A New Quantum Behaved Particle Swarm Optimization

Transcription

1 A New Quantum Behaved Partcle Swarm Optmzaton Mlle Pant Department of Paper Technology IIT Roorkee, Saharanpur Inda Radha Thangaraj Department of Paper Technology IIT Roorkee, Saharanpur Inda Ajth Abraham QS, Norwegan Center of Excellence NTNU, Trondhem Norway ABSTRACT Ths paper presents a varant of Quantum behaved Partcle Swarm Optmzaton (QPSO) named Q-QPSO for solvng global optmzaton problems. The Q-QPSO algorthm s based on the characterstcs of QPSO, and uses nterpolaton based recombnaton operator for generatng a new soluton vector n the search space. The performance of Q-QPSO s compared wth Basc Partcle Swarm Optmzaton (BPSO), QPSO and two other varants of QPSO taken from lterature on sx standard unconstraned, scalable benchmark problems. The expermental results show that the proposed algorthm outperforms the other algorthms qute sgnfcantly. Categores and Subject Descrptors D.3.3 [Programmng Languages]: Language Contructs and Features abstract data types, polymorphsm General Terms Algorthms, Performance, Relablty, Expermentaton Keywords Partcle swarm optmzaton, Interpolaton, Global optmzaton, Quantum behavor.. INTRODUCTION Partcle Swarm Optmzaton (PSO) s relatvely a newer addton to a class of populaton based search technque for solvng numercal optmzaton problems. The partcles or members of the swarm fly through a multdmensonal search space lookng for a potental soluton. In classcal (or orgnal PSO), developed by Kennedy and Eberhart n 995 [], each partcle adjusts ts poston n the search space from tme to tme accordng to the flyng experence of ts own and of ts neghbors (or colleagues). For a D-dmensonal search space the poston of the th partcle s represented as X = (x, x,, x D ). Each partcle mantans a memory of ts prevous best poston P best = (p, p p D ). Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. GECCO 8, July --6, 8, Atlanta, Georga, USA. Copyrght 8 ACM /8/7 $5.. The best one among all the partcles n the populaton s represented as P gbest = (p g, p g p gd ). The velocty of each partcle s represented as V = (v, v, v D ). In each teraton, the P vector of the partcle wth best ftness n the local neghborhood, desgnated g, and the P vector of the current partcle are combned to adjust the velocty along each dmenson and a new poston of the partcle s determned usng that velocty. The two basc equatons whch govern the workng of PSO are that of velocty vector and poston vector gven by: vd = wvd + cr ( pd xd ) + cr ( pgd xd ) () x d = xd + vd () The frst part of equaton () represents the nerta of the prevous velocty, the second part s the cognton part and t tells us about the personal experence of the partcle, the thrd part represents the cooperaton among partcles and s therefore named as the socal component. Acceleraton constants c, c and nerta weght w are the predefned by the user and r, r are the unformly generated random numbers n the range of [, ]. PSO has undergone a plethora of changes snce ts development. One of the recent developments n PSO s the applcaton of Quantum laws of mechancs to observe the behavor of PSO. Such PSO s are called Quantum PSO (QPSO). Some varants of QPSO nclude mutaton based PSO [], [3], where mutaton s appled to the mbest (mean best) and gbest (global best) postons of the partcle, also n one of the varants of QPSO a perturbaton constant called Lyapunov constant s added. However to the best of our knowledge no has used the concept of recombnaton operator n QPSO. Ths paper presents a QPSO called Q-QPSO whch uses the quantum theory of mechancs to govern the movement of swarm partcles along wth an nterpolaton (quadratc nterpolaton) based recombnaton operator. The concept of quadratc nterpolaton as a recombnaton operator was ntroduced by us [4], [5], for mprovng the performance of classcal PSO, where t gave sgnfcantly good results. Ths motvated us to apply ths concept for QPSO also to mprove ts performance. The remanng of the paper s organzed as follows: Secton brefly descrbes the Quantum Partcle Swarm Optmzaton. Secton 3, explans the proposed Q-QPSO, Secton 4, gves the expermental settngs and numercal results of some selected unconstraned benchmark problems. The paper fnally concludes wth Secton 5. 87

2 . QUANTUM PARTICLE SWARM OPTIMIZATION The development n the feld of quantum mechancs s manly due to the fndngs of Bohr, de Brogle, Schrödnger, Hesenberg and Bohn n the early twenteth century. Ther studes forced the scentsts to rethnk the applcablty of classcal mechancs and the tradtonal understandng of the nature of motons of mcroscopc objects [6]. As per classcal PSO, a partcle s stated by ts poston vector x and velocty vector v, whch determne the trajectory of the partcle. The partcle moves along a determned trajectory followng Newtonan mechancs. However f we consder quantum mechancs, then the term trajectory s meanngless, because x and v of a partcle cannot be determned smultaneously accordng to uncertanty prncple. Therefore, f ndvdual partcles n a PSO system have quantum behavor, the performance of PSO wll be far from that of classcal PSO [7]. In the quantum model of a PSO, the state of a partcle s depcted by wavefuncton Ψ ( x, t), nstead of poston and velocty. The dynamc behavor of the partcle s wdely dvergent from that of the partcle n tradtonal PSO systems. In ths context, the probablty of the partcle s appearng n poston x from probablty densty functon Ψ ( x, t), the form of whch depends on the potental feld the partcle les n []. The partcles move accordng to the followng teratve equatons [8], [9]: x( t + ) = p + β * mbest x( t) *ln(/ u) f k. 5 x( t + ) = p β * mbest x( t) *ln(/ u) f k <. 5 (3) where p = c p + c P ) /( c ) (4) ( d gd + c M M mbest = P = P M M, M M P n,..., P M d Mean best (mbest) of the populaton s defned as the mean of the best postons of all partcles, u, k, c and c are unformly dstrbuted random numbers n the nterval [, ]. The parameter β s called contracton-expanson coeffcent. The flow chart of QPSO algorthm s shown n Fgure. (5) recombnaton. The Q-QPSO algorthm starts lke the usual QPSO usng equatons (3), (4) and (5). At the end of each teraton, the quadratc nterpolaton recombnaton operator s nvoked to generate a new swarm partcle. The new partcle s accepted n the swarm only f t s better than the worst partcle (.e. the partcle havng maxmum ftness) present n the swarm. Ths process s repeated teratvely untl a better soluton s obtaned. The quadratc crossover operator s a nonlnear operator whch produces a new soluton vector lyng at the pont of mnma of the quadratc curve passng through the three selected swarm partcles. The selecton of partcles, say {a, b, c} s done as follows: a = X mn, (the swarm partcle havng mnmum (or best) ftness functon value) {b, c} = {randomly chosen partcles from the remanng members of the swarm. (a, b and c should be three dfferent partcles) The dea behnd the ncluson of an nterpolaton operator s to facltate the Q-QPSO wth a recombnaton operator whch wll help n fndng a new soluton pont n the search space. Snce, we are always holdng the partcle havng the best ftness functon value to take part n recombnaton process, the probablty of generatng a new tral vector wth ftness functon whch s better than at least one of the exstng soluton vectors n the swarm ncreases. Thus the new partcle s accepted n the swarm only f t s better than the worst partcle. ~ ~ ~ ~ n Mathematcally, the new partcle x = ( x, x,..., x ), s gven as: ~ ( b c )* f ( a) + ( c a )* f ( b) + ( a b )* f ( c) x = (6) ( b c )* f ( a) + ( c a )* f ( b) + ( a b )* f ( c) The computatonal steps of Q-QPSO algorthm are gven by: Step : Intalze the swarm wth unformly dstrbuted random numbers. Step : Calculate mbest usng equaton (5) Step 3: Update partcles poston usng equaton (3) Step 4: Evaluate the ftness value of each partcle Step 5: If the current ftness value s better than the best ftness value (Pbest) n hstory Then Update Pbest by the current ftness value. Step 6: Update Pgbest (global best) Step 7: Fnd a new partcle usng equaton (6) Step 8: If the new partcle s better than the worst partcle n the swarm Then replace the worst partcle by the new partcle. Step 9: Go to step untl maxmum teratons reached. Fgure Flow of QPSO Algorthm 3. PROPOSED Q-QPSO ALGORITHM The proposed Q-QPSO algorthm s a smple and modfed verson of QPSO n whch we have ntroduced the concept of 4. EXPERIMENTAL SETTINGS AND BENCHMARK PROBLEMS In Q-QPSO algorthm a lnearly decreasng contracton-expanson coeffcent (β) s used whch starts at. and ends at.5. the acceleraton constants are taken as.. In order to check the compatblty of the proposed Q-QPSO algorthm we have tested 88

3 t on benchmark problems (unconstraned) gven n Table. All the test problems are hghly multmodal and scalable n nature and are consdered to be startng pont for checkng the credblty of any optmzaton algorthm. There s not much lterature avalable n whch QPSO s used extensvely for solvng a varety of test problems. Therefore, for the present study, we have consdered test problems out of whch the frst three problems are the ones that have been tested wth some varants of QPSO. The remanng 6 problems we have solved wth our verson and wth QPSO and BPSO. As n [3], for functons f, f and f3, three dfferent dmenson szes are tested. They are, and 3. The maxmum number of generatons s set as, 5 and correspondng to the dmensons, and 3 respectvely. Dfferent populaton szes are used for each functon wth dfferent dmensons. The populaton szes are, and 8. We have tested the functons f4 f wth dmensons, 3 and 5. A total of 3 runs for each expermental settng are conducted and the average ftness of the best solutons throughout the run s recorded. Tables, 3 and 4 show the mean best ftness of Q-QPSO, BPSO, QPSO and ts two varants n lterature for functons f, f and f3 respectvely. Table 5 shows the mean best ftness values of Q- QPSO, BPSO and QPSO for the functons f4 f. Tables 6, 7 and 8 shows the T-test values for the benchmark problems f, f and f3 n comparson wth the other algorthms. Fgures, 3 and 4 depct the performance wth a focus on mean best ftness for some selected functons. In all the Tables, Pop represents populaton, Dm represents dmenson and Gen represents Generaton. Table. Numercal benchmark problems Functon Range Optmum f ( x) = ( x cos(π x ) + ) = [.56,5.] n ( ) n x f x = x cos( ) + = + [3,6] n f ( ) ( ) ( ) 3 x = x + = + x x [5,3] f 4 ( x) x sn( x ) [-5,5] *n = n n 4 5 ( x) = ( ( + ) x ) + rand[,] f [-.8,.8] f n n 6 ( x) = + e exp(. x n ) exp( cos(π )) [-3,3] n f ( x) = max x, < n [-,] 7 n 8 ( x) = x + / f [-,] n 9 ( x) = n f x + x [-,] n ( x) = ( x ) = j = f [-,] x 89

4 Pop Dm Gen Q-QPSO 8 Table. Comparson results of functon f (Mean best) QPSO Mutaton [3] BPSO [] QPSO [] QPSO Mutaton [] Gbest mbest gbest mbest 4.95e e e e e e e Table 3. Comparson results of functon f (Mean best) Pop Dm Gen Q-QPSO BPSO[] QPSO[] QPSO Mutaton [3] QPSO Mutaton[] gbest mbest gbest mbest Pop Dm Gen Q-QPSO 8 Table 4. Comparson results of functon f3 (Mean best) BPSO [] QPSO QPSO Mutaton [3] QPSO Mutaton [] [] Gbest mbest gbest mbest

5 Table 5. Comparson results of functons f4 f (Mean best) Functon Dm Gen BPSO QPSO Q-QPSO f4 f5 f6 f7 f8 f9 f e- 4.7e-5.e e e- 5.88e e e e-6.986e e e-4.58e e e e e e e-.558e e-54.95e e e-.999e e e-7 Table 6. T-test* value for the functon f: comparson of Q-QPSO wth other algorthms Pop Dm Gen 8 BPSO [] QPSO [] QPSO Mutaton [3] QPSO Mutaton [] gbest mbest gbest mbest

6 Table 7. T-test* value for the functon f: comparson of Q-QPSO wth other algorthms Pop Dm Gen BPSO [] QPSO [] QPSO Mutaton[3] QPSO Mutaton [] gbest mbest gbest mbest Po p Table 8. T-test* value for the functon f3: comparson of Q-QPSO wth other algorthms QPSO Mutaton [3] QPSO Mutaton [] Dm Gen BPSO [] QPSO [] gbest mbest gbest mbest * The t value of 9 degrees of freedom s sgnfcant at a level a.5 level of sgnfcance by a two-taled t-test 9

7 (a) Fgure. Performance for Rastrngn functon (a) dmenson (b) dmenson (b) (a) Fgure 3. Performance for Grewank functon (a) dmenson (b) dmenson (b) (a) Fgure 4. Performance for Rosenbrock functon (a) dmenson (b) dmenson (b) 93

8 5. CONCLUSIONS Ths paper presents a varant of Quantum PSO called Q-QPSO, ncorporatng the concept of recombnaton operator. The proposed Q-QPSO s tested on three standard unconstraned benchmark test problems and the results are compared wth some of the exstng QPSO (contanng mutaton operator) and standard PSO. For the frst test problem, whch s, Rastrngn s functon (a hghly multmodal functon), the proposed Q-QPSO algorthm outperformed all the gven algorthms, qute sgnfcantly and gave the near optmum soluton (whch s ) n all the test cases. Smlarly for the second test problem (Grewank functon), Q- QPSO gave better results than the other algorthms n seven out of the nne test cases tred. For the test functons f3 f also Q- QPSO gave superor results n all the cases. The domnance of the proposed Q-QPSO algorthm s also apparent from the two taled t-test gven n Tables 5, 6 and 7. However, we would lke to add that the collecton of test problems taken n ths paper s not exhaustve and therefore makng any concrete concluson on the performance of Q-QPSO do not sound justfed. Moreover, t s very much possble that the ncluson of a crossover operator may have an adverse effect on the dversty of the algorthm whch n turn may deterorate ts performance for solvng problems havng large number of varables. Also, t would be nterestng to do a theoretcal analyss on quadratc nterpolaton operator and ts applcaton to optmzaton problems. However, on the postve sde t s qute evdent from the numercal results that for the hghly multmodal problems havng varables up to 5, Q-QPSO s defntely a better choce over the contemporary optmzaton algorthms. For the future work, we shall be apply Q-QPSO for solvng more complex unconstraned and constraned optmzaton problems. Also we shall be studyng the combned effect of mutaton and recombnaton on a Quantum behaved PSO. 6. REFERENCES [] Kennedy, J. and Eberhart, R. Partcle Swarm Optmzaton. IEEE Internatonal Conference on Neural Networks (Perth, Australa), IEEE Servce Center, Pscataway, NJ, IV: , 995. [] Lu J, Sun J, Xu W, Quantum-Behaved Partcle Swarm Optmzaton wth Adaptve Mutaton Operator. ICNC 6, Part I, Sprnger-Verlag: , 6. [3] Lu J, Xu W, Sun J. Quantum-Behaved Partcle Swarm Optmzaton wth Mutaton Operator. In Proc. of the 7 th IEEE Int. Conf. on Tools wth Artfcal Intellgence, Hong Kong (Chna), 5. [4] Mlle Pant, Radha Thangaraj and Ajth Abraham, A New PSO Algorthm wth Crossover Operator for Global Optmzaton Problems, Second Internatonal Symposum on Hybrd Artfcal Intellgent Systems (HAIS 7), Advances n Softcomputng Seres, Sprnger Verlag, Germany, E. Corchado et al. (Eds.): Innovatons n Hybrd Intellgent Systems, Vol. 44, pp. 5-, 7. [5] Mlle Pant, Radha Thangaraj and Ajth Abraham, A New Partcle Swarm Optmzaton Algorthm Incorporatng Reproducton Operator for Solvng Global Optmzaton Problems, 7th Internatonal Conference on Hybrd Intellgent Systems, Kaserslautern, Germany, IEEE Computer Socety press, USA, ISBN , pp , 7. [6] Pang XF, Quantum mechancs n nonlnear systems. Rver Edge (NJ, USA): World Scentfc Publshng Company, 5. [7] Bn Feng, Wenbo Xu, Adaptve Partcle Swarm Optmzaton Based on Quantum Oscllator Model. In Proc. of the 4 IEEE Conf. on Cybernetcs and Intellgent Systems, Sngapore: 9 94, 4. [8] Sun J, Feng B, Xu W, Partcle Swarm Optmzaton wth partcles havng Quantum Behavor. In Proc. of Congress on Evolutonary Computaton, Portland (OR, USA), 35 33, 4. [9] Sun J, Xu W, Feng B, A Global Search Strategy of Quantum-Behaved Partcle Swarm Optmzaton. In Proc. of the 4 IEEE Conf. on Cybernetcs and Intellgent Systems, Sngapore: 9 94, 4. 94