MODIFIED PARTICLE SWARM OPTIMIZATION FOR OPTIMIZATION PROBLEMS

Transcription

1 Journal of Theoretcal and Appled Informaton Technology 3 st ecember 0. Vol. No JATIT & LLS. All rghts reserved. ISSN: EISSN: MIFIE PARTICLE SARM PTIMIZATIN FR PTIMIZATIN PRBLEMS ZHA PENGJUN, LIU SANYANG AN 3 CHENG GU epartment of Mathematcs and Computatonal Scence, Shangluo Unversty, Shangluo 7000, P. R. Chna epartment of Appled Mathematcs, Xdan Unversty, X'an 7007, P. R. Chna 3 epartment of Mathematcs and Computatonal Scence, Shangluo Unversty, Shangluo 7000, P. R. Chna Emal: pengjunzhao@.com, lusanyang@.com, 3 chengguo3805@3.com ABSTRACT In the paper a modfed partcle swarm optmzaton (M s proposed where concepts from frefly algorthm (FA are borrowed to enhance the performance of partcle swarm optmzaton (. The modfcatons focus on the velocty vectors of the, whch fully use benefcal nformaton of the poston of partcles and ncrease randomzaton tem n the. Fnally, the performance of the proposed algorthm s compared wth that of the. Smulaton results demonstrate the effectveness of the proposed algorthm. Keywords: Partcle Swarm ptmzaton; Frefly Algorthm; MetaHeurstc Algorthm. INTRUCTIN The optmzaton problems frequently arse n almost every feld of the natural scences and the engneerng technology. urng the last few decades, Naturenspred Metaheurstc algorthms have been proposed for solvng the optmzaton problems. There are many dfferent metaheurstc algorthms for the optmzaton problems, such as dfferental evoluton (E [], ant colony optmzaton (AC [], frefly algorthm [3], and so on. Partcle swarm optmzaton (, proposed by Kennedy and Eberhart [5] n 995 s a new, selfadaptve global optmzaton algorthm based on the swarm behavor of brds and fsh, In the, a potental soluton for a gven problem s consdered as a partcle, a partcle fles through a dmensonal, realvalued search space and adjusts ts poston vector accordng to ts own experence and other partcles. The approach s becomng very popular due to ts smplcty of mplementaton and ablty to qucly converge to a reasonably good soluton; t has been successfully appled n a vast range of problems [58].To mprove the performance of the, Hongq L et al. [9] proposed a novel hybrd partcle swarm optmzaton algorthm combned wth harmony search for hgh dmensonal optmzaton problems,. Begambre et al. [0] proposed a hybrd partcle swarm optmzaton smplex algorthm for structural damage dentfcaton, Changsheng Zhang et al. [] proposed a novel hybrd dfferental evoluton and partcle swarm optmzaton algorthm for unconstraned optmzaton. Frefly algorthm (FA s a new metaheurstc algorthm whch s nspred from socal behavor of frefles n nature. Ths algorthm was developed recently by XnShe Yang at Cambrdge Unversty. It uses three dealzed rules: All frefles are unsex and can be attracted by other frefles; attractveness of each frefly s proportonal to ther brghtness and brghtness of each frefly s determned by evaluatng objectve functon. Further detals about the FA are gven n [3]. In order to mprove the search capablty of, the purpose of ths paper s to present a based on the part thought of frefly algorthm (M. To show the performance of ths algorthm, M s appled to four standard benchmar functons. Numercal results reveal that the proposed algorthm s a powerful search algorthm for optmzaton problems. 0

2 Journal of Theoretcal and Appled Informaton Technology 3 st ecember 0. Vol. No JATIT & LLS. All rghts reserved. ISSN: EISSN: The remander of the paper s organzed as follows: Secton descrbes the. The proposed approach (M s presented n Secton 3. Results of the experments are presented and dscussed n Secton. Fnally, Secton 5 concludes the paper.. THE PARTICLE SARM ALGRITHM In, every partcle has a poston vector x = ( x, x,, x and a velocty vector v = ( v, v,, v. At each tme step t, the velocty of partcle s updated accordng to Eq. and then ts poston s updated accordng to Eq.. v ( t + = ωv ( t + c r ( pbest x ( t + c r ( gbest x ( t x ( t + = x ( t + v ( t + ( ( here w s the nertal weght, and c and c are postve acceleraton coeffcents used to scale the contrbuton of cogntve and socal components, respectvely. pbest s the best poston that partcle has been vsted. gbest s the best poston found by all partcles n the swarm. r and r are unform random number n [0,], vd [ V, V ], and max max Vmax specfy maxmum of velocty. can be summarzed as the pseudo code shown n Fgure. begn ntalze the partcle populaton x (=,,,n and v whle (t <Max number of Generatons evaluate the ftness f (x, x = (x,..., x update pbest and gbest calculate new velocty accordng to Eq. update the poston accordng to Eq. end whle end Fgure Partcle Swarm ptmzaton 3. THE MIFIE PARTICLE SARM ALGRITHM In ths secton, the part thought of FA s used n the to accelerate convergence speed and also to enhance ts capablty for handlng optmzaton problems. The M has exactly the same steps as the wth the excepton that velocty vector s modfed as follows: In the M, the dstance between x and pbest, respectvely, s the Cartesan dstance r = pbest x ( t px = ( pbest x =,, (3 The dstance between x and gbest, respectvely, s the Cartesan dstance r = gbest x ( t gx = ( gbest x =, ( The velocty vectors v of the s randomly mutated by usng Eq.5. ωv ( t + cr ( pbest x ( t + cr ( gbest x ( t, r3 pa ( rpx v ( t + = ω v ( t + r e ( pbest x ( t ( r gx + r e ( gbest x ( t + α( r, else (5 here the thrd term s randomzaton wth the control parameter α, whch maes the exploraton of the search space more effcent. p a s a mutaton probablty, r 3 s unform random number n [0,], the proposed algorthm fully uses benefcal nformaton of the solutons to modfy the velocty vector v. Intutvely, ths modfcaton allows the M to wor effcently n both contnuous and dscrete problems.. EXPERIMENTS. Benchmars In ths secton, four well nown benchmar functons for mnmzaton are chosen to test the performance of M n comparson wth [7]. The test functons are lsted below: Sphere functon x = f ( x = (

3 Journal of Theoretcal and Appled Informaton Technology 3 st ecember 0. Vol. No JATIT & LLS. All rghts reserved. ISSN: EISSN: Rosenbroc functon ( = [00( + + ( ] = f x x x x Rastrgrn functon ( = [ 0 cos( π + 0] = f x x x Grewan functon ( = x cos( = = f x x (7 (8 (9 Search space ranges of the above benchmar functons for the experments are lsted n Table. Table Search Space For Each Test Functons Functon Search space Sphere Rosenbroc 30 x 30 Rastgrn 5. x 5. Grewan 00 x 00. Algorthm s Settngs And Expermental Results To evaluate the performance of the proposed, all common parameters of [7] and M are set the same to have a far comparson. All functons were mplemented n 30 dmensons. The results reported n ths secton are mean and standard dev. over 50 smulatons. The maxmum number of generatons (Ng was set to,000 for two algorthms, t s good to lmt the V max to the upper value of the range of search, =0. For the M, p a =0.9, Ng t w = ( , Ng c = c =. Table summarzes value data obtaned by applyng the two approaches to the benchmar x functons. As seen, for Sphere functon, Rosenbroc functon and Grewan functon, the result generated by M s better than those generated by, for Rastgrn functon, M slghtly outperformed. It can be concluded that the M outperformed n all four benchmar functons when the predefned number of generatons s completed. The modfed that combnes the dstance nformaton and randomzaton term s proved to be correct and effectve n convergng to the global optmal. Functo n Sphere Rosenb roc Rastgr n Grewa n Table Shows The Mean And STEV. f The Benchmar Functon ptmzaton Results Met hod Mean e e e Stdev e e e e

4 Journal of Theoretcal and Appled Informaton Technology 3 st ecember 0. Vol. No JATIT & LLS. All rghts reserved. ISSN: EISSN: M Mean FtnessValue(log Number of Generatons (a 0 8 M Mean FtnessValue(log Number of Generatons (b. M 5.8 Mean FtnessValue(log Number of Generatons (c 7 M 5 Mean FtnessValue(log Number of Generatons (d Fgure Varaton f The Mean That Are Best Ft th Tme. (A Sphere Functon. (B Rosenbroc Functon. (C Rastgrn Functon. ( Grewan Functon. 3

5 Journal of Theoretcal and Appled Informaton Technology 3 st ecember 0. Vol. No JATIT & LLS. All rghts reserved. ISSN: EISSN: Fgure show the search progress of the average values found by the two algorthms over 50 runs four functons, whch plot the ftness values (log aganst the number of generatons. From Fgure, t s clear that the M converges sgnfcantly faster than for Sphere functon, Rosenbroc functon and Grewan functon, the M converges slghtly faster than for Rastgrn functon. 5. CNCLUSINS Ths paper proposes a new smple but effectve and effcent modfed for contnuous optmzaton problems. The results obtaned show that by usng the M may yeld better solutons than those obtaned by usng, and demonstrate the effectveness and robustness of the proposed algorthm. In concluson, my research wor, therefore, suggests that the M s potentally a powerful search and optmzaton technque for solvng complex problems. In ths wor, we only consder the unconstraned functon optmzaton. ur future wor conssts on addng the dversty rules nto M for constraned optmzaton problems. ACKNLEGMENT The authors would le to than the anonymous referees for ther constructve and useful comments that have helped me to mprove the presentaton of ths paper. Ths wor s supported by Research Program Project of Educaton epartment of Shaanx Provncal Government (Grant No.JK057 and Scentfc Research Foundaton of Shangluo Unversty (Grant No. 0SKY0. REFERENCES [5] J. Kennedy and R. C. Eberhart, Partcle swarm optmzaton. Proceedngs of the 995 IEEE Internatonal Conference on Neural Networs, 995, vol., pp [] R. C. Eberhart and J. Kennedy, New optmzer usng partcle swarm theory, Proceedngs of the th Internatonal Symposum on Mcro Machne and Human Scence, 995, pp [7] Y.Sh, Emprcal study of partcle swarm optmzaton. Proceedngs of IEEE Internatonal Congress on Evolutonary Computaton, 999, vol. 3, pp.00. [8] A. Ratnaweeta, S. K. Halgamuge and H. C. atson, Selforganzng herarchcal partcle swarm optmzer wth tmevaryng acceleraton coeffcents. IEEE Transactons on Evolutonary Computaton, 00, Vol. 8, No. 3, pp.055. [9] Hongq L and L L, A Novel Hybrd Partcle Swarm ptmzaton Algorthm Combned wth Harmony Search for Hgh mensonal ptmzaton Problems. 007 Internatonal Conference on Intellgent Pervasve Computng, 007, pp.997. [0]. Begambre and J. E. Laer, A hybrd Partcle Swarm ptmzaton Smplex algorthm (S for structural damage dentfcaton. Advances n Engneerng Software, Vol.0, No. 9, 009, pp [] Changsheng Zhang, Jaxu Nng and Shua Lu, et al. A novel hybrd dfferental evoluton and partcle swarm optmzaton algorthm for unconstraned optmzaton. peratons Research Letters, Vol.37, No., 009, pp.7. [] R. Storn and K. Prce, fferental evoluton a smple and effcent heurstc for global optmzaton over contnuous spaces. Journal of Global ptmzaton, 997, vol., No., pp [] M. orgo and T. St ǚ utzle, Ant colony optmzaton, MIT Press, Cambrdge, MA, 00. [3] X.S. Yang, Frefly Algorthms for multmodal optmzaton, Lecture Notes n Computer Scence, 009,vol. 579, pp [] X.S. Yang, Frefly Algorthm, Stochastc Test Functons and desgn optmzaton, Internatonal Journal of BoInspred computaton, 00,Vol., No., pp. 788.