A Multimodal Firefly Optimization Algorithm Based on Coulomb s Law

Transcription

1 A Multmodal Frefly Optmzaton Algorthm Based on Coulomb s Law TAYMAZ RAHKAR-FARSHI Computer engneerng Istanbul Aydn Unversty Sefaköy-Küçükçekmece, Istanbul TURKEY Abstract: - In ths paper, a multmodal frefly algorthm named the CFA (Coulomb Frefly Algorthm) has been presented based on the Coulomb s law. The algorthm s able to fnd more than one optmum soluton n the problem search space wthout requrng any addtonal parameter. In ths proposed method, less brght frefles would be attracted to frefles whch are not only brghter, but accordng to the Coulomb s law pose the hghest gravty. Approachng the end of teraton, frefles' moton steps are reduced whch fnally results n a more accurate result. Wth lmted number of teratons, groups of frefles gather around global and local optmal ponts. After the fnal teraton, the frefly whch has the hghest ftness value, would be survved and the rest would be omtted. Experments and comparsons on the CFA algorthm show that the proposed method has successfully reacted n solvng multmodal optmzaton problems. Key-Words: - Swarm Intellgence, multmodal frefly algorthm, multmodal optmzaton, frefly algorthm Introducton Optmzaton s fndng an optmum soluton from a set of avalable optons wth the purpose of optmzng crtera for the problem n a lmted tme. The man challenge wth sngle soluton optmzaton algorthms, however, s that they are only able to fnd one optmum soluton from a set of avalable optons whle most real-world problems have more than one optmum soluton []. Hence, multmodal optmzaton algorthms whch are among the novel nventons of evolutonary algorthms, have been desgned to fnd a set of possble solutons from avalable optons. Unlke unmodal optmzaton algorthms whch try to avod local optmal ponts, multmodal optmzaton algorthms recognze these ponts as a soluton. Although normally the algorthms have not been bascally desgned to merely solve these problems, several algorthms have recently tred to solve these problems by modfyng exstng unmodal optmzaton algorthms. The majorty of these algorthms are based on partcle swarm optmzaton algorthms [-6] and genetc algorthms [7-0]. The frefly optmzaton algorthm has been used successfully to optmze dfferent knds of problems, but all of them have been wthn the span of unmodal optmzaton problems. In ths paper, the Coulomb s law has been appled to the frefly optmzaton algorthm n order to turn t nto a multmodal algorthm. 2 Related Work 2.. EPSO EPSO algorthm [3] was ntroduced by J. Barbara and Carlos A. C. n In ths method, the selecton of global optmum mechansm, n PSO algorthm, was changed usng Coulomb's law. Then, the partcles that are to be selected as the global optmum can be separately calculated for each partcle. In fact, partcles may move towards dfferent partcles as the global optma. In other words, the global optma for every partcle could vary from one partcle to another. Hence, partcles not only do not surround the global optma, but they also surround ther local optma. It s evdent that a partcle wth a more desrable cost functon s surrounded by more partcles. It s ths mechansm's property that partcles tend to move towards a pont that has both an approprate cost functon value and an approprate dstance from the partcle FERPSO FERPSO [2] s a well-known algorthm that has been proposed for solvng multmodal optmzaton problems whch was ntroduced by Xaodong L n In terms of nature, ths algorthm could be E-ISSN: Volume 7, 208

2 vewed as: more brds wll gather where there s more food. In fact, f they fnd a good resource near themselves, they wll not use farther resources. In FERPSO, the partcles that are to be selected as global optmal pont are selected for each partcle regardng the Eucldan dstance between partcles. In essence, the overall structure of FERPSO and EPSO are hghly smlar, and they both have the same level of complexty LPSO B. Y. Qu et al have combned a local searchng technque wth some exstng multmodal PSO optmzaton algorthms that have used nchng [2,, 2] method tryng to solve such problems. In ths method, the personal best for partcles are mproved sgnfcantly by usng a local searchng method. In fact, the personal best s mproved by generatng a random pont between the partcle and the nearest pont, that s, f the newer pont s more desrable than the current personal best, the new pont wll replace the former, otherwse, the orgnal pont stays ntact SNPSO J.Zhang et al. [3] proposed a modfed algorthm called the sequental nchng partcle swarm optmzaton (SNPSO). Ths algorthm dvdes the whole populaton nto several sup-populatons whch can be located around optmal solutons n multmodal problems. They use space convergence rate (SCR), n whch each sub-populaton detects global and local optmal solutons untl the end of teraton SPSO Xaodong l. [] proposed an mproved PSO algorthm called the (SPSO). In ths method, the dea of speces s used to specfy each speces best value of neghborhood. The algorthm dvdes the whole populaton nto several populatons called speces wth regard to ther smlarty. Each speces gather around a partcle called speces seed. 3 Frefly Algorthm 3.. The Behavor of Frefles n Nature There are almost two thousand known speces of frefly n nature, most of whch emt flashes of lght wth a certan rhythm n order to attract a matng partner or bat. In addton to these reasons, frefles can protect themselves aganst the attackers usng the flashes whch can also attract the opposte sexes. The dstance between the frefles and the envronment, where the lght s emtted, s somehow effectve on the ntensty of lght receved by frefles. As the lght ntensty obeys the nverse square law at a partcular dstance r (I /r2), and because lght s absorbed by ar, most frefles can just be vsble to a lmted dstance Frefly Algorthm The frefly algorthm s one of the novel optmzaton algorthms based on swarm ntellgence whch was frst ntroduced by X.Yang n 2008 [4]. It was nspred by the natural behavor of frefles. The frefly algorthm randomly dstrbutes a number of artfcal frefles n the search space at the begnnng. All of the frefles are unsexual and thus regardless of gender, each frefly can be attracted by any other frefly. Each frefly produces a lght whose ntensty depends on the optmalty of ts poston and s proportonal to ts ftness value. The next step s comparng constantly the ntensty of the lght of each frefly wth that of other frefles and less brght frefles moves towards brghter ones. Evdently, dependng on the dstance, frefles receve lghts wth varyng ntenstes; however, the brghtest frefly moves randomly n search of space to ncrease ts chance of fndng the global optmum soluton. Movement of the less brght frefly towards the brghter one s expressed through equaton (). 2 r j x x 0e x j x rand 2 () Where β 0 s the maxmum coeffcent of attracton between th and jth frefles, α s the coeffcent of random dsplacement vector, γ s the lght absorpton coeffcent for the envronment, and r j s the Eucldean dstance between two frefles. Each frefly s compared to all others and f ts ftness value s less than that of another one, t wll be attracted accordng to equaton (). Ths trend contnues to the last algorthm teraton when fnally the optmum soluton s obtaned as the fnal soluton. Man steps of the frefly algorthm can be expressed n the form of the pseudo-code brefed n Algorthm (FA). E-ISSN: Volume 7, 208

3 Algorthm Pseudo-code for FA man steps Objectve functon f(x), x = (x,..., x d) T Generate ntal populaton of frefles x ( =, 2,..., n) Lght ntensty I at x s determned by f(x ) Defne lght absorpton coeffcent γ : whle (t <MaxGeneraton) 2: for = : n all n frefles 3: for j = : all n frefles 4: f (I j > I ) 5: Move frefly towards j n d-dmenson; 6: end f 7: Attractveness vares wth dstance r va exp[ γr] 8: Evaluate new solutons and update lght ntensty 9: end for j 0: end for : Rank the frefles and fnd the current best 2: end whle Postprocess results and vsualzaton 4 Multmodal Optmzaton Constrants such as physcal, temporal and economc lmtatons can prevent achevement of actual results; however, havng knowledge of multmodal optmzaton solutons s very useful n engneerng felds. In such cases, f multple local and global solutons are avalable, the optmum system performance s obtaned by swtchng between solutons. Snce there are several solutons to many real-world problems, multmodal optmzaton algorthms are useful for solvng these problems. Not only are these algorthms able to locate multple optma n a sngle run, but they also preserve ther populaton dversty. The reason why classc optmzaton technques are not used to fnd multple solutons shows ther unrelablty n fndng more than one soluton n multple runs [5]. Evolutonary algorthms ncludng Genetc Algorthms (GAs), Dfferental Evoluton (DE), Partcle Swarm Optmzaton (PSO), and Evoluton Strategy (ES) are knds of algorthms whch has been tred to solve multmodal optmzaton problems. Referred algorthms [7, 0, 5-23] are among algorthms desgned to the aforementoned crtera. 5 Multmodal Frefly Algorthm Studes on multmodal optmzaton have mostly focused on the PSO and genetc algorthms. In ths paper, lke other meta-heurstc algorthms whch have used unmodal algorthms for solvng multmodal optmzaton problems, some changes are made on FA algorthm wthout the need for any addtonal parameter, and t has been utlzed to solve multmodal optmzaton problems. In the proposed algorthm, the Coulomb s law n equaton (2) has been used to calculate the electrostatc nteracton between two frefles. Ths technque was successfully used by J. Barrera and A. Coello n [3] to obtan a multmodal PSO algorthm. They have used ths method to calculate forces between two partcles whch has also been used n the present paper to calculate the attracton between two frefles. F Q. Q. j j, 2 4π 0 r (2) In ths equaton, s the proportonalty constant 4πƐ 0 (Coulomb constant); Q and Q j denote the magntude of two charged partcles, and r s the dstance between two charges. Accordng to ths formula, force magntude s proportonal to magntude of charges but t obeys nverse square law for dstance. Hence, the attracton between two frefles s calculated through the followng equaton. f ( p ). f ( p ) j F., j 2 p p j (3) In ths equaton, α s equal to and f(p ) s the ftness value for the frefly whch wll be attracted to one of the present frefles. f(p j ) s the vector of the ftness value of other frefles to whch the th frefly s beng compared. Fnally, the destnaton frefly s obtaned by usng equaton (4). E-ISSN: Volume 7, 208

4 Algorthm 2 Pseudo-code for CFA man steps Objectve functon f(x), x = (x,..., x d) T Generate ntal populaton of frefles x ( =, 2,..., n) Lght ntensty I at x s determned by f(x ) Defne lght absorpton coeffcent γ : whle (t <MaxGeneraton) 2: γ= γ +t / (MaxGeneraton*5); 3: α = α * (-t / )MaxGeneraton*2); 4: for = : n all n frefles 5: for j = : all n frefles 6: f (I j > I ) 7: push(f,( I * I j)/norm(x()-x(j))^2); */ n d-dmenson 8: end f 9: end for j 0: Move frefly towards j max (f) n d-dmenson; : Attractveness vares wth dstance r va exp[ γr] 2: Evaluate new solutons and update lght ntensty 3: end for 4: end whle Postprocess results and vsualzaton Fmax argmax F F, j (4) Calculatng the above equaton yelds to the maxmum value of F (,j).fnally, the ndex of the j th frefly, wth the hghest value of F, s calculated and equaton (5) s obtaned by changng equaton (). r 2 Fmax j Fmax 0 j x x e x x rand (5) 2 As a result, the th frefly s attracted to the frefly whch has the hghest value of F. Therefore, destnaton frefles are not selected just based on the value of ftness value but from calculatng the electrostatc nteracton between other frefles. Ths method prevents the attracton of other frefles by the best frefly. Instead, frefles are attracted by frefles whch besdes havng suffcent ftness value, must be at a close dstance snce dstance s an effectve parameter n ther attracton. In each teraton n ths case, each frefly compares ts electrostatc nteracton wth others and then moves toward the frefly whch has the hghest electrostatc nteracton. As t was mentoned before, α s the coeffcent of random dsplacement wth a value consdered to be [0.- ] at the begnnng. Ths makes the frefly s movements to be random to some extent and to search for new sources; however, ths value of α results n a less precse soluton at the end of teraton. To prevent t, the coeffcent of random movement s reduced n each teraton so as to reduce the randomness of the movement of the frefly toward the destnaton. Moreover, the value of γ s ncreased n each teraton so that frefles take smaller steps at the end of the teraton. These two actons take place usng equatons (6) and (7). Iteraton α α MaxGeneraton 2 (6) Iteraton γ γ MaxGeneraton 5 (7) Man steps of CFA can be summarzed nto the pseudo-code shown n Algorthm 2. 6 Expermental result 6.. Test Functons The experments have been performed on benchmark functons common n multmodal optmzaton. Specfcatons of these algorthms are presented n Table () Confguratons All algorthms were mplemented n Matlab 203 and were run n a computer equpped wth an Intel Core(TM) QM 2.2 GH processor and 8 ggabytes of RAM. E-ISSN: Volume 7, 208

5 Table Test Functon Test functon cos 2π cos 2π x x 2 20 e f x, x 2 20 ( 20.exp x x 2 e f x, x x 0cos 2 x x 0cos 2 x f x, x 220 cos( x ). cos( x ). 2 2 x f x x x x x x x x , f x x 2 x x 2 x x 2, 2500 ( 7 ) 6 f x sn 6 x 7 Range 5 < x < 5 5 < x 2 < < x < < x 2 < < x < < x 2 < < x <.9. < x 2 <. 6 < x < 6 6 < x 2 < 6 Peaks Global/Local < x < 5/5 2log f x 5 e sn 5 x 2 x 0. / / 4/20 2/6 4/4 0 < x < /5 f = Ackley, f2 = Rastrgn, f3 = Shubert, f4 = Sx-hump camel back, f5 = Hmmelblau, f6 = Equal maxma, f7 = Decreasng maxma 6.3. Performance measures To assess the performance of aforementoned algorthms n secton (6.4), the followng 7 crtera are consdered and measured 50 runs.. Rates (SR): The percentage of performances n whch all the optmum ponts have been found successfully. To calculate success rate, a user specfed parameter, called the Level of Accuracy (LOA), s consdered. Ths parameter s usually between (0,] and s used to measure the dfference between found solutons and the real optmum ponts n functons, so that f the dfference between found soluton and the real soluton s less than the amount of LOA, then the found soluton s counted as a successful soluton [2]. 2. Number Of Optma Found (ANOF): The average of optmum ponts found consderng LOA for 50 runs 3. Global Number of Optma Found: The average of global optma found consderng LOA for 50 runs 4. Functon Evaluaton s the average number of ftness functon callng for 50 runs []. 5. Performance (SP): Ths parameter s computable when the amount of SR s not zero [24]. SP s calculated from equaton (8) : SP Avarage Number Of Functon Evaluaton ANOF (8) SR Based on the fact that algorthms wth less ANOF and hgher SR amount can be consdered as a better one, t can be concluded that less SP amount s more acceptable. 6. Maxmum Peak Rato (MPR): The qualty of optma s tested wthout consderng the populaton dstrbuton, and the performance metrc, whch s called the maxmum peak rato statstc (MPR), s adopted. The MPR s defned as follows: C f MPR C (9) q F q c: The number of found optmum pont n the soluton q: The number of real optmum ponts n the soluton f : The quantty of ftness functon obtaned n the fnal populaton F j : The quantty of real ftness value n objectve functon 7. Precson : the rato of found optmum ponts to the number of real optmum ponts. E-ISSN: Volume 7, 208

6 Precson c q (0) 6.4. Test and comparson results The results of the experments are shown n the tables 2-4 whch have the accuraces of 0-, 0-2 and 0-3 respectvely. Frst columns of all three tables represent test functons; second columns are equvalent of mplemented algorthms and the other columns are as wrtten on top of each column. The best performance was reported n boldface. As t can be seen from the results, wth the ncrease of the level of accuracy, the proposed algorthm has a better performance compared to FER-PSO [2], EPSO [3] and LS-FER-PSO [] algorthms. It s shown n the tables that the presented algorthm and LS-FER-PSO algorthm have better performance compared to other algorthms. Comparng LS-FER- PSO algorthm and presented algorthm, t s shown that sometmes one performance s better than the other and vce versa. However t should be mentoned that the rato of average functon evaluaton of LS-FER-PSO algorthm s.67 tmes hgher than that of the presented algorthm. The performance of ths algorthm proved ts usefulness n solvng optmzaton problems. Fg.. shows the search space of f4. Fg. 2 also ndcates the poston of frefles durng the runnng of the proposed algorthm wth 60 frefles and 60 teratons usng the f4 functon. Fg. 2. A-D shows the poston of partcles n the st, 0 th, 20 th and 30 th teratons. In fact, 4 out of the 6 avalable ponts were successfully found n 800 functon evaluaton. However all optmum ponts can be found by ncreasng the number of frefles. As t can be clearly observed n Fg. 2, by gettng close to the end of the teratons, frefles around optmum solutons become gradually more and more concentrated. Snce the frefly algorthm was desgned for maxmum optmzaton problems, the average value of cost functons of frefles ncreased n each teraton Fg. 3. Moreover, as seen n Fg. 4, as the concentraton of frefles around optma (.e. around each other) ncreased, the standard devaton of cost functons decreased. The reason was that the more the process of the algorthm got closer to the end of the teraton, the more the frefles and ther cost functons got closer to form neghborhoods. algorthm was successful n solvng multmodal optmzaton problems. Results of experments ndcated that ths unmodal optmzaton algorthm was successfully turned nto a multmodal optmzaton algorthm through modfcatons. Two of the advantages of ths algorthm are quckly yeldng optmal results and not requrng addtonal parameter for beng turned nto a multmodal algorthm. Accordng to the results, ths algorthm can be consdered as a relable multmodal optmzaton algorthm. Fg.. search landscape of f4 7 Concluson Ths paper proposed CFA multmodal frefly algorthm based on the Coulomb s law. Ths E-ISSN: Volume 7, 208

7 Table2 The results of the experments - accuracy: 0 - Functon Func- Func-2 Func-3 Func-4 Func-5 Func-6 Func-7 Algorthm Rate Optma Found Global Optma Found Mean Peak Raton Precson Performance Functon Evaluaton CFA Inf FER-PSO Inf EPSO Inf LS-FER-PSO Inf CFA Inf FER-PSO Inf EPSO Inf LS-FER-PSO CFA Inf FER-PSO Inf EPSO Inf LS-FER-PSO Inf CFA FER-PSO EPSO LS-FER-PSO CFA FER-PSO EPSO LS-FER-PSO CFA FER-PSO EPSO 0 Inf 500 LS-FER-PSO CFA FER-PSO EPSO Inf 500 LS-FER-PSO Table3 The results of the experments - accuracy: 0-2 Functon Func- Func-2 Func-3 Func-4 Func-5 Func-6 Func-7 Algorthm Rate Optma Found Global Optma Found Mean Peak Raton Precson Performance Functon Evaluaton CFA Inf FER-PSO Inf EPSO Inf LS-FER-PSO Inf CFA Inf FER-PSO Inf EPSO Inf LS-FER-PSO Inf CFA Inf FER-PSO Inf EPSO Inf LS-FER-PSO Inf CFA FER-PSO Inf 0200 EPSO Inf 0200 LS-FER-PSO Inf CFA FER-PSO Inf 0200 EPSO Inf 0200 LS-FER-PSO CFA FER-PSO EPSO 0 Inf 500 LS-FER-PSO CFA Inf 600 FER-PSO EPSO Inf 500 LS-FER-PSO E-ISSN: Volume 7, 208

8 Table4 The results of the experments - accuracy: 0-3 Functon Func- Func-2 Func-3 Func-4 Func-5 Func-6 Func-7 Algorthm Rate Optma Found Global Optma Found Mean Peak Raton Precson Performance Functon Evaluaton CFA Inf FER-PSO Inf EPSO Inf LS-FER-PSO Inf CFA Inf FER-PSO Inf EPSO Inf LS-FER-PSO Inf CFA Inf FER-PSO Inf EPSO Inf LS-FER-PSO Inf CFA Inf 2200 FER-PSO Inf 0200 EPSO Inf 0200 LS-FER-PSO Inf CFA FER-PSO Inf 0200 EPSO Inf 0200 LS-FER-PSO CFA FER-PSO EPSO 0 Inf 500 LS-FER-PSO CFA Inf 600 FER-PSO EPSO Inf 500 LS-FER-PSO A: teraton B: teraton 0 E-ISSN: Volume 7, 208

9 C: teraton 20 D: teraton 30 Fg. 2. the poston of partcles n dfferent teratons (begnnng to end) Fg. 3. The average value of cost functon n each teraton Fg. 4. The standard devaton of cost fucntons of frefles n each teraton References: [] B.-Y. Qu, J. J. Lang, and P. N. Suganthan, "Nchng partcle swarm optmzaton wth local search for mult-modal optmzaton," Informaton Scences, vol. 97, pp. 3-43, 202. [2] X. L, "A multmodal partcle swarm optmzer based on ftness Eucldean-dstance rato," n Proceedngs of the 9th annual conference on Genetc and evolutonary computaton, 2007, pp [3] J. Barrera and C. A. C. Coello, "A partcle swarm optmzaton method for multmodal optmzaton based on electrostatc nteracton," n MICAI 2009: Advances n Artfcal Intellgence, ed: Sprnger, 2009, pp [4] M. L, D. Ln, and J. Kou, "A hybrd nchng PSO enhanced wth recombnaton-replacement crowdng strategy for multmodal functon optmzaton," Appled Soft Computng, vol. 2, pp , 202. [5] E. Özcan and M. Yılmaz, "Partcle swarms for multmodal optmzaton," n Adaptve and Natural Computng Algorthms, ed: Sprnger, 2007, pp [6] T. Rahkar-Farsh, S. Behjat-Jamal, and M.-R. Fez-Derakhsh, "An mproved multmodal PSO method based on electrostatc nteracton usng n- nearest-neghbor local search," arxv preprnt arxv: , 204. [7] T. Grünnger and D. Wallace, "Multmodal optmzaton usng genetc algorthms," Master's thess, Stuttgart Unversty, 996. [8] E. Dlettoso and N. Salerno, "A self-adaptve nchng genetc algorthm for multmodal optmzaton of electromagnetc devces," Magnetcs, IEEE Transactons on, vol. 42, pp , [9] R. K. Ursem, "Multnatonal GAs: Multmodal Optmzaton Technques n Dynamc Envronments," n GECCO, 2000, pp E-ISSN: Volume 7, 208

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