STUDY ON VELOCITY CLAMPING IN PSO USING CEC`13 BENCHMARK

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1 STUDY ON VELOCITY CLAMPING IN PSO USING CEC`13 BENCHMARK Michal Pluhacek, Roman Senkerik, Adam Vikorin and Tomas Kadavy Tomas Baa Universiy in Zlin, Faculy of Applied Informaics Nam T.G. Masaryka 5555, Zlin, Czech Republic {pluhacek, senkerik, avikorin, KEYWORDS Paricle swarm opimizaion, PSO, Velociy clamping, Opimizaion ABSTRACT In his paper, we perform a new sudy of he imporance of using clamping of he velociy of paricles in he Paricle Swarm Opimizaion algorihm. The velociy clamping is used o preven he paricles from rapid acceleraion. We presen resuls of esing differen seings of maximal velociy on he exensive CEC 13 benchmark se and discuss he resuls, alongside he overall imporance of he velociy clamping mehod. INTRODUCTION The Paricle Swarm Opimizaion algorihm (PSO) (Kennedy and Eberhar 1995, Kennedy 1997) is one of he mos prominen members of Swarm inelligence based caegory of evoluionary opimizaion algorihms (Volna and Koyrba 2014). The PSO is very popular and widely applied. Many researchers focus on analyzing he inner principles of he algorihm in order o improve he undersaing of he inner dynamics and propose performance improvemens (Shi and Eberhar 1998a, Van Den Bergh, and Engelbrech 2006, Nickabadi e al. 2012). Like any oher mehod, he PSO suffers from several drawbacks, and he coninuous research focuses on addressing hese problems. Velociy clamping is a popular approach for dealing wih one of he main drawbacks of PSO he rapid paricle acceleraion. In his mehod, a maximal velociy value (v max) is se. The popular choice for he seing of maximal velociy (Shi and Eberhar 1998b) is 20% of he search space range. However, i is no always favorable o use his value or o use velociy clamping a all. In his sudy, we choose o invesigae he impac of differen seings of maximal velociy on he performance of PSO algorihm on he complex CEC`13 benchmark se ha represens a large variey of finess landscapes. Based on he resul we choose o re-evaluae he significance of velociy clamping in PSO. We provide evidence ha he popular seing of he v max = 20% of range migh no be he bes choice for many problems and ha he velociy clamping migh be omied in some cases. The paper is srucured as follows: In he second secion, a brief descripion of original PSO algorihm is given. In he nex secion, he experimen seup is deailed. Following is an exensive presenaion of colleced daa, and he resuls are discussed in he following secion. PARTICLE SWARM OPTIMIZATION The PSO algorihm is inspired by he naural swarm behavior of animals (such as birds and fish). I was firsly inroduced by Eberhar and Kennedy in 1995 (Kennedy and Eberhar, 1995). Each paricle in he populaion represens a possible soluion of he opimizaion problem, defined by he cos funcion (CF). In each ieraion of he algorihm, a new locaion (combinaion of CF parameers) of he paricle is calculaed based on he previous locaion and velociy vecor (velociy vecor conains paricle velociy for each dimension). The velociy calculaion formula is given in (1). v 1 2 wv c Rand ( gbes c Rand ( pbes 1 j x ) x Where: v +1 - New velociy of he ih paricle in ieraion +1. (componen j of he dimension D). w Ineria weigh value. v - Curren velociy of he ih paricle in ieraion. (componen j of he dimension D). c 1, c 2 - Acceleraion consans. pbes Personal bes soluion found by he ih paricle. (componen j of he dimension D). gbes j - Bes soluion found in a populaion. (componen j of he dimension D). x - Curren posiion of he ih paricle (componen j of he dimension D) in ieraion. Rand Pseudo random number, inerval (0, 1). The new posiion of each paricle is hen given by (2), where x i +1 is he new paricle posiion: EXPERIMENT SETUP 1 1 i i i ) (1) x x v (2) In he experimen, 11 differen seings of maximal velociy were used. The maximal velociy is ypically se as a fracional muliplicaion of he search space range. This paern was followed in his sudy wih v max se o values <0.1 Range ; Range> by sep 0.1. In addiion, he varian wih no velociy clamping was esed (noed N/A). Therefore, 11 differen seings of PSO were esed. The performance of PSO algorihm wih differen seings of maximal allowed velociy was esed on he IEEE CEC 2013 benchmark se (Liang e al. 2013) for dimension seing (dim) = 10 and 30. According o he benchmark rules, 51 separae runs were performed for Proceedings 32nd European Conference on Modelling and Simulaion ECMS Lars Nolle, Alexandra Burger, Chrisoph Tholen, Jens Werner, Jens Wellhausen (Ediors) ISBN: / ISBN: (CD)

2 each algorihm, and he maximum number of cos funcion evaluaions (CFE) was se o dim. The populaion size was se o 40. According o lieraure (Shi and Eberhar 1998, Shi and Eberhar 1999), he values of conrol parameers were se o popular values as follows: c 1, c 2 = ; w = The resuls were esed for saisical significance using he Friedman rank es (α=0.05), followed by he Nemenyi pos-hoc es. RESULTS In his secion, he resuls overview is presened. Firsly, he resuls for dim = 10 are presened. The Figures 1 8 depic seleced examples of mean gbes hisory for various benchmark funcions. Table 1 conains he median resul values for all esed seings. The ranking according o he Friedman rank es is given in Fig. 9. The criical disance from he lowes rank is displayed. Furher, he resuls for dim = 30 are presened in a similar way. The median resul values are given in Table 2. Seleced examples of mean gbes hisory are displayed in Figures Finally, he Friedman ranking is presened in Fig. 14. Figure 4. Mean gbes value hisory f 13 dim = 10 Figure 5. Mean gbes value hisory f 16 dim = 10 Figure 1. Mean gbes value hisory f 2 dim = 10 Figure 6. Mean gbes value hisory f 21 dim = 10 Figure 2. Mean gbes value hisory f 4 dim = 10 Figure 7. Mean gbes value hisory f 24 dim = 10 Figure 3. Mean gbes value hisory f 6 dim = 10

3 Figure 8. Mean gbes value hisory f 26 dim = 10 Figure 9. Friedman rank wih Nemenyi criical disance dim = 10 TABLE I. MEDIAN OF RESULTS, DIM = 10; vmax N/A f1-1.40e e e e e e e e e e e+03 f2 1.11E E E E E E E E E E E+05 f3 1.29E E E E E E E E E E E+07 f4 5.69E E E E E E E E E E E+03 f5 f6 f7 f8 f9 f10 f11 f12 f e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e+02 f E E E E E E E E E E E+02 f E E E E E E E E E E E+02 f E E E E E E E E E E E+02 f E E E E E E E E E E E+02 f E E E E E E E E E E E+02 f E E E E E E E E E E E+02 f E E E E E E E E E E E+02 f E E E E E E E E E E E+02 f E E E E E E E E E E E+03 f E E E E E E E E E E E+03 f E E E E E E E E E E E+03 f E E E E E E E E E E E+03 f E E E E E E E E E E E+03 f E E E E E E E E E E E+03 f E E E E E E E E E E E+03

4 TABLE II. MEDIAN OF RESULTS, DIM = 30; vmax N/A f1-1.40e e e e e e e e e e e+03 f2 1.49E E E E E E E E E E E+07 f3 7.69E E E E E E E E E E E+10 f4 3.82E E E E E E E E E E E+04 f5 f6 f7 f8 f9-1.00e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e+02 f E E E E E E E E E E E+02 f11 f e e e e e e e e e e e e e e e e e e e e e e+01 f E E E E E E E E E E E+01 f E E E E E E E E E E E+03 f E E E E E E E E E E E+03 f E E E E E E E E E E E+02 f E E E E E E E E E E E+02 f E E E E E E E E E E E+02 f E E E E E E E E E E E+02 f E E E E E E E E E E E+02 f E E E E E E E E E E E+03 f E E E E E E E E E E E+03 f E E E E E E E E E E E+03 f E E E E E E E E E E E+03 f E E E E E E E E E E E+03 f E E E E E E E E E E E+03 f E E E E E E E E E E E+03 f E E E E E E E E E E E+03 Figure 10. Mean gbes value hisory f 1 dim = 30 Figure 12. Mean gbes value hisory f 25 dim = 30 Figure 11. Mean gbes value hisory f 4 dim = 30 Figure 13. Mean gbes value hisory f 30 dim = 30

5 Figure 14. Friedman rank wih Nemenyi criical disance dim = 30 RESULTS DISCUSSION According o he saisical daa presened in Fig. 9, he seings of v max = 0.3 and 0.6 are bes performing for dim = 10. The value of 0.1 does no perform well and so does he algorihm when velociy clamping is no used a all. Oher seings do no perform significanly worse (according o saisical es). As is displayed in Figures 1 8, he overall shape of he convergence hisory does no differ significanly regardless he velociy clamping value. A similar rend can be observed in Figures for he dim = 30. This migh also mean ha he v max seing does no direcly affec he diversiy of he swarm bu a fuure sudy will be needed o suppor such conclusion. According o saisics (Fig. 14) in higher dimensions (dim = 30), he value 0.1 is he only significanly ouperformed seing by all oher. However, he PSO wihou velociy clamping is no longer underperforming. For researchers ineresed in comparing oher approaches wih our sudy he median resul values are presened in Tables 1 and 2. CONCLUSION In his sudy, we esed he performance of sandard PSO algorihm wih differen seing of maximal velociy and wihou velociy clamping on an exensive benchmark suie conaining 28 differen funcions. The observaions can be summarized as follows: A very low value of v max (0.1) is no advisable. In lower dimension, he velociy clamping mus no be omied. However, in higher dimensions (dim = 30), he performance of PSO wih velociy clamping (v max > 0.1) and PSO wihou he velociy clamping is comparable. Fine uning of v max value does no seem o be necessary as he performance does no differ wih saisical significance for values over 0.1. The popular (according o lieraure) seing v max = 0.2 does no seem o be favorable across his exensive benchmark suie The overall convergence behavior seems no affeced by he maximal velociy value. I seems ha he mos favorable values for he maximum velociy are in he region from 0.3 o 0.6. However, more daa is needed o suppor his claim. Given he iniial resuls and findings presened in his sudy, we will coninue o sudy he mechanics of velociy clamping for a beer undersanding of his issue. Fuure sudy will focus on he relaion beween v max seing and dimensionaliy of he problem. In addiion, he relaion beween favorable v max seing and finess landscape parameers (such as ruggedness, ec.) will be closely invesigaed. In addiion, he relaion of v max and oher adjusable parameers of PSO will be aken ino consideraion. ACKNOWLEDGMENT This work was suppored by he Minisry of Educaion, Youh and Spors of he Czech Republic wihin he Naional Susainabiliy Programme Projec no. LO1303 (MSMT-7778/2014), furher by he European Regional Developmen Fund under he Projec CEBIA-Tech no. CZ.1.05/2.1.00/ and by Inernal Gran Agency of Tomas Baa Universiy under he Projecs no. IGA/CebiaTech/2018/003. This work is also based upon suppor by COST (European Cooperaion in Science & Technology) under Acion CA15140, Improving Applicabiliy of Naure-Inspired Opimisaion by Joining Theory and Pracice (ImAppNIO), and Acion IC1406, High-Performance Modelling and Simulaion for Big Daa Applicaions (chipse). The work was furher suppored by resources of A.I.Lab a he Faculy of Applied Informaics, Tomas Baa Universiy in Zlin (ailab.fai.ub.cz). REFERENCES Kennedy J. and Eberhar R., Paricle swarm opimizaion, in Proceedings of he IEEE Inernaional Conference on Neural Neworks, 1995, pp Kennedy J., The paricle swarm: social adapaion of knowledge, in Proceedings of he IEEE Inernaional Conference on Evoluionary Compuaion, 1997, pp Liang JJ, Qu B-Y., Suganhan PN, Hernández-Díaz AG (2013) Problem Definiions and Evaluaion Crieria for he CEC 2013 Special Session and Compeiion on Real-Parameer Opimizaion, Technical Repor 2012, Compuaional Inelligence Labora-ory, Zhengzhou Universiy, Zhengzhou China and Technical Repor, Nanyang Technological Universiy, Singapore. Kennedy J. and Eberhar R., Paricle swarm opimizaion, in Proceedings of he IEEE Inernaional Conference on Neural Neworks, 1995, pp Nickabadi A., Ebadzadeh M. M., Safabakhsh R., A novel paricle swarm opimizaion algorihm wih adapive ineria weigh, Applied Sof Compuing, Volume 11, Issue 4, June 2011, Pages , ISSN Shi Y. and Eberhar R. C., A modified paricle swarm opimizer, in Proceedings of he IEEE Inernaional Conference on Evoluionary Compuaion (IEEE World Congress on Compuaional Inelligence), 1998a, pp I. S.

6 Shi Y., Eberhar R.C., Parameer selecion in paricle swarm opimizaion, in: Proceedings of he Sevenh Annual Conference on Evoluionary Programming, New York, USA, 1998b, pp Shi Y., Eberhar R.C., Empirical sudy of paricle swarm opimizaion, in: Proceedings of he IEEE Congress on Evoluionary Compuaion, IEEE Press, 1999, pp Van Den Bergh, F., and Engelbrech, A. P. (2006). A sudy of paricle swarm opimizaion paricle rajecories. Informaion sciences, 176(8), Volna, E., and Koyrba, M. (2014) A comparaive sudy o evoluionary algorihms, In Proceedings 28h European Conference on Modelling and Simulaion, ECMS 2014, Brescia, Ialy, 2014, pp AUTHOR BIOGRAPHIES MICHAL PLUHACEK was born in he Czech Republic, and wen o he Faculy of Applied Informaics a Tomas Baa Universiy in Zlín, where he sudied Informaion Technologies and obained his MSc degree in 2011 and Ph.D. in 2016 wih he disseraion opic: Modern mehod of developmen and modificaions of evoluionary compuaional echniques. He now works as a researcher a he same universiy. His address is: pluhacek@ub.cz ROMAN SENKERIK was born in he Czech Republic, and wen o he Tomas Baa Universiy in Zlin, where he sudied Technical Cyberneics and obained his MSc degree in 2004, Ph.D. degree in Technical Cyberneics in 2008 and Assoc. prof. in 2013 (Informaics). He is now an Assoc. prof. a he same universiy (research and courses in: Evoluionary Compuaion, Applied Informaics, Crypology, Arificial Inelligence, Mahemaical Informaics). His address is: senkerik@ub.cz ADAM VIKTORIN was born in he Czech Republic, and wen o he Faculy of Applied Informaics a Tomas Baa Universiy in Zlín, where he sudied Compuer and Communicaion Sysems and obained his MSc degree in He is sudying his Ph.D. a he same universiy and he field of his sudies are: Arificial inelligence, daa mining and evoluionary algorihms. His address is: avikorin@ub.cz TOMAS KADAVY was born in he Czech Republic, and wen o he Faculy of Applied Informaics a Tomas Baa Universiy in Zlín, where he sudied Informaion Technologies and obained his MSc degree in He is sudying his Ph.D. a he same universiy and he fields of his sudies are: Arificial inelligence and evoluionary algorihms. His address is: kadavy@ub.cz