Research Article Adaptive Parameters for a Modified Comprehensive Learning Particle Swarm Optimizer

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1 Hndaw Publshng Corporaton Mathematcal Problems n Engneerng Volume 2012, Artcle ID , 11 pages do: /2012/ Research Artcle Adaptve Parameters for a Modfed Comprehensve Learnng Partcle Swarm Optmzer Yu-Jun Zheng, 1 Ha-Feng Lng, 2 and Qu Guan 1 1 College of Computer Scence and Technology, Zhejang Unversty of Technology, Hangzhou, Zhejang , Chna 2 Engneerng Insttute of Engneerng Corps, PLA Unversty of Scence and Technology, Nanjng, Jangsu , Chna Correspondence should be addressed to Yu-Jun Zheng, yujun.zheng@computer.org Receved 5 October 2012; Accepted 25 November 2012 Academc Edtor: Sheng-yong Chen Copyrght q 2012 Yu-Jun Zheng et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Partcle swarm optmzaton PSO s a stochastc optmzaton method senstve to parameter settngs. The paper presents a modfcaton on the comprehensve learnng partcle swarm optmzer CLPSO, whch s one of the best performng PSO algorthms. The proposed method ntroduces a self-adaptve mechansm that dynamcally changes the values of key parameters ncludng nerta weght and acceleraton coeffcent based on evolutonary nformaton of ndvdual partcles and the swarm durng the search. Numercal experments demonstrate that our approach wth adaptve parameters can provde comparable mprovement n performance of solvng global optmzaton problems. 1. Introducton The complexty of many real-world problems has made exact soluton methods mpractcal to solve them wthn a reasonable amount of tme and thus gves rse to varous types of nonexact metaheurstc approaches 1 3. In partcular, swarm ntellgence methods, whch smulate a populaton of smple ndvduals evolvng ther solutons by nteractng wth one another and wth the envronment, have shown promsng performance on many dffcult problems and have become a very actve research area n recent years Among these methods, partcle swarm optmzaton PSO, ntally proposed by Kennedy and Eberhart 4, s a populaton-based global optmzaton technque that nvolves algorthmc mechansms smlar to socal behavor of brd flockng. The method enables a number of ndvdual solutons, called partcles, to move through the soluton space and towards

2 2 Mathematcal Problems n Engneerng the most promsng area for optmal soluton s by stochastc search. Consder a D-dmensonal optmzaton problem as follows: mn f x, x x 1,x 2,...x D T s.t. x L x x U, 0 < D. 1.1 In the D-dmensonal search space, each partcle of the swarm s assocated wth a poston vector x x 1,x 2,...,x D and a velocty vector v v 1,v 2,...,v D, whch are teratvely adjusted by learnng from a local best pbest found by the partcle tself and a current global best gbest found by the whole swarm: v t 1 v t c 1 r 1 pbest t x t 1 x t x t c 2 r 2 gbest t x t, 1.2 v t 1, 1.3 where c 1 and c 2 are two acceleraton constants reflectng the weghtng of cogntve and socal learnng, respectvely, and r 1 and r 2 are two dstnct random numbers n 0, 1. Its recommended that c 1 c 2 2 snce t on average makes the weghts for cogntve and socal parts both to be 1. To acheve a better balance between the exploraton global search and explotaton local search, Sh and Eberhart 12 ntroduce a parameter named nerta weght w to control velocty, whch s currently the most wdely used form of velocty update equaton n PSO algorthms: v t 1 wv t c 1 r 1 pbest t x t c 2 r 2 gbest t x t. 1.4 Emprcal studes have shown that a large nerta weght facltates exploraton and a small nerta weght one facltates explotaton and a lnear decreasng nerta weght can be effectve n mprovng the algorthm performance: w t w mn t max t t max w max w mn, 1.5 where t s the current teraton number, t max s the maxmum number of allowable teratons, and w max and w mn are the ntal value and the fnal value of nerta weght, respectvely. It s suggested that w max can be set to around 1.2 and w mn around 0.9, whch can result n a good algorthm performance and remove the need for velocty lmtng. PSO s conceptually smple and easy to mplement, and has been proven to be effectve n a wde range of optmzaton problems Furthermore, It can be easly parallelzed by concurrently processng multple partcles whle sharng the socal nformaton 21, 22. Kwok et al. 23 present an emprcal study on the effect of randomness on the control coeffcents of PSO, and the results show that the selectve and unformly dstrbuted random coeffcents perform better on complcate functons. In recent years, PSO has attracted a hgh level of nterest, and a number of varant PSO methods e.g., have been proposed to accelerate convergence speed and avod local

3 Mathematcal Problems n Engneerng 3 optma. In partcular, Lang et al. develop a comprehensve learnng partcle swarm optmzer CLPSO 26, whch uses all other partcles hstorcal best nformaton nstead of pbest and gbest to update a partcle s velocty: v t,d w t v t,d cr,d pbest t f d,d x t, 1.6,d where pbest f d,d can be the dth dmenson of any partcle s pbest ncludng pbest tself, and partcle f d s selected based on a learnng probablty P C. The authors suggest a tournament selecton procedure that randomly chooses two partcles and then select one wth the best ftness as the exemplar to learn from for that dmenson. Note that CLPSO has only one acceleraton coeffcent c whch s normally set to 1.494, and t lmts the nerta weght value n the range of 0.4, 0.9. Accordng to emprcal studes 29, 30, 33, CLPSO has been shown to be one of the best performng PSO algorthms, especally for complex multmodal functon optmzaton. In 34 a self-adaptaton technque s ntroduced to adaptvely adjust the learnng probablty, and the hstorcal nformaton s used n the velocty update equaton, whch effectvely mprove the performance of CLPSO on sngle modal problems. Wu et al. 35 adapt the CLPSO algorthm by mprovng the search behavor to optmze the contnuous externalty for antenna desgn. In 36 L and Tan present a hybrd strategy to combne CLPSO wth Broyden-Fletcher-Goldfarb-Shanno BFGS method, whch defnes a local dversty ndex to ndcate whether the swarm enters nto an optmalty regon n a hgh probablty. They apply the method to dentfy multple local optma of the generalzaton bounds of support vector machne parameters and obtan satsfyng results. However, to the best of our knowledge, modfcatons of CLPSO based on adaptve nerta weght and acceleraton coeffcent have not been reported. In ths paper we propose an mproved CLPSO algorthm, named CLPSO-AP, by ntroducng a new adaptve parameter strategy. The algorthm evaluates the evolutonary states of ndvdual partcles and the whole swarm, based on whch values of nerta weght and acceleraton coeffcent are dynamcally adjusted to search more effectvely. Numercal experments on test functons show that our algorthm can sgnfcantly mprove the performance of CLPSO. In the rest of the paper, Secton 2 presents our PSO method wth adaptve parameters, Secton 3 presents the computatonal experments, and Secton 4 concludes wth dscusson. 2. The CLPSO-AP Algorthm 2.1. Adaptve Inerta Weght and Acceleraton Coeffcent To provde an adaptve parameter strategy, we frst need to determne the stuaton of each partcle at each teraton. In ths paper, two concepts are used for ths purpose. The frst one consders whether or not a partcle mproves ts personal best soluton at the tth teraton n the paper we assume the problem s to mnmze the objectve functon wthout loss of generalty : s t 1 f f pbest t 0 else. <f pbest t 1 2.1

4 4 Mathematcal Problems n Engneerng And the second consders the partcle s rate of growth from the t 1 th teraton to the tth teraton: γ t f x t x t f x t 1 x t 1, 2.2 where x t x t 1 denotes the Eucldean dstance between x t and x t 1 : x t x t 1 D k 1 x t k x t 1 k Based on 2.1, we can calculate the percentage of partcles that successfully mprove ther personal better solutons: ρ t p 1 s t, 2.4 p where p s the number of partcles n the swarm. Ths measure has been utlzed n 33 and some other evolutonary algorthms such as 37. Generally, n PSO a hgh ρ ndcates a hgh probablty that the partcles have converged to a nonoptmum pont or slowly movng toward the optmum, whle a low ρ ndcates that the partcles are oscllatng around the optmum wthout much mprovement. Consderng role of nerta weght n the convergence behavor of PSO, n the former case the swarm should have a large nertal weght and n the latter case the swarm should have a small nertal weght. Here we use the followng nonlnear functon to map the values of ρ to w: w t e ρ t It s easy to derve that w ranges from about 0.36 to 1. The nonlnear and nonmonotonous change of nertal weght can mprove the adaptvty and dversty of the swarm, because the search process of PSO s hghly complcated on most problems. Most PSO algorthms use constant acceleraton coeffcents n 1.4. But t deserves to note that Ratnaweera et al. 38 ntroduce a tme-varyng acceleraton coeffcent strategy where the cogntve coeffcent s lnearly decreased and the socal coeffcent s lnearly ncreased. The basc CLPSO also uses a constant acceleraton coeffcent n 1.6, where c reflects the weghtng of stochastc acceleraton term that pulls each partcle towards the personal best poston of partcle f d at each dmenson d. Consderng the measure defned n 2.2, a large value of γ t ndcates that at t tme, partcle falls rapdly n the search space and gets a consderable mprovement on the ftness functon; thus t s reasonable to antcpate that the partcle may also gan much mprovement at least at the next teraton. On the contrary, a small value of γ t ndcates that partcle progresses slowly and thus needs a large acceleraton towards the exemplar.

5 Mathematcal Problems n Engneerng 5 From the prevous analyss, we suggest that the acceleraton coeffcent should be an ncreasng functon of γ t /Γ t, where Γ t s the SRSS of the rates of growth of all the partcles n the swarm: Γ t p j 1 γ t j Based on our emprcal tests, we use the followng functon to map the values of γ to c : c t γ t. 2.7 Γ t 2.2. The Proposed Algorthm Usng the adaptve parameter strategy descrbed n the prevous secton, the equaton for velocty update for the CLPSO-AP algorthm s v t,d w t v t c t,d r,d pbest t f d,d x t, 2.8,d where w t and c t are calculated based on 2.5 and 2.7, respectvely. Now we present the flow of CLPSO-AP algorthm as follows. 1 Generate a swarm of p partcles wth random postons and veloctes n the range. Let t 0 and ntalze w t 1 and each c t 1. 2 Generate a learnng probablty P C for each partcle based on the followng equaton suggested n 10 : P C exp 10 1 / p exp Evaluate the ftness of each partcle and update ts pbest and then select a partcle wth the best ftness value as gbest. 4 For each partcle n the swarm do the followng. 4.1 For d 1toD do the next Generate a random number rand n the range 0, If rand >P C,letf d Else, randomly choose two other dstnct partcles 1 and 2,andselectthe one wth better ftness value as f d Update the dth dmenson of the partcle s velocty accordng to Update the partcle s poston accordng to Calculate s t and r t for the partcle accordng to 2.1 and 2.2, respectvely.

6 6 Mathematcal Problems n Engneerng 5 Calculate w t of the swarm based on 2.4 and Calculate Γ t based on 2.5, and then calculate c t for each partcle based on Let t t 1. If t t max or any other termnaton condton s satsfed, the algorthm stops and returns gbest. 8 Go to step 3. In Step 7, other termnaton condtons can be that a requred functon value s obtaned, or all the partcles converge to a stable pont. 3. Numercal Experments In order to evaluate the performance of the proposed algorthm, we choose a set of wellknown test functons as benchmark problems, the defntons of whch are lsted n the Appendx secton. The search ranges, optmal ponts and correspondng ftness values, and requred accuraces are shown n Table 1. We comparatvely execute the basc PSO algorthm, CLPSO algorthm, and our CLPSO-AP algorthm on the test functons wth 10 and 30 dmensons, where each experment s run for 40 tmes. The parameter settngs for the algorthms are gven n Table 2. We use the mean best ftness value and the success rate wth respect to the requred accuracy shown n Table 1 as two crtera for measurng the performance of the algorthms. The expermental results averaged over 40 runs of whch are presented n Tables 3, 4, 5,and 6 respectvely. As we can see from the expermental results, for all 10D and 30D dmensonal problems, CLPSO-AP performs better than CLPSO n terms of both mean best values and success rates. Among the seven test functons, Ackley functon s the only one on whch CLPSO performs no better than the basc PSO However, CLPSO-AP performs better than basc PSO on both the 10D and 30D Ackley functons, and thus CLPSO-AP also outperforms basc PSO on all benchmark problems. It also deserves to note the 10D Rosenbrock functon, for whch CLPSO and most of the other PSO varants hardly succeed 26, 39, 40 whle our CLPSO-AP algorthm gans a 10% success rate. Except for the 30D Rosenbrock functon, CLPSO-AP successfully obtans the global optmum for all the other functons. In summary, our algorthm performs very well and overwhelms the other two algorthms on all of the test problems. 4. Concluson CLPSO has been shown to be one of the best performng PSO algorthms. The paper proposes a new mproved CLPSO algorthm, named CLPSO-AP, whch uses evolutonary nformaton of ndvdual partcles to dynamcally adapt the nerta weght and acceleraton coeffcent at each teraton. Expermental results on seven test functons show that our algorthm can sgnfcantly mprove the performance of CLPSO. Ongong work ncludes applyng our algorthm to ntellgent feature selecton and lghtng control n robotcs and extendng the adaptve strategy to other PSO varants, ncludng those for fuzzy and/or multobjectve problems 44, 45.

7 Mathematcal Problems n Engneerng 7 Table 1: Detaled nformaton of the test functons used n the paper. No. Functon Search range x f x Requred accuracy f 1 Sphere 100, 100 D 0, 0,..., f 2 Rosenbrock 2.048, D 1, 1,..., f 3 Schwefel 500, 500 D , ,..., f 4 Rastrgn 5.12, 5.12 D 0, 0,..., f 5 Grewank 600, 600 D 0, 0,..., f 6 Ackley , D 0, 0,..., f 7 Weerstrass 0.5, 0.5 D 0, 0,..., Table 2: Parameter settngs of the algorthms. Algorthm Inertal weght Acceleraton coeffcent s 10D functons 30D functons p t max p t max PSO 0.4, 1.2 c 1 c CLPSO 0.4, 0.9 c CLPSO-AP N/A N/A Table 3: The mean best values obtaned by the algorthms for 10D problems. Functon PSO CLPSO CLPSO-AP Sphere 3.239E E E 96 Rosenbrock 2.172E E E 00 Schwefel 1.048E Rastrgn 5.774E Grewank 8.132E E E 03 Ackley 5.951E E E 15 Weerstrass Table 4: The success rates obtaned by the algorthms for 10D problems. Functon PSO CLPSO CLPSO-AP Sphere 100% 100% 100% Rosenbrock % Schwefel 5% 100% 100% Rastrgn 12.5% 100% 100% Grewank % 75% Ackley 100% 100% 100% Weerstrass 100% 100% 100% Table 5: The mean best values obtaned by the algorthms for 30-D problems. Functon PSO CLPSO CLPSO-AP Sphere 4.900E E E 75 Rosenbrock 5.164E E E 01 Schwefel 1.425E E E 01 Rastrgn 4.726E E 02 0 Grewank 2.745E E E 12 Ackley 1.106E E E 14 Weerstrass 1.155E E E 14

8 8 Mathematcal Problems n Engneerng Table 6: The success rates obtaned by the algorthms for 30-D problems. Functon PSO CLPSO CLPSO-AP Sphere 100% 100% 100% Rosenbrock Schwefel 0 50% 55% Rastrgn 0 95% 100% Grewank 37.5% 100% 100% Ackley 100% 100% 100% Weerstrass 100% 100% 100% Appendx Defntons of the Test Functons 1 Sphere functon: f 1 x D x 2. 1 A.1 2 Rosenbrock s functon: D 1 2 f 2 x 100 x 2 x 1 x 1 2. A Schwefel s functon: D f 3 x D x sn x 1/2. A Rastrgn s functon: f 4 x D x 2 10 cos 2πx 10. A Grewank s functon: f 5 x D x 2 D 4000 x cos 1. A Ackley s functon: f 6 x 20 exp D D x D exp cos 2πx 20 e. D 1 A.6

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