Weightless Swarm Algorithm (WSA) for Dynamic Optimization Problems

Transcription

1 Weighless Swarm Algorihm (WSA) for Dynamic Opimizaion Problems T.O. Ting 1,*, Ka Lok Man 2, Sheng-Uei Guan 2, Mohamed Nayel 1, and Kaiyu Wan 2 1 Dep. Elecrical and Elecronic Eng. 2 Dep. Compuer Science and Sofware Eng. Xi an Jiaoong-Liverpool Universiy, Suzhou, China {oing,ka.man,seven.guan,mohamed.nayel, kaiyu.wan}@xjlu.edu.cn Absrac. In his work he well-known Paricle Swarm Opimizaion (PSO) algorihm is applied o some Dynamic Opimizaion Problems (DOPs). The PSO algorihm is improved by simplificaion insead of inroducing addiional sraegies ino he algorihm as done by many oher researchers in he aim of improving an algorihm. Several parameers (w, V max, V min and c 2 ) are being excluded from he convenional PSO. This algorihm is called Weighless Swarm Algorihm (WSA) as he prominen parameer, ineria weigh w does no exis in his proposed algorihm. Ineresingly, WSA sill works effecively via swapping sraegy found from counless rials and errors. We hen incorporae he proven clusering echnique from lieraure ino he framework of he algorihm o solve he six dynamic problems in lieraure. From he series of abulaed resuls, we proved ha WSA is compeiive as compared o PSO. As only one parameer exiss in WSA, i is feasible o carry ou parameer sensiiviy o find he opimal acceleraion coefficien, c 1 for each problem se. Keywords: dynamic opimizaion, swapping, weighless swarm. 1 Inroducion Paricle swarm opimizaion (PSO) [1] is one of he prominen algorihms in he caegory of naure-inspired algorihms and i has been one of he mos successful numerical opimizaion algorihms applied in many fields [2]. One of he advanageous feaures of Paricle Swarm Opimizaion is is abiliy o converge quickly o a poenial soluion. In oher words, PSO is faser compared o many evoluionary algorihms such as Geneic Algorihm (GA) [3], Evoluionary Programming (EP) [4], Evoluionary Sraegies (ES) [5], Differenial Evoluion (DE) [6] ec. This is how PSO works. Firsly, candidae soluions (or commonly known as paricles) are seeded ono he search space in a random manner. These paricles will hen move hrough he problem space in he aim of finding he global opimum. The movemen is guided by he essenially imporan ingredien formulas: V i = w V 1 i + 2r 1 Pbes ( i X i )+ 2r 2 Gbes ( i X i ) (1) * Corresponding auhor. J.J. Park e al. (Eds.): NPC 2012, LNCS 7513, pp , IFIP Inernaional Federaion for Informaion Processing 2012

2 Weighless Swarm Algorihm (WSA) for DOPs X + i = Xi + Vi (2) where by: V i is he velociy for ih dimension a ime w is he ineria weigh, usually se o 0.5 X i Pbes i Gbes i r1, r2 is he curren posiion of ih dimension a ime is he bes posiion for ih dimension a ime of a paricle, also known as personal bes is he bes soluion among all paricipaing paricles for ih dimension a ime, also known as global bes These are independen uniform random numbers wihin [0, 1] In each ieraion, all he paricles will be evaluaed hrough a similar cos funcion. Then, he updae of Pbes and Gbes values are performed insanly. In oher words, he asynchronous updae is adoped here. The reason for asynchronous updae is ha he informaion of Pbes and Gbes can be feedback ino he whole populaion insanly wihou delay and his will accelerae he convergence rae. In lieraure, many works prefer asynchronous updae [7, 8]. During he updae of velociy, V hrough (1), he limi of Vmax and Vmin is imposed, usually wihin 10%, 50% or 100% of search space. The value chosen for Vmax and Vmin is no really crucial and does no affec he performance of PSO drasically. Also, afer he updae hrough (2), checking is done o ensure ha paricles only explore he predefined search space. There are many echniques o handle hese boundary limis, which are beyond he scope of his paper. By simply se he value o boundary limi is one of he alernaives. Anoher alernaive will be o impose reiniializaion wihin he search space upon violaion. The laer alernaive is preferred as his will increase he diversiy of he enire populaion and hence assiss in avoiding local opima. The similar boundary handling echnique is adoped in his work. As he number of ieraion increases, paricles accelerae owards hose wih beer finess unil maximum ieraion is reached. By careful inspecion of (1) and (2), he following inerpreaions are valid in regards o PSO: i. The velociy somehow acs as shor-erm memory reenion and plays a crucial role in he updae process. ii. The updae of a dimensional value is guided by Pbes and Gbes. Simply, his means ha a paricle moves beween Pbes and Gbes. iii. The independen random numbers r 1 and r 2 conrol he raio of movemen owards Pbes and Gbes. In his work, insead of using convenional Paricle Swarm Opimizaion (PSO), a much simpler ye robus varian is presened o solve Dynamic Opimizaion Problems

3 510 T.O. Ting e al. (DOPs). This novel algorihm is given he name, Weighless Swarm Algorihm (WSA) as he ineria weigh inroduced by Shi and Eberhar in he year 1998 [9] is no presen in his algorihm. The work on WSA is novel and horough work will be carried ou in fuure o sabilize is performance. The exclusion of ineria weigh reduces several oher parameers such as Velociy, V max and V min. Due o his, WSA is faser compared o is original form. As he complexiy of he algorihm is reduced, he uning of he algorihm is much easier in his work. The res of he paper is organized as follows. Secion 2 describes he essence of WSA; his includes he explanaion of he sraegies incorporaed o enable PSO o work wihou ineria weigh. Parameer seings and experimenal resuls are given in Secion 3 and 4 respecively. Lasly is he conclusion in Secion 5. 2 The Essence of Weighless Swarm Algorihm Weighless Swarm Algorihm (henceforh abbreviaed as WSA) has he same form as he canonical PSO. Wihou he ineria weigh, he updaed equaion is simplified from wo-line equaion o a single line: ( ) ( ) X + = X + rc Gbes X + rc Pbes X (3) 1 id, id, 1 1 id, id, 2 2 id, id, whereby Xi,d is he posiion of dh dimension of ih paricle. Pbes is he bes posiion found in he search hisory of a paricle whereas Gbes is he bes soluion found in he enire search hisory. r 1 and r 2 are wo independen uniform random number generaors wihin [0, 1]. The acceleraion coefficien, c 1 and c 2 are boh se o 1.7. Following he heoreical analysis by Clerc and Kennedy [10], a consricion facor K = is inroduced on he basis of c 1 +c If we assume c 1 =c 2 =2.05 and K=0.729, he new coefficiens will have he value of = Furhermore, new resuls presened in [11] based on he heory of dynamic sysems for analysis of a paricle rajecory have been carried ou wih differen parameer se (w=0.6 and c 1 =c 2 =1.7) which showed slighly superior performance. The WSA inroduced here agreed o hese parameer seings even wihou he presen of ineria weigh. By seing c 1 =c 2 =1.7, resuls are slighly improved as compared o 1.5. Ineresingly, he defaul seing for c 1 and c 2 of PSO in EAlib [12] is also 1.7. From he resuls on saic numerical problems, we found ha equaion (1) can be furher simplified as: ( ) X + = X + rc Gbes X (4) 1 id, id, 1 1 id, id, Using Equaion (4) works effecively as when swapping is done during he updae of Pbes and Gbes values; many X values are acually he previous Pbes values. Hence, i is no really necessary o learn from oneself. Therefore, in WSA, several parameers prominen in PSO are omied. The wellknown ineria weigh, w is now no presen. Hence, i means ha he velociy, V is also unnecessary. Wihou V, a user also discards he concern of he bound for his parameer, namely V max and V min. Also, by adoping (4) in WSA, one of he acceleraion coefficiens is auomaically discarded. Thus, he proposed algorihm has a much simpler form compared o canonical PSO. By his form of algorihm, he complexiy presen is

4 Weighless Swarm Algorihm (WSA) for DOPs 511 grealy reduced and we only need o une c 1 for opimal performance; his has been done successfully in his work. The reducion in he complexiy of he algorihm hereby resuls in a lower compuaional cos of he relevan compuer program. By running boh programs (PSO and WSA) concurrenly, i is observed ha a he poin whereby WSA is compleing 20 runs, he PSO is only a is 4 h run. I means ha he proposed mehod solves dynamic opimizaion problems five imes faser compared o PSO. This is due o simpler code and more resource effecive as memory allocaion for boh w and V are commened in he exising C++ EAlib program, available from [12]. 2.1 The Trick in WSA The secre of WSA is exremely simple and his is indeed he core in his proposed mehodology. In he canonical PSO, seing w o zero value resuled in sagnan search; he algorihm does no seem o work. During he updae of Pbes and Gbes, i has been a radiional pracice ha hese values are replaced by beer paricles. However, in our proposed WSA, insead of doing replacemen, swapping of values is adoped. By swapping, we increase he diversiy of he populaion and accelerae he convergence rae. This is due o he reason ha when swapping is imposed, he probabiliy of paricle X equal o Pbes or Gbes is significanly reduced. For insance, in he case of replacemen scenario in PSO, for a given ieraion, if Pbes is updaed for 5 imes, here are 5 ineffecive posiional updaes as he erm ( Pbesid, X id, ) = 0. These ineffecive updaes are avoided in WSA, resuling in beer accuracy and faser convergence given he same number of funcion evaluaions. 2.2 Implemenaion of WSA The implemenaion of WSA is prey simple and can be implemened ino any exising PSO algorihm wih he following seps: i. Se ineria weigh = 0, ii. iii. Discard he Pbes erm by seing c 1 in equaion (1) o zero. Swapping is done during Pbes updae. The swapping for Gbes may no be necessary as in many algorihm implemenaions; one of he Pbes values is acually he Gbes. The above hree seps are simple ye hey improve he performance of he algorihm drasically wihou he need of ineria weigh. This simple sraegy can be implemened easily ino any exising PSO algorihm. 3 Parameer Seings In his work, as PSO is he bes algorihm in undeecable dynamic environmens from he resuls presened in [13, 14], i is adoped for comparison in his work. The seings for boh algorihms are as follows:

5 512 T.O. Ting e al. 3.1 PSO The ineria weigh, w is linearly decreased from 0.6 o 0.3 and acceleraion coefficiens, c 1 = c 2 = WSA Ineria weigh is no presen; herefore i is in fac zero. The acceleraion coefficien, c 1 is found hrough parameer sensiiviy analysis and c 2 = 0. 4 Experimenal Resuls Six dynamic problems from [15] are adoped as es bed in his work. Problems such as Sphere, Rasrigin, Griewank and Ackley are well-known problems in he area of numerical opimizaion. The descripions of he differen change sraegies T1, T2, T3 are available from [15]. Simulaion resuls are presened in Tables 1-4 wih Tables 5 recording he mean and sandard deviaion for resuls in Table 4. From Table 2, i is observed ha he performance of WSA is close o PSO in Table 1. As from Table 1, he PSO recorded oal overall performance of whereas WSA recorded a figure of I means ha he soluion of WSA is compeiive compared o is predecessor. As he naure of WSA may no be he same as PSO, he parameer sensiiviy analysis is carried ou o obain he opimal seings for boh diversiy and overlapping raio. From he analysis, he diversiy: α=0.6 and overlapping raio: β=0.9 are suggesed for opimal performance. Resuls of parameer sensiiviy analysis are no included o avoid exended paper. Resuls using hese seings are recorded in Table 3, now wih overall performance of (even closer o PSO s). I is ineresing o noe ha wih he opimal parameer seing, he performance of each problem se is slighly improved. As WSA can perform effecively even wihou learning from Pbes, equaion (2) is used for he resuls depiced in Table 4, now wih independen values of acceleraion coefficien. These values are depiced in he second column of he able, The oal overall performance is now improved up o From our analysis, i is found ha c1 ranges from 1.9 o 3.5. Wih differen changing raio for he case of, i is ineresing o noe ha he value of c1 increases in a similar paern; he greaer changing raio favors greaer value of c1. From his behavior, he feasibiliy of adapive c1 is observed. This will be one of he promising direcions for fuure work. The composiion of Rasrigin problem se seems o be mos challenging in his sudy as his is a mulimodal problem wih many local opima residing close o one and oher. For such a case, long jump or sep is favored by seing c1=3.5 and his helps in reducing chances of falling ino a local opimum in he ligh of dynamic environmen. Again, i is ineresing o noe ha he performance of each problem se is slighly improved compared o he one in Table 4. Mean values and sandard deviaion (STD) are abulaed in Table 5.

6 Weighless Swarm Algorihm (WSA) for DOPs 513 Table 1. Performance of PSO on -F6, Overlapping raio = 0.1, Diversiy = 0.3 and Popsize = 40 / 10 Problem raio F F F F F Toal overall performance = Table 2. Performance of WSA on -F6 (c 1 =c 2 =1.7) Overlapping raio = 0.1, Diversiy = 0.3 and Popsize = 40 / 10 Problem raio F F F F F The overall performance = Table 3. Performance of WSA on -F6 (c 1 =c 2 =1.7) Overlapping raio = 0.9, Diversiy = 0.6 and Popsize = 40 / 10 Problem raio F F F F F The overall performance =

7 514 T.O. Ting e al. Table 4. Performance of WSA on -F6 (c 1 varies) Overlapping raio = 0.9, Diversiy = 0.6 and Popsize = 40 / 10 Problem c 1 raio F F F F F Problem raio The overall performance = Table 5. Mean Values and STD for Resuls in Table ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.71 F ± ± ± ± ± ±2.34 F ± ± ± ± ± ±108 F ± ± ± ± ± ±12.4 F ± ± ± ± ± ±2.55 F ± ± ± ± ± ± Conclusions In his work, i is proven ha PSO works effecively even wihou he presen of he prominen ineria weigh on Dynamic Opimizaion Problems. Thus, he proposed algorihm is called Weighless Swarm Algorihm (WSA). The sraegy can be incorporaed ino any exising PSO algorihm by discarding he ineria weigh and changing he updae sraegy o swapping insead of replacemen. From he series of resuls, i is shown ha he naure of WSA is differen from PSO and herefore parameer sensiiviy analysis is done o obain he opimal parameers (overlapping raio and diversiy). The performance of WSA is only slighly less compared o PSO wihou ineria weigh. The simpliciy of WSA allows he uning of acceleraion consan, c1 independenly in order o obain beer resuls. Thus, i is eviden ha WSA has superior properies alongside exremely simple sraegy and cheaper compuaional coss. Fuure work will be done o find he underlying properies ye o be discovered.

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