A Binary Quantum-behaved Particle Swarm Optimization Algorithm with Cooperative Approach

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1 IJCSI Internatonal Journal of Computer Scence Issues, Vol., Issue, No, January 3 ISSN (Prnt): ISSN (Onlne): A Bnary Quantum-behave Partcle Swarm Optmzaton Algorthm wth Cooperatve Approach Jng Zhao,, Jun Sun an Wenbo Xu 3 School of Dgtal ea, Jangnan Unversty Wux, 4, Jangsu, Chna School of Informaton, Shanong Polytechnc Unversty, Jnan, 5353, Shanong, Chna 3 School of Internet of Thngs Engneerng, Jangnan Unversty Wux, 4, Jangsu, Chna Abstract A novel bnary Quantum-behave Partcle Swarm Optmzaton algorthm wth cooperatve approach () s ntrouce. In the propose algorthm, the upatng metho of partcle s prevous best poston an swarm s global best poston are performe n each menson of soluton vector to avo loss some components that have move closer to the global optmal soluton n the vector. Fve test functons are use to test the performance of. The results of experments show that the propose technque can ncrease versty of populaton an converge more raply than other bnary algorthms. Keywors: Quantum-behave Partcle Swarm Optmzaton, Bnary, Cooperatve Approach, Test Functons.. Introucton Partcle Swarm Optmzaton (PSO) s an evolutonary computaton technque evelope by Dr. Eberhart an Dr. Kenney n 995 [], nspre by socal behavor of br flockng or fsh schoolng. The optmal soluton s obtane by exchangng nformaton between nvuals. owever, the algorthm cannot converges to the global mnmum pont wth probablty one uner sutable conton []. Jun Sun et al have propose a global convergence-guarantee PSO algorthm [3], Quantum-behave Partcle Swarm Optmzaton (QPSO) algorthm, whch s nspre by quantum mechancs. It has been shown that QPSO outperforms PSO on several aspects, such as smple evoluton equatons, more few control parameters, fast convergence spee, smple operaton an so on [4,5]. In 997, Kenney propose the bnary verson of PSO () [6], an Jun Sun et al propose the bnary verson of QPSO () n 7 [7]. Ths paper wll focus on evelopng the bnary verson of QPSO wth cooperatve metho (). In the propose algorthm, each menson of partcle s new soluton vector replaces n turn the corresponng menson of partcle s prevous best poston an swarm s global best poston to calculate the ftness value. The rest structure of ths paper s as follows. In secton, a bref ntroucton of the s presente. The s escrbe n secton 3. Next, the novel s epcte n secton 4. Then the experment results are gven n secton 5. Fnally, the concluson s put forwar n secton 6.. Bnary Partcle Swarm Optmzaton In PSO, the populaton wth nvuals, whch s treate as a partcle, s calle a swarm X n the D- mensonal space. The poston vector an velocty vector of partcle at the generaton t represente as x t) ( x, x,, x ( )) an ( D t ( v ( t), v,, vd( t v )).The partcle moves accorng to the equatons: v ( t ) wv cr ( x ) () c r ( gbest x ) x ( t ) x v ( t ) () Where,,, ;,,, D, w s the nerta weght, whose value s typcally setup to vary lnearly from.9 to.4. c an c are calle the acceleraton coeffcents whch usually are set as c c. r an r are ranom number unformly strbute n (,).Vector (,,, D) s the best prevous poston of partcle wth the name personal best poston( ), whle the global best poston( gbest ), gbest ( gbest, gbest,, gbest D), s the best partcle poston among all the partcles n the populaton. Copyrght (c) 3 Internatonal Journal of Computer Scence Issues. All Rghts Reserve.

2 IJCSI Internatonal Journal of Computer Scence Issues, Vol., Issue, No, January 3 ISSN (Prnt): ISSN (Onlne): In [6,8], Eq. (3) replaces Eq. (). f ran () S( V )) then X else X (3) ( Where S (v) s a sgmo lmtng transformaton functon( s( v) v ( e ) ), an ran () s a ranom number selecte from a unform strbuton n (,). 3. Bnary Quantum-behave Partcle Swarm Optmzaton 3. Quantum-behave Partcle Swarm Optmzaton In PSO algorthm, the state of partcle s epcte by ts poston vector an velocty vector, whch etermne the trajectory of the partcle. The partcle moves along a etermne trajectory n Newtonan mechancs, but ths s not the case n quantum mechancs. In quantum worl, the term trajectory s meanngless, because poston an velocty of a partcle cannot be etermne smultaneously accorng to uncertanty prncple. Therefore, f nvual partcles n a PSO system have quantum behavor, the PSO algorthm s boun to work n a fferent fashon. In quantum tme-space framework, Jun Sun et al. ntrouce QPSO algorthm. The equatons are as follows: mbest (4) (,,, D) p x ( ) gbest (5) ( t ) p mbest x *ln( ) (6) u where s a ranom number unformly strbute n (,). mbest s mean best poston of the populaton. Parameter s calle the Contracton-Expanson coeffcent, whch can be tune to control the convergence spee of the algorthm. From the results of stochastc smulaton [9], t can be conclue that n QPSO, when. 78, the partcles wll converge. In the process of teraton, s ece by the ranom number, when t s bgger than.5, mnus sgn ( - ) s propose, others plus sgn ( + ) s propose. 3. Bnary Quantum-behave Partcle Swarm Optmzaton In ths secton, a screte bnary verson of QPSO () s propose. Because the teraton equatons of QPSO are far fferent from those of PSO, the methoology of oes not apply to QPSO. In QPSO, there are no veloctes an trajectores concepts but poston an stance. In, the poston of the partcle s represente as a bnary strng. The stance s efne as the ammng stance between two bnary strngs. That s X Y ( X, Y) (7) Where X an Y are two bnary strngs an represent two postons. The functon () s to get the ammng stance between X an Y. The ammng stance s the count of bts fferent n the two strngs. The jth bt of the mbest s etermne by the states of the jth bts of all partcles n. If more partcles take on at the jth bt of ther own, the jth bts of mbest wll be ; otherwse the bt wll be. owever, f half of the partcles take on at the jth bt of ther, the jth bt of mbest wll be set ranomly to be or, wth probablty.5 for ether state. In, the pont operaton on mult-pont crossover operaton on p s obtane by crossover an gbest. Frstly make one-pont or an gbest to generate two offsprng. Then ranomly select one of the offsprng an output t as the pont Conser teratve Eq. (6) an transform t as b ( x, p ) ( x, mbest) ln( ) (8) u We can obtan the new strng x by the transformaton n whch each bt n p s mutate wth the probablty compute by b l c (9) b f l Where l s the length of the th menson of partcle. In the process of teraton, f ran () c, the corresponng bt n the poston of partcle wll be reverse, otherwse remans t. p. Copyrght (c) 3 Internatonal Journal of Computer Scence Issues. All Rghts Reserve.

3 IJCSI Internatonal Journal of Computer Scence Issues, Vol., Issue, No, January 3 ISSN (Prnt): ISSN (Onlne): Wth the above efnton an mofcatons of teratve equatons, the algorthm s escrbe as the followng proceure: Step Intalze an array of bnary bts for all partcles, partcle s personal best postons an swarm s global best poston gbest ; Step For each partcle, etermne the mbest an get a stochastc poston p by exertng crossover operaton on an gbest ; Step 3 For each menson, compute the mutaton probablty c an then upate the partcle s new poston x by c ; Step 4 Evaluate the objectve functon value of the partcle, an compare t wth the objectve functon value of an gbest. If the current objectve functon value s better than that of an gbest, then upate an gbest ; Step 5 Repeat step ~4 untl the stoppng crteron s satsfe or reaches the gven maxmal teraton. 4. Bnary Quantum-behave Partcle Swarm Optmzaton wth Cooperatve Approach As an escrbe, each partcle represents a complete soluton vector for the objectve functon f ( X ) f X, X,, X N. Each upate step s also performe on a full D-mensonal vector. Then t may be appear the possblty that some menson n the soluton vector have move closer to the global optmum, whle others move away from the global optmum. Whereas the objectve functon value of the soluton vector s worse than the former value. an take the new soluton vector for a complete vector an neglect the eterorate components urng the teratons. As long as the current objectve functon value s better than the former value, then upate an gbest. Therefore, the current soluton vector can be gve up n next teraton an the valuable nformaton of the soluton vector s lost unknowngly. In orer to make full use of the benefcal nformaton, the cooperatve metho [,] s ntrouce to. In the propose metho, we expect that the operaton can avo the unesrable behavor, whch s a case of takng two steps forwar (some menson mprove), an one step back (some menson eterorate). 4. Cooperatve Approach We expect that once for every tme a component n the vector has been upate, resultng n much qucker feeback. Thus, a cooperatve metho for ong just ths s presente. In the new metho each menson of the new soluton vector replaces n turn the corresponng menson of an gbest, an then compare the new objectve functon value to ece whether to upate an gbest. The process s as follows: Step For each partcle, ntalze c ; cgbest gbest, Step For each menson of partcle, replace the menson of c an cgbest by the corresponng menson of the partcle; Step 3 Evaluate the new objectve functon value of c an cgbest, an compare them wth the objectve functon value of an gbest. If the current objectve functon value s better than that of an gbest, then upate an gbest ; Step 4 Repeat step ~3 untl all the menson of the partcle s compare. 4. Wth above mofcatons, the teraton process of s escrbe step-by-step below. Step Intalze an array of bnary bts for all partcles, partcle s personal best postons an swarm s global best poston gbest ; Step Upate the partcle s new poston x by ; Step 3 Evaluate the objectve functon value of the partcle, an compare them wth the objectve functon value of an gbest. If the current objectve functon value s better than that of an gbest, then upate an gbest ; Step 4 Use cooperatve strategy to upate an gbest ; Copyrght (c) 3 Internatonal Journal of Computer Scence Issues. All Rghts Reserve.

4 IJCSI Internatonal Journal of Computer Scence Issues, Vol., Issue, No, January 3 ISSN (Prnt): ISSN (Onlne): Step 5 Repeat step ~4 untl the stoppng crteron s satsfe or reaches the gven maxmal teraton. The propose algorthm tres to mprove convergence precson by comparng each menson of soluton vector. It must exten the search space an then ncrease the tme consumpton. Two aaptve control methos are propose. Frstly, the cooperatve strategy s aopte n a certan nterval. In our metho, t set to 5. Then the cooperatve strategy s performe when the bt of the partcle s fferent from the corresponng bt of an gbest. 5. Experments In ths secton, the performance of algorthm s teste on the followng fve fferent stanar functons [7] to be maxmze. Then the results are compare wth an. 3 f ( X ) x ( 5. x 5.) f ( X ) (( x (.48 x x ).48) f3( X ) 5 ( x x x3 x4 x5) x Z, ( 5. 5.) x 3 ( x ) 4 f 4( X ) 48. x (.8 x.8) 5 f5( X ) 5. 6 ( ) j j x aj a ( x ) In the numercal experments, the algorthms parameters settngs are escrbe as follow: for, the acceleraton coeffcents are set to c c an the nerta weght w s ecreasng lnearly from.9 to.4. In experments for an, the value of s.4 []. All experments are run 5 nepenent tmes respectvely wth a populaton of, an partcles on an Intel(R) Xeon(R) GB RA computer wth the software envronment of ATLAB9a. All the algorthms termnate when the number of teratons succees. The best ftness value (BFV), maxmum value an mnmum value are recore after the algorthm termnates at each run. The performance of all the algorthms s ) evaluate by average BFV (Avg. BFV) an Stanar Devaton (St. Dev.). All the measurements are lste on Table. Fg. llustrates the convergence process of average BFV of three algorthms over 5 runs wth partcles on fve test functons. The optma of functon f, whose ftness value s, can be fn out by, an. As can be seen from Table, the average BFV an St. Dev. of s best. An outperforms. As of soluton qualty, an wth partcles make successful searches out of 5 tral runs, whereas fn out the optma for 7 tmes. An the corresponng tmes s 4, 3 an respectvely wth partcles. When the populaton number s, the optma are foun out for 9, an 4 tmes corresponng wth, an. On the functon f, all the algorthms can be foun the optmum ftness value. owever generates best average BFV an St. Dev.. An takes secon place. As can be seen from Table, has the worst performance than other two algorthms wth partcles. Note that the St. Dev. of wth partcles s better than that of. The thr functon f 3 s a smple nteger functon wth an optmum of., an wth partcles ht the optma for 5 tmes out of 5 runs. an have better qualty of soluton than wth an partcles. In orer to measure the average ftness value over the entre populaton, Gaussan nose s ntrouce nto f 4 functon. In ths functon, the average BFV of s nferor to but superor to. owever the St. Dev. of s the best results. The last functon f 5 has an optmum 5. All the algorthms can be foun out the best value wth an partlces s able to ht the optmum beyon 47 tmes out of 5 runs. The number of successful searches of s better than. owever the average BFV an St. Dev. of s nferor to. As s llustrate n Fg., we can see that the effectveness of the propose. can converge to the optmum more raply than an on three functons except f an f 5. On f, converges more quckly but generates worse soluton than. On f 5, converges raply than other two Copyrght (c) 3 Internatonal Journal of Computer Scence Issues. All Rghts Reserve.

5 IJCSI Internatonal Journal of Computer Scence Issues, Vol., Issue, No, January 3 ISSN (Prnt): ISSN (Onlne): algorthms at the early stage of runnng, but excees soon an generates a slghtly better soluton. Compare wth an, expermental results show the effectveness of the propose. Functon f f f 3 f 4 f 5 Partcles Table : Results of, an on fve testng functons ean (St.Dev.) AX (IN) ean (St.Dev.) AX (IN) ean (St.Dev.) AX (IN) functon f functon f teratons teratons (a) f (b) f Copyrght (c) 3 Internatonal Journal of Computer Scence Issues. All Rghts Reserve.

6 IJCSI Internatonal Journal of Computer Scence Issues, Vol., Issue, No, January 3 ISSN (Prnt): ISSN (Onlne): functon f3 6 functon f teratons (c) f 3 () f 4 5 functon f5 teratons teratons (e) f 5 Fg. The convergence process of three algorthms wth partcles. 6. Conclusons In, an mprovement n two components wll overrule a potentally goo value for a sngle component. In ths paper, a screte bnary verson of Quantumbehave Partcle Swarm Optmzaton algorthm wth cooperatve metho () s ntrouce to mprove the unesrable behavor by ecomposng the soluton vector. In the propose algorthm, each menson upate of partcle can fee back to personal best postons an swarm best poston. The results of experment have showe that the algorthm performs better than other algorthm on global convergence an has stronger ablty to escape from the local optmal soluton urng the search process. owever t can be exten the search space wth the ncreasng complexty of the problem, tme consumpton s the man efcency of. References [] Kenney J., an Eberhart R., "Partcle Swarm Optmzaton", n IEEE Internatonal Conference on Neural Networks, 995, Vol.4, pp [] Frans Van Den Bergh, "An Analyss of Partcle Swarm Optmzers", Ph.D. thess, Unversty of Pretora, South Afrca,. [3] Jun Sun, Bn Feng, an Wenbo Xu, "Partcle Swarm Optmzaton wth Partcles avng Quantum Behavor", n IEEE Congress on Evolutonary Computaton, 4, Vol., pp [4] Jun Sun, We Fang, Xaojun Wu, Vasle Palae, an Wenbo Xu, "Quantum-behave Partcle Swarm Optmzaton: Analyss of Invual Partcle Behavor an Parameter Selecton", Evolutonary Computaton, Vol., No. 3,, pp [5] We Fang, Jun Sun, Yanru Dng, Xaojun Wu, an Wenbo Xu, "A Revew of Quantum-behave Partcle Swarm Optmzaton", IETE Techncal Revew, Vol.7, No.4,, pp Copyrght (c) 3 Internatonal Journal of Computer Scence Issues. All Rghts Reserve.

7 IJCSI Internatonal Journal of Computer Scence Issues, Vol., Issue, No, January 3 ISSN (Prnt): ISSN (Onlne): [6] Kenney, J., an Eberhar, R. C, "A Dscrete Bnary Verson of the Partcles Swarm Algorthm". In IEEE Internatonal Conference on Systems, an an Cybernetcs, 997, Vol.5, pp [7] Jun Sun, Wenbo Xu, We Fang, an Zhle Cha, "Quantumbehave Partcle Swarm Optmzaton wth Bnary Encong", n Internatonal Conference on Aaptve an Natural Computng Algorthms, 7, Vol., pp [8] Seyye Al ashem,an Behrouz Nowrouzan, "A Novel Dscrete Partcle Swarm Optmzaton for FR FIR Dgtal Flters", Journal of Computers, Vol.7, No.6,, pp [9] Jun Sun, Xaojun Wu, Vasle Palae, We Fang, Cho-ong La, an Wenbo Xu, "Convergence Analyss an Improvements of Quantum-behave Partcle Swarm Optmzaton", Journal of Informaton Scence, Vol.93,, pp.8-3. [] Van en Bergh F., an Engelbrecht A. P., "A Cooperatve Approach to Partcle Swarm Optmzaton", IEEE Transactons on Evolutonary Computaton, Vol.8, No.3, 4, pp [] eh Neshat, Shma F. Yaz, Daneyal Yazan, an eh Sargolzae, "A New Cooperatve Algorthm Base on PSO an K-eans for Data Clusterng", Journal of Computers Scence, Vol.8, No.,, pp [] Jun Sun, "Partcle Swarm Optmzaton wth Partcles avng Quantum", Ph.D. thess, Jangnan Unversty, Wux, Chna, 9. Copyrght (c) 3 Internatonal Journal of Computer Scence Issues. All Rghts Reserve.