IEEE Swarm Intellgence Symposum, 7, pp

5 [4] L C, Lu Y., Zhao A, Kang L, Wang H., A Fast Partcle Swarm Algorthm wth Cauchy Mutaton and Natural Selecton Strategy,,LNCS 4683, 7, pp , Sprnger-Verlag [5] Stacey A, Jancc M, Grundy I,Partcle swarm optmzaton wth mutaton, n Proc. Congr. Evol. Comput, 3, pp [6] Tang J, Zhao X, Partcle Swarm Optmzaton wth Adaptve Mutaton, WASE Internatonal Conference on Informaton Engneerng,9 [7] Wu X, Zhong M, Partcle Swarm Optmzaton Based on Power Mutaton, ISECS Internatonal Colloquum on Computng, Communcaton, Control, and Management,9 [8] Gao Y, Ren Z, Adaptve Partcle Swarm Optmzaton Algorthm Wth Genetc Mutaton Operaton,Thrd Internatonal Conference on Natural Computaton(ICNC 7). [9] We J, Guangbn L,ong L, Elte Partcle Swarm Optmzaton wth Mutaton, In Proc. Asa Smulaton Conference - 7th Internatonal Conference on System Smulaton and Scentfc Computng, 8. ICSC 8. [1] Nebro A.J, SMPSO: A New PSO-based Metaheurstc for Mult-obectve Optmzaton, eee symposum on Computatonal ntellgence n mult-crtera decson-makng,9 [11] eb K, Mult-Obectve Optmzaton Usng Evolutonary Algorthms,John Wley and Sons,1. [1] Pant M, Thangara R, Abraham A., Partcle Swarm Optmzaton usng Adaptve Mutaton, n Proc. 19th Internatonal Conference on atabase and Expert Systems Applcaton, 8, pp [13] Esquvel S. C, Coello Coello C. A., On the use of partcle swarm optmzaton wth multmodal functons,n Proc. Congr. Evol.Comput., 3, pp , [14] Chen J, Ren Z, Fan X, Partcle swarm optmzaton wth adaptve mutaton and ts applcaton research n tunng of PI parameters, n Proc. 1st Internatonal Symposum on Systems and Control n Aerospace and Astronautcs, 6, pp [15] Sh Y, Eberhart R.C. A modfed partcle swarm optmzer, n proc. IEEE World Congr. Comput.Intell.,1998,pp [16] X. Yao, Y. Lu and G. Ln, Evolutonary programmng made faster,ieee Transactons on Evolutonary Computaton.,vol. 3,1999,pp. 8-1 [17] S T, Jana N., Sl J. Partcle Swarm Optmzaton wth Adaptve Polynomal Mutaton, n proc. World Congress on Informaton and Communcaton Technologes(WICT 11),Mumba,Inda,11,pp