A Self-Adaptive Chaos Particle Swarm Optimization Algorithm

Transcription

1 TELKOMNIKA, Vol.13, No.1, March 2015, pp. 331~340 ISSN: , accredted A by DIKTI, Decree No: 58/DIKTI/Kep/2013 DOI: /TELKOMNIKA.v A Self-Adaptve Chaos Partcle Swarm Optmzaton Algorthm Yaln Wu*, Shupng Zhang Faculty of Informaton Engneerng, Jangx Unversty of Scence and Technology, Ganzhou, Jangx, Chna *Correspondng author, e-mal: Abstract As a new evolutonary algorthm, partcle swarm optmzaton (PSO) acheves ntegrated evoluton through the nformaton between the ndvduals. All the partcles have the ablty to adjust ther own speed and remember the optmal postons they have experenced. Ths algorthm has solved many practcal engneerng problems and acheved better optmzaton effect. However, PSO can easly get trapped n local extremum, makng t fal to get the global optmal soluton and reducng ts convergence speed. To settle these defcences, ths paper has proposed an adaptve chaos partcle swarm optmzaton (ACPSO) based on the dea of chaos optmzaton after analyzng the basc prncples of PSO. Ths algorthm can mprove the populaton dversty and the ergodcty of partcle search through the property of chaos; adjust the nerta weght accordng to the premature convergence of the populaton and the ndvdual ftness; consder the global optmzaton and local optmzaton; effectvely avod premature convergence and mprove algorthm effcency. The expermental smulaton has verfed ts effectveness and superorty. Keywords: chaotc theory, partcle swarm optmzaton, self-adapton 1. Introducton PSO s a new evolutonary algorthm developed n recent years and as a novel swarm ntellgent search technque, t has attracted the attenton of a great number of researchers who have appled t n every feld actvely because of the features of smple computaton form, few parameter settng and quck search speed snce t was proposed. As a knd of evolutonary algorthm, PSO starts from a random soluton; searches the optmal solutonva teratons and t evaluates the soluton qualty through ftness, but t s smpler than the rules of genetc algorthm (GA). It doesn t have theoperatonsof crossover and mutaton of GA and t searches the global optmum by followng the current optmal value [1]. Ths knd of algorthm has drawn the attenton of the academc crcle for the advantages of ease to realze, hgh accuracy and fast convergence and t has demonstrated ts superorty n solvng practcal problems. In practcal applcatons, snce the ntalzaton of the basc PSO s random, t has certan blndness. Although the random ntalzaton can bascally guarantee the unform dstrbuton of the ntal populaton, t can t guarantee the qualty of every partcle and t may cause some partcles get far away from the optmal soluton and affect the convergence speed of the algorthm [2]. PSO can easly get trapped n local extremum and t can t have global optmal soluton. The convergence speed of PSO s qute slow. It usually takes some tme to reach the correspondng accuracy n solvng practcal problems. Sometmes t s not worthy to spend a long tme to get a feasble soluton[3]. The reason to cause ths problem s that PSO doesn t make full advantage of the nformaton from the computaton; nstead, t only uses the nformaton of global optmum and ndvdual optmum n every teraton. Besdes, the algorthm tself doesn t have an optmzaton mechansm to elmnate the bad canddate soluton so that t has a slow convergence speed [4],[5]. To solve the above-mentoned shortcomngs of basc PSO, ths paper has proposed adaptve chaos partcle swarm optmzaton (ACPSO) based on the theory of chaos optmzaton. Generally, chaos refers to a random moton state to get from a determnstc formula and t s a common phenomenon n non-lnear system. It s complcated and t has the characterstc smlar to randomness. However, the seemngly chaotc chaos change s not totally chaos, on the contrary, t has certan nherent law. ACPSO performs chaotc ntalzaton to the basc partcle swarm wth the unque ergodcty of chaos optmzaton; generates a seres of ntal soluton groups and select the ntal populaton from them, n ths way, t effectvely Receved October 18, 2014; Revsed December 21, 2014; Accepted January 10, 2015

2 332 ISSN: mproves the dversty of PSO and the ergodcty of partcle search.the nerta weght n PSO adjusts tself accordng to the ftness value of every partcle ndvdual so as to enhance ts global coarse search and local coarse search capactes and mprove ts convergence speed and accuracy. Determne the good or bad of the current partcle accordng to the ftness value and conduct chaotc operaton on some selected partcles to help nerta partcle jump out from the local soluton regon and quckly search the global optmal soluton. The last part of ths paper has verfed the effectveness and advancement of the algorthm proposed n ths paper through experment smulaton. 2. Overvew of Standard Partcle Swarm Optmzaton The ntal purpose of Doctor Ebethart and Doctor Kennedy was to establsh a model n a 2D space to schematze the moton of the brd flock. Assumng such a scenaro that a group of brds randomly search food n a regon where there s only a pece of food. However, all of the brds don t know where the food s. All they know s how far they are from the food. Then what s the optmal strategy to fnd the food? The smplest and the most effectve strategy s the search the surroundng area of the brd whch s the closest to the food. PSO s nspred from ths knd of model and t s used to settle optmzaton problems. The brd has been abstracted as the partcle wth no qualty and volume and t has been extended to N-dmensonal space where the X x, x,, x and ts flyng speed as vector poston of partcle s expressed as vector 1 2 N V ( V 1, V2,, VN). Every partcle has a ftness value determned by the objectve functon and t knows the current optmal poston p and the current poston, whch can be seen as ts own flyng experence. In addton, every partcle also knows the optmal poston p g ( p g s the optmal value of p ) all partcles n the group has found. The partcle decdes ts next moton accordng to ts own experence and the optmal experence of ts companons. For the kth teraton, every partcle n the PSO changes accordng to the formula below: v v c rand p x c rand p x k 1 k k k d d 1 d d 2 gd d x x v k1 k k1 d d d (1) (2) In Formulas (1) and (2), 1, 2,, M, M s the total partcles n the group. v k d s the dth sub-vector of the poston vector of the teraton partcle n the kth teraton. pd s thedth component of the optmal poston p of the partcle. pgd s the dth component of the optmal poston pg n the group. c 1 & c 2 are weght factors and rand() s the random functon to generate a random number wthn the range of (0,1). Formula (1) manly computes the new speed of partcle through 3 parts: the speed of partcle at a former moment; the dstance between the current poston of partcle and ts optmal poston as well as the dstance between the current poston of partcle and the optmal poston of the group. Partcle computes the coordnate of ts new poston through Formula (2). Partcle decdes ts next moton poston through Formulas (1) and (2). From the angle of socology, the frst part of Formula (1) s called memory tem, standng for the nfluence of the last speed and drecton, the second part s the self cognton tem, a vector pontng from the current pont to the optmal pont of the partcle, representng that the partcle moton comes from ts own experence and the thrd part s called group cognton tem, namely a vector from the current pont to the optmal pont of the group, reflectng the coordnaton and nformaton sharng between the partcles. Therefore, the partcle determnes ts next moton through ts own experence and the best experence of ts partners [6],[7]. In Fgure 1, the example of 2D space descrbes the prncple that the partcle moves from poston X K K 1 to X accordng to Formulas (1) and (2). TELKOMNIKA Vol. 13, No. 1, March 2015 :

3 TELKOMNIKA ISSN: Fgure 1. Schematc dagram of partcle moton w. In 1998, Sh and other people revsed Formula (1) and ntroduced nerta weght factor v wv c rand p x c rand p x k 1 k k k d d 1 d d 2 gd k (3) w, non-negatve, s called nerta factor. The bgger w s, the stronger the global optmzaton ablty s and the local optmzaton ablty s, f w s small, and t s good for local search. Intally, Sh took w as constant, but n the later experment, t can be found that the dynamc w can obtan an optmzaton result better than the fxed value. Dynamc w can change lnearly n the search process of PSO and t can also change accordng to certan measure functon dynamcally of the performance of PSO. At present, the lnearly decreasng weght strategy (LDW) proposed by Sh has been used frequently, namely k wmax wmn w wmn N k N (4) In ths formula, N s the maxmum teratons and w max & w mn are the maxmum and mnmum nerta weghts respectvely. The ntroducton of nerta weght factor can greatly mprove the performance of PSO and t can also adjust the global and local search capactes accordng to dfferent search problems, makng PSO solve many practcal problems. Formulas (2) and (3) are called standard PSO [8]. 3. Chaos Optmzaton Algorthm and Its Ergodc Property In non-lnear system, chaos s a common moton phenomenon wth such excellent characterstcs as ergodcty, randomness and regularty.chaotc moton can experence all the states n the state space wthout repetton accordng to certan rule wthn certan moton range. Even a tny change n the ntal value can cause huge changes n the post moton, whch s called the strong senstvty of the ntal value. The researches have used these features of chaos and proposed chaos optmzaton algorthm, whch can easly jump out of the local soluton regon and whch can have hgh computaton effcency [9]. The most typcal chaotc system s sought from Logstc formula and ts teratve formula s as follows: 1, 0,1,,0 1, 0,4 x n 1 S x n xn n N x 0 (5), the orbt When 4 xn n 0,1,..., N s chaotc. It can be seen from Fg.2 that the two orbt ponts close to each other n the ntal state dverge after only 3 teratons. After several A Self-Adaptve Chaos Partcle Swarm Optmzaton Algorthm (Yaln Wu)

4 334 ISSN: teratons, snce the errors of these two orbts ncrease contnuously, t s dffcult to dentfy whch state the system s n and Logstc map s totally n the chaotc state [10]. Fgure 2. The logstc map when 4 Randomly take an ntal pont x0 0,1, the orbt of Logstc map s chaotc among (0,1), that s to say, regardless of N formerly known teratve ponts, the poston of the ( N 1)th x, there s a sub-orbtx of x to make teratve pont can t be predcted. In fact, for 0 0,1 xnk x. Although the chaotc dynamc system has a complcated chaotc state, t can be found that the certan chaos has regularty and ergodcty n statstcs by observng ths system wth the perspectve of statstc. As ndcated n Fgure 3, perform 1500 teratons wth x and x as the ntal values and get two chaotc sequencesx 1n and 2n x. A dot n the fgure stands for a pont n the space wth ts coordnate as ( x 1, x 2 ). nk j n j Fgure 3. Ergodcty of logstc map From the above fgure, t s clear that chaotc varable s senstve to the ntal value, as evdenced by the fact that the chaotc trajectores of these two close ntal values n the state space are completely dfferent. Chaotc varable can experence all the states n the state space wthout repetton accordng to ts own rule. Chaotc varable s as chaotc as random varable. When 4, the total chaotc teratve formula map of Logstc moves unstably n the range of [0,1] and when t experences a long-tme dynamc moton, t wll reflect the characterstc of TELKOMNIKA Vol. 13, No. 1, March 2015 :

5 TELKOMNIKA ISSN: randomness [11],[12]. Although chaotc varable s random, wth the ntal value determned, ts chaotc varable has also been determned and the hghlghted randomness s the nherent regularty of chaotc moton. The specfc steps of chaos optmzaton algorthm can be ndcated by Fgure 4: Perform the frst carrer by usng Logstc or Tent map to produce chaos sequence Search the chaos sequence after the carrer YES Whether the frst search has acheved the set teratons or met certan error requrements Whether the small perturbaton has reached certan teratons NO YES Conduct small perturbaton search NO Whether to reach the termnaton condton of the algorthm YES Output the computaton result Fgure 4. Procedures of chaos optmzaton algorthm 4. Adaptve Chaos Partcle Swarm Optmzaton & Example Test Because PSO uses the randomly-generated partcle, t may have unsearched dead zones for multmodal functon. Snce chaos optmzaton algorthm has the excellent feature of ergodcty, ths paper ntalzes by usng chaos optmzaton n the early phase of the optmzaton of the partcle swarm and selects the ntal partcle populaton. The specfc process ncludes two parts: the 1st part s to perform global search wth the basc partcle swarm optmzaton whle the 2nd part s to mplement local search accordng to the result of PSO by usng chaos optmzaton Chaotc Local Search Chaotc local search (CLS) s manly to mprove the search performance of the partcle and avod the partcle to get trapped n local extremum. Take Logstc map as example. CLS frsts maps the partcle varable nto chaotc varable and ten transforms the chaotc varable nto partcle varable after the teraton. The basc formulas of these two transformatons are as follows: k k x xmn, xmax, xmn, (6) A Self-Adaptve Chaos Partcle Swarm Optmzaton Algorthm (Yaln Wu)

6 336 ISSN: k1 k1 mn, max, mn, x x x x x (7) Here, xmn, and x max, represent the upper and lower bounds of the th partcal k respectvely. x s the decson varable (namely the ndvdual extremum of the partcle swarm) wthn the range of xmn,, x max, whle k s the chaotc varable wth ts value n the range of (0,1). The specfc steps of CLS are as follows: k k (1) Assume that k 0 and transform x nto through Formula (6). (2) Accordng to the current k, determne the chaotc varable k 1 of the next teraton. (3) Accordng to Formula (7), transform k 1 nto the decson varable x k 1. k 1 (4) Evaluate the new x. (5) If the new soluton s superor to the optmal soluton before the local search of the partcle swarm or t reaches the preset maxmum teratons, output the soluton whch s found from chaotc local search; otherwse, make k k 1 and return to Step (2) [13],[14] The Steps and Procedures of Adaptve Partcle Swarm Optmzaton By ntegratng the search process of the two phases of chaos partcle swarm optmzaton, the overall search steps of ths algorthm are as follows: (1) Set the sze of the partcle populaton N and the maxmum number of teratons and ntalze the poston and speed of the partcle randomly wthn the feasble value range. (2) Evaluate the ftness of every partcle; set the objectve ftness value of the 1st partcle as the global optmal value and ts ntal poston s ts own ndvdual extremum value. (3) Update the partcle speed and poston accordng to Formulas (2) and (3). (4) Evaluate the partcle ftness; compare the partcle ftness value wth the prevous value and update the superor objectve functon ftness value as the current ndvdual extremum value; compare the current optmal partcle ftness wth the prevous and update the superor objectve ftness value as the current global optmal value. (5) Keep the former N/5 partcles of the populaton. (6) Update these partcle postons by usng chaos local search and the CLS results. If t meets the termnaton standard, output the currently optmal soluton. (7) Narrow down the search space and randomly generate 4N/5 new partcles n the narrowed search space. (8) Form a new populaton wth the partcles through CLS update and the new partcles of 4N/5. (9) Make k k 1 and return to Step (3) Algorthm Testng & Result Analyss Ths paper has used four classcal testng functons n the experments so as to test the performance of ACPSO and t also compares the testng result of ACPSO wth those of DE and PSO. In order to nvestgate the algorthm expandablty, the varable dmensons for every functon to be used n the testng are 20 dmensons. In the experment, operates every crcumstance 30 tmes and seek the mean optmal value as the bass of performance comparson. The parameters selecton n these experments are for PSO and ACPSO, w reduces from 0.8 to 0.3 wth the evolutonary algebra, the scalng factor and crossover factor of DE are 0.5 and 0.9 respectvely. (1) Functon f 1 (Grewank Functon) TELKOMNIKA Vol. 13, No. 1, March 2015 :

7 TELKOMNIKA ISSN: n n 1 2 x f x x cos 1, x mn f x* f 0, 0, 0 0 Grewank functon reaches ts global mnmal pont when x 0 and t reaches the local mnmal ponts when x k, 1,2,, n, k 1,2,, n. The computaton results of these algorthms are demonstrated n Table 1. Table 1. The Mean Ftness Value for Grewank Functon Algorthm PSO DE ACPSO Varable Dmenson n=20 n=20 n=20 Mean Optmal Value When the varable dmenson n 20, the changng curve of the mean ftness value for Grewank functon wth the teratons s as ndcated n Fgure 5. Fgure 5. Iteratons on Grewank functon (2) Functon f 2 (Rastrgrn Functon) n 2 f x x 10 cos 2 x 10, x mn f x* f 0,0,,0 0 As a multmodal functon, Rastrgrn functon reaches the global mnmal pont when S x 5.12,5.12, 1,2,, n. See x 0.There are about 10n local mnmal ponts wthn the computaton results of these algorthms n Table 2. Table 2. The Mean Ftness Value for Rastrgrn Functon Algorthm PSO DE ACPSO Varable Dmenson n=20 n=20 n=20 Mean Optmal Value A Self-Adaptve Chaos Partcle Swarm Optmzaton Algorthm (Yaln Wu)

8 338 ISSN: When the varable dmenson n 20, the changng curve of the mean ftness value for Rastrgrn functon wth the teratons s as ndcated n Fgure 6. Fgure 6. Iteratons on Rastrgrn functon (3) Functon f 3 (Ackley Functon) n 1 n 2 1 f x 20exp 0.2 x exp cos2 x 20 e, x 32 n 1 n 1 mn f x* f 0, 0, 0 0 Ackley functon reaches the global mnmal pont when x 0. The computaton results of these algorthms are shown n Table 3. Table 3. The Mean Ftness Value for Ackley Functon Algorthm PSO DE ACPSO Varable Dmenson n=20 n=20 n=20 Mean Optmal Value When the varable dmenson n 20, the changng curve of the mean ftness value for Ackley functon wth the teratons s as ndcated n Fgure 7. Fgure 7. Iteratons on Ackley functon TELKOMNIKA Vol. 13, No. 1, March 2015 :

9 TELKOMNIKA ISSN: (4) Functon f 4 (Rosenbrock Functon) n f x 100 x x x 1, x 50 mn f x* f 1,1, 1 0 Rosenbrock s a non-convex pathologcal quadratc functon and ts mnmal pont s easy to fnd, but t s greatly dffcult to converge to global mnmum. The computaton results of the several algorthms can be seen n Table 4. Table 4. The Mean Ftness Value for Rosenbrock Functon Algorthm PSO DE ACPSO Varable Dmenson n=20 n=20 n=20 Mean Optmal Value e-3 When the varable dmenson n 20, the changng curve of the mean ftness value for Rosenbrock functon wth the teratons s as ndcated n Fgure 8. Fgure 8. Iteratons on Rosenbrock functon It can be seen from the above expermental test that ACPSO s superor to PSO and DE n all test functon performance. It not only has fast convergence speed, but also has strong global search ablty. In the four testng functons, the mean optmal ftness of ACPSO s the mnmum, therefore, t has hgher accuracy and stablty than PSO and DE and ts advantage s more obvous n the cases of hgher dmensons and relatvely sharp functon value changes whle the effect of PSO falls sgnfcantly. From the mean expermental result, t s clear that the mean ftness and standard devaton of DE are bg, therefore, ths algorthm has bgger fluctuatons, the bgger the dmensons, the more unstable t s. It can be seen from the above expermental result that ACPSO s a sutable tool to solve the global optmzaton problems of complcated functons. For the hgh-dmensonal and mult-extreme-pont functons, the global mnmum of certan accuracy can be obtaned at fewer computaton cost. 5. Concluson Ths paper has analyzed the consttuton and optmzaton prncple of partcle swarm optmzaton and t has ponted out that the basc partcle swarm optmzaton can easly get trapped n local optmal soluton n the optmum teraton and that ts convergence speed s very A Self-Adaptve Chaos Partcle Swarm Optmzaton Algorthm (Yaln Wu)

10 340 ISSN: slow n the late stage snce t generates the partcles representng varable value randomly. Therefore, ths paper ntegrates chaos optmzaton algorthm n the partcle swarm optmzaton and proposes adaptve partcle swarm optmzaton to solve objectve optmzaton problems. Ths algorthms uses the chaos prncple, enhances the dversty of varable value; computes the correspondng ftness value to every partcle, namely the objectve functon value of the problem, through partcle swarm optmzaton; selects varable value to perform chaos optmzaton accordng to ts value n order to help the varable to jump out from the local extremal regon and self-adaptvely adjusts ts nerta weght coeffcent accordng to the objectve functon value of the problem to mprove the global and local search capactes of the algorthm. References [1] Masoud JS, Ramn V, Mehrdad B, Hassan S, Ara A. Applcaton of Partcle Swarm Optmzaton n Chaos Synchronzaton n Nosy Envronment n Presence of Unknown Parameter Uncertanty. Communcatons n Nonlnear Scence and Numercal Smulaton. 2012; 17(2): [2] Amr HG, Gun JY, Xn SY, Samak T. Chaos-enhanced Accelerated Partcle Swarm Optmzaton. Communcatons n Nonlnear Scence and Numercal Smulaton. 2013; 18(2): [3] We ZR, Jng C. A Chaos Partcle Swarm Optmzaton Rangng Correcton Locaton n Complex Envronment. The Journal of Chna Unverstes of Posts and Telecommuncatons. 2012; 19(1): 1-5. [4] Na D, Chun HW, Wa HI, Zeng QC, Ka LY. Chaotc Speces Based Partcle Swarm Optmzaton Algorthms and Its Applcaton n PCB Components Detecton. Expert Systems wth Applcatons. 2012; 39(15): [5] Shjun Z, Tng J. A Novel Partcle Swarm Optmzaton Traned Support Vector Machne for Automatc Sense-through-folage Target Recognton System. Knowledge-Based Systems. 2014; 65: [6] MR Tanweer, S Suresh, N Sundararajan. Self Regulatng Partcle Swarm Optmzaton Algorthm. Informaton Scences. 2015; 294(10): [7] Oguz ET, Mert ST, Mustafa TC. Chaotc Quantum Behaved Partcle Swarm Optmzaton Algorthm for Solvng Nonlnear System of Equatons. Computers & Mathematcs wth Applcatons. 2014; 68(4): [8] Kej T, TakeruIbuk, Tetsuzo T. A Chaotc Partcle Swarm Optmzaton Explotng A Vrtual Quartc Objectve Functon Based on The Personal and Global Best Solutons. Appled Mathematcs and Computaton. 2013; 219(17): [9] Yaoyao H, Qfa X, Shanln Y, L L. Reservor Flood Control Operaton Based on Chaotc Partcle Swarm Optmzaton Algorthm. Appled Mathematcal Modellng. 2014; 38(17): [10] Shen X, Lu J, Lu Y. A New Partcle Swarm Optmzaton Soluton to Power Flow Regulaton for Unsolvable Case. Energy Proceda. 2012; 14: [11] We FG, San YL, Lng LH. Partcle Swarm Optmzaton wth Chaotc Opposton-based Populaton Intalzaton and Stochastc Search Technque. Communcatons n Nonlnear Scence and Numercal Smulaton. 2012; 17(11): [12] Humn J, CK Kwong, Zengqang C, YC Ysm. Chaos Partcle Swarm Optmzaton and Fuzzy Modelng Approaches to Constraned Predctve Control. Expert Systems wth Applcatons. 2012; 39(1): [13] Omd MN, Anvar B, Para J. Dynamc Dversty Enhancement n Partcle Swarm Optmzaton (DDEPSO) Algorthm for Preventng from Premature Convergence. Proceda Computer Scence. 2013; 24: [14] Mehd S, Hassan S, Ara A. Mnmum Entropy Control of Chaos Va Onlne Partcle Swarm Optmzaton Method. Appled Mathematcal Modellng. 2012; 36(8): TELKOMNIKA Vol. 13, No. 1, March 2015 :