Modified Seeker Optimization Algorithm for Unconstrained Optimization Problems

Transcription

1 Modfed Seeker Optmzaton Algorthm for Unconstraned Optmzaton Problems Ivona BRAJEVIC Mlan TUBA Faculty of Mathematcs Faculty of Computer Scence Unversty of Belgrade Megatrend Unversty Belgrade Studentsk trg 16 Bulevar umetnost 29 SERBIA SERBIA Abstract: - Seeker optmzaton algorthm (SOA) s a novel search algorthm based on smulatng the act of human searchng, whch has been shown to be a promsng canddate among search algorthms for unconstraned functon optmzaton. In ths artcle we propose a modfed seeker optmzaton algorthm. In order to enhance the performance of SOA, our proposed approach uses two search equatons for producng new populaton and employs modfed nter-subpopulaton learnng phase of algorthm. Ths modfed algorthm has been mplemented and tested on fourteen multmodal benchmark functons and proved to be better on majorty of tested problems. Key-Words: - Unconstraned optmzaton, Seeker optmzaton algorthm, Artfcal bee colony, Nature nspred heurstcs 1 Introducton 1Optmzaton problems have the objectve to fnd mnmum or mamum of the functon under consderaton. Unconstraned optmzaton problems can be formulated as a D-dmensonal mnmzaton or mamzaton problem: mn (or ma) f(), ( 1,..., D ) R (1) where D s the number of the parameters to be optmzed. There are many populaton based metaheurstcs appled to unconstraned optmzaton problems. Genetc algorthms (GA) [1], partcle swarm optmzaton (PSO) [2], artfcal bee colony (ABC) algorthm [3], dfferental evoluton (DE) [4] and recently proposed seeker optmzaton algorthm (SOA) [5] are among the most popular metaheurstcs whch employ a populaton of ndvduals to solve the problem. The success of a populaton based method depends on ts ablty to produce proper balance between eploraton and eplotaton. The eplotaton refers to the ablty of algorthm to apply the knowledge of the prevous good solutons to better gude the search towards mprovng regons of the search space. The eplora- Ths research s supported by Mnstry of Scence, Republc of Serba, Project No D ton refers to the ablty of algorthm to nvestgate the unknown regons n the search space to dscover the global optmum. A poor balance between eploraton and eplotaton may result n a weak optmzaton method whch may suffer from premature convergence,.e. the algorthm may be trapped n a local optmum. In order to enhance the performance of SOA, we propose a modfed seeker optmzaton algorthm named MSO, whch uses two search equatons for producng new populaton and employs modfed nter-subpopulaton learnng phase of algorthm. In each teraton of the proposed algorthm a new possble soluton s produced by search equaton of artfcal bee colony (ABC) algorthm or search equaton of seeker optmzaton algorthm (SOA). The ABC algorthm developed by Karaboga and ts modfed versons were successfully appled to unconstraned and constraned optmzaton problems [3], [6], [7], [8]. Seeker optmzaton algorthm was analyzed wth a challengng set of benchmark problems for functon optmzaton, where ts performance was compared wth the performance of DE and three versons of PSO algorthms. SOA has shown better global search ablty and faster convergence speed for the most chosen benchmark problems, especally for unmodal benchmarks. For multmodal test functons the results were not very satsfactory because t was notced that for ths type of problems ISBN:

2 algorthm may be stuck at a local optmum. Snce ts nventon, SOA has been appled to solve the other knds of problems besde numercal functon optmzaton. In [9], the applcaton of the SOA to tunng the structures and parameters of artfcal neural networks s presented, whle n [10] SOAbased evolutonary method s proposed for dgtal IIR flter desgn. Also, a new optmzed model of proton echange membrane fuel cell (PEMFC) was proposed by usng SOA [11]. In ths work, our MSO algorthm was appled to multmodal test functons and ts performance was compared wth the performance of SOA. Modfed seeker optmzaton algorthm showed better performance than SOA for the majorty of tested problems. Ths paper s organzed as follows. Secton 2 presents the seeker optmzaton algorthm. Secton 3 descrbes our proposed approach. Secton 4 descrbes benchmark functons. Secton 5 presents the epermental setup adopted and test results. Our concluson s provded n secton 6. 2 Seeker optmzaton algorthm Seeker optmzaton algorthm (SOA) mmcs the behavor of human search populaton based on ther memory, eperence, uncertanty reasonng and communcaton wth each other. SOA s a populaton-based heurstc algorthm. The algorthm operates on a set of solutons called search populaton. Each ndvdual of the populaton s called a seeker or agent. The total populaton s categorzed nto three subpopulatons accordng to the ndees of the seekers. All the seekers n the same subpopulaton consttute a neghborhood whch represents the socal component for the socal sharng of nformaton. Each seeker has the followng attrbutes: the current poston t) [,,...,,..., ], where j s the j th ( 1 2 D dmenson, D s the dmenson sze and t s the number of teraton, the personal best poston p, so far, and the neghborhood best poston best g best so far. The man characterstc features of ths algorthm are the followng: 1. The algorthm uses search drecton and step length to update the postons of seekers 2. The calculaton of the search drecton s based on a compromse among egotstc behavor, altrustc behavor and pro-actveness behavor 3. Fuzzy reasonng s used to generate the step length because the uncertan reasonng of human searchng could be the best descrbed by natural lngustc varables and a smple f-else control rule: If {objectve functon value s small} (.e., condton part), then {step length s short} (.e., acton part) A search drecton d (t) and a step length (t) are separately computed for each ndvdual on each dmenson j at each teraton t, where ( t) 0 and d ( t) { 1,0,1 }. 2.1 Calculaton of the search drecton In swarm dynamc there are two etreme types of cooperatve behavor: egotstc whch s entrely pro-self and altrustc whch s entrely pro-group. Every seeker s unformly egotstc f he beleves that he should go toward hs personal best poston p, through cogntve learnng. The egotstc best behavor of each seeker may be modeled by vector called egotstc drecton d, by: d ego p ( ) (2), ego, best t In altrustc behavor, seekers want to communcate wth each other and adjust ther behavors n response to the other seekers n the same neghborhood regon for achevng the desred goal. Ths atttude of each seeker may be modeled by vector called altrustc drecton d, by: d alt g ( ) (3), alt best t where gbest represents the neghborhood best poston so far. Besde of egostc and altrustc behavor, the future behavor can be predcted and guded by the past behavor. Due to ths, the seeker may be proactve to change ts search drecton and ehbt goal-drected behavor accordng to hs past behavor. The pro-actve behavor of each seeker may be modeled by vector called pro-actveness drecton d, by: d pro, pro ( 1 t2 t ) ( ) (4) where t 1, t 2 { t, t 1, t 2}, ( t 1 ) and ( t 2 ) are the best and the worst postons n the set { ( t 2), ( t 1), ( t)} respectvely. The epresson of search drecton for the th seeker s set to the stochastc combnaton of egotstc drecton, altrustc drecton and pro-actveness drecton by: d( t) sgn( w d, pro 1 d, ego 2 d, alt ) (5) ISBN:

3 where the functon sgn ( ) s a sgnum functon on each dmenson of the nput vector, ω s the nerta weght and 1 and 2 are real numbers chosen unformly and randomly n the range [0,1]. Inerta weght s used to gradually reduce the local search effect of pro-actveness drecton d, and provde pro a balance between global and local eploraton and eplotaton. Inerta weght s lnearly decreased from 0.9 to 0.1 durng a run. 2.2 Calculaton of the step length From the vew pont of human searchng behavor, t s understood that one may fnd the near-optmal solutons n a narrower neghborhood of the pont wth lower objectve functon value and on the other hand, n a wder neghborhood of the pont wth hgher objectve functon value. To desgn a fuzzy system to be applcable to a wde range of optmzaton problems, the objectve functon values of each subpopulaton are sorted n descendng order and turned nto the sequence numbers from 1 to S as the nputs of fuzzy reasonng, where S denotes the sze of the subpopulaton to whch the seekers belong. The epresson s presented as: S I ma ( ma mn ) (6) S 1 where I s the sequence number of (t) after sortng the objectve functon values n descendng order, ma s the mamum membershp degree value whch s equal to or a lttle less than / 2 2 Bell membershp functon f ( ) e s well utlzed n the lterature to represent the acton part of the control rule. Because the membershp functon value of beyond [ 3,3 ] are less than , a mnmum membershp degree mn = s set. Moreover, the parameter δ of the Bell membershp functon s determned by : abs( mn avg ) (7) In Eq. (7), the absolute value of the nput vector as the correspondng output vector s represented by the symbol abs ( ), mn s the poston of the best seeker n the subpopulaton to whch the th seeker belongs, and s the averaged poston of the avg seekers n the same subpopulaton. The nerta weght s used to decrease the step length wth ncreasng tme step, whch mproves the search precson. In order to ntroduce the randomness n each varable and to mprove the local search capablty, the followng equaton s ntroduced to convert nto a vector wth elements as gven by: rand,1), j 1,2,... D (8) ( The acton part of the fuzzy reasonng gves the step length for each seeker by : ln( ), j 1,2,... D (9) 2.3 Implementaton of seeker optmzaton algorthm At each teraton the poston of each seeker s updated by: ( t 1) ( t) ( t) d ( t) (10) where 1,2,... SN; j 1,2,... D (SN s the number of seekers) Also, at each teraton, the current postons of the worst two ndvduals of each subpopulaton are echanged for both of the best one n each of the other two subpopulatons, whch s called ntersubpopulaton learnng. Short pseudo code of the SOA algorthm s gven below: 1. Generatng s postons unformly and randomly n search space; 2. cycle = 0; 3. Repeat 4. For = 1 to s do Computng d (t) by Eqs. (2), (3), (4) and (5); Computng (t) by Eqs. (6), (7), (8) and (9); Updatng each seeker s poston usng Eq. (10); 5. End of For 6. Evaluatng all the seekers and savng the hstorcal best poston; 7. Implementng the nter-subpopulaton learnng operaton; 8. cycle = cycle+1; 9. Untl the termnaton crteron s satsfed 3 Our proposed approach: MSO In order to enhance the performance of SOA, MSO algorthm uses two search equatons for producng new populaton: search equaton of ABC algorthm and the search equaton of seeker optmzaton algorthm. Also, MSO algorthm mplements the modfed nter-subpopulaton learnng usng the bnomal crossover operator as n [12]. At the frst step, MSO algorthm generates a randomly ISBN:

4 dstrbuted ntal populaton of SN solutons, where SN denotes the number of seekers (solutons). Each soluton X ( = 1, 2,..., SN ) s a D-dmensonal vector and D s the number of optmzaton parameters. Total populaton s dvded nto K subpopulatons accordng to the ndees of the seekers. After ntalzaton, the populaton of the solutons s subjected to repeated cycles of the search processes. Any teraton of MSO algorthm can be descrbed as followng: 1. Perform an update process for each soluton n the search populaton usng randomly selected search equaton. MSO chooses between search equaton (10) whch s used n SOA and the varant of ABC search equaton whch can be descrbed as: v ( ), f R 0.5 kj, j otherwse (11) where k s randomly chosen nde of soluton from the subpopulaton to whch the th seeker belongs, and k has to be dfferent from, j 1,2,..., D and s a random number between [-1, 1). The MSO ncluded a new control parameter whch s called behavor rate (BR) n order to select the search equaton n the followng way: If a random number between [0, 1) s less then BR the SOA search equaton s used, otherwse Eq.(11) s performed. 2. Evaluatng all the seekers and savng the hstorcal best poston. 3. The modfed nter-subpopulaton learnng s mplemented as follows: The postons of seekers wth the hghest objectve functon values of each subpopulaton l are combned wth the postons of seekers wth the lowest objectve functon values of (l+t) mod K subpopulatons respectvely, where t=1,2,.. NSC. NSC denotes the number of the worst seekers of each populaton whch are combned wth the best seekers. The approprate seekers are combned usng the followng bnomal crossover operator as epressed n: ln j, worst In Eq. (12), wthn [0, 1), jbest, f R j 0.5 (12) l j worst, otherwse n, R j s a unformly random real number, s denoted as the j th varable j l n worst of the n th worst poston n the l th subpopulaton, s the j th varable of the best poston n the th j best subpopulaton. Addtonally, we ncluded a new parameter whch we named nter-subpopulaton learnng ncrease perod (ILIP). After ILIP teratons the number of the worst seekers of each subpopulatons whch are combned wth the best seekers s ncreased to 2* NSC. We can conclude that n MSO algorthm we have three new control parameters n comparson wth the orgnal SOA: the behavor rate (BR), the number of seekers of each subpopulaton for combnaton (NSC) and the nter-subpopulaton learnng ncrease perod (ILIP). BR parameter s used to control whch of the search equatons for producng new populaton wll be used. In MSO algorthm as n SOA, each subpopulaton s searchng for the optmal soluton usng ts own nformaton and hence the subpopulaton may trap nto local optma yeldng a premature convergence. Hence the ntersubpopulaton learnng, whch ensures that good nformaton acqured by each subpopulaton s shared among the subpopulatons, s ncluded n the algorthm. But, n the nter-subpopulaton learnng of SOA t was notced that t may not always brng the benefts for multmodal functons snce t may attract all agents toward a local optmal soluton. In MSO algorthm we ncreased the dversty of populaton by usng the crossover operator for producng new solutons. In order to provde better balance between eplotaton and eploraton abltes of algorthm, the dversty of populaton s addtonally ncreased by NSC and ILIP parameters, whch ncrease the number of seekers for combnng. 4 Benchmark functons Fourteen benchmark functons from [5] were used to test the performance of our MSO algorthm. The characterstcs of benchmark functons are gven n Table 1. D denotes the dmensonalty of the test problem, S denotes the ranges of the varables, and f mn s a functon value of the global optmum. Functons f1 - f 6 (Schwefel, Rastrgn, Ackley, Grewank, Penalzed - two functons) are multmodal functons where the number of local mnma ncreases eponentally wth the problem dmenson. Functons f 7 - f 14 (Shekel foholes, Kowalk, S-hump Camel-Back, Brann, Goldsten-Prce, Hartman famly - two functons, Shekel) are low-dmensonal functons whch have only a few local mnma. For multmodal functons, the fnal results are much more mportant than the convergence rates, snce they reflect the algorthm's ablty to escape from poor local optma and locate a good near-global optmum. ISBN:

5 D S f mn f1 30 [-500, 500] D f2 30 [-5.12, 5.12] D 0 f3 30 [-32, 32] D 0 f4 30 [-600, 600] D 0 f5 30 [-50, 50] D 0 f6 30 [-50, 50] D 0 f7 2 [-65.54, 65.54] D f8 4 [-5, 5] D 3.075E-4 f9 2 [-5, 5] D f10 2 [-5, 15] D f11 2 [-2, 2] D 3 f12 3 [0, 1] D f13 6 [0, 1] D f14 4 [0, 10] D Table 1 Characterstcs of benchmark functons 5 Parameter settngs, results and dscusson We appled our modfed seeker optmzaton (MSO) algorthm to the set of benchmark multmodal optmzaton problems. The proposed algorthm has been mplemented n Java programmng language. Tests for fourteen benchmark problems were done on a Intel(R) Core(TM) 2 Duo CPU E8500@4-GHz personal computer wth 4 GB RAM memory. The obtaned results are compared to the results obtaned by seeker optmzaton algorthm (SOA) and ts results are taken from [5]. In all eperments for the both algorthms the same sze of populaton of 100 seekers s used and the algorthms were repeated 50 runs. The mamum number of generatons (G) for each tested problem s the same for the both algorthms and ther values are lsted n Table 2. In our proposed approach the number of subpopulatons (SubpopNum) s taken 10 and the values of new control parameters are: the behavor rate (BR) s 0.4, the number of seekers of each subpopulaton for combnaton (NSC) s taken 0.2* SP/SubpopNum and the nter-subpopulaton learnng ncrease perod (ILIP) s taken 0.4*G. The comparatve results of the best, mean values and standard devatons over 50 ndependent runs of the SOA and MSO algorthm are summarzed n Table 2. The dmensons of multmodal functons wth many local mnma f1 - f 6 were all set to 30 for both algorthms. As t can be seen from Table 2, MSO algorthm acheved better results for functons f 1, f 5 and f 6, whle for functons f 2 and f 3, SOA performed better than MSO algorthm. For functon f 4, SOA has better mean value and the same best value as MSO algorthm. For functons f 7 - f 14, the number of local mnma and the dmenson are small. Functon Stats SOA MSO Best f1 Mean (G = 9 000) St. dev f2 (G = 5 000) f3 (G = 1 500) f4 (G = 2 000) f5 (G = 1 500) f6 (G = 1 500) f7 f8 (G = 4 000) f9 f10 f11 f12 f13 (G = 200) f14 Best E-00 Mean E+01 St. dev E+01 Best -4.44E E-15 Mean -4.44E E-14 St. dev E-14 Best Mean E-16 St. dev E-17 Best 3.04E E-31 Mean 1.28E E-03 St. dev 7.62E E-02 Best 2.77E E-31 Mean 1.89E E-02 St. dev 1.30E E-01 Best Mean St. dev 5.30E E-01 Best E E-04 Mean E E-04 St. dev 1.58E E-04 Best Mean St. dev 6.73E E-16 Best Mean St. dev E-14 Best 3 3 Mean St. dev 1.17E E-15 Best Mean St. dev 6.69E E-16 Best Mean St. dev E-02 Best Mean St. dev 4.72E E-09 Table 2 Comparson of SOA and MSO algorthm The results from Table 3 showed that SOA s outperformed by MSO algorthm for functons f 7, ISBN:

6 f 10, f 11, f 12 and f 14, whle SOA acheved better results for functons f 8 and f 13. Also, MSO algorthm s not statstcally dfferent from SOA for functon f 9. In [5] t was notced that for multmodal functons SOA may be stuck at a local optmum. From Table 2 t can be seen that the performance of MSO algorthm s better for the majorty of multmodal benchmarks, half of hgh-dmensonal multmodal functons and for the most of lowdmensonal multmodal functons. 6 Concluson Seeker optmzaton algorthm has been tested for a challengng set of benchmark problems for functon optmzaton. The smulaton results showed that the proposed algorthm s compettve and a promsng canddate among swarm algorthms for numercal functon optmzaton. In ths paper we presented the modfed seeker optmzaton (MSO) algorthm for unconstraned functon optmzaton. In order to avod the algorthm to trap at some local attractors MSO algorthm uses two search equatons for producng new populaton and bnomal crossover operator n the nter-subpopulaton learnng phase of algorthm. The epermental results tested on fourteen multmodal benchmark functons show that the proposed approach mproved the performance of SOA for majorty of tested functons. References: [1] J. Holland, Adaptaton n Natural and Artfcal Systems, MIT Press, Cambrdge, MA, 1992 [2] Kennedy, J., Eberhart, R.C., Partcle swarm optmzaton, Proceedngs of the 1995 IEEE Internatonal Conference on Neural Networks, Pscataway, NJ: IEEE Servce Center, 1995, pp [3] Karaboga D., Akay B., A comparatve study of Artfcal Bee Colony algorthm, Appled Mathematcs and Computaton, 2009, Volume 214, Issue 1, pp [4] K.V. Prce, R.M. Storn, J.A. Lampnen, Dfferental Evoluton: A Practcal Approach to Global Optmzaton, Sprnger-Verlag, Berln, Germany, 2005 [5] Da C., Chen W., Song Y., Zhu Y., Seeker optmzaton algorthm: a novel stochastc search algorthm for global numercal optmzaton, Journal of Systems Engneerng and Electroncs, Vol. 21, No. 2, 2010, pp [6] Zhu G, Kwong S., Gbest-guded artfcal bee colony algorthm for numercal functon optmzaton, Appled Mathematcs and Computaton, Volume 217, Issue 7, 2010, pp [7] Stanarevc N., Tuba M., Bacann N., Modfed artfcal bee colony algorthm for constraned problems optmzaton, Internatonal Journal of Mathematcal Models and Methods n Appled Scences, Volume 5, Issue 3, 2011, pp [8] Brajevc I., Tuba M., Subotc M., Performance of the mproved artfcal bee colony algorthm on standard engneerng constraned problems, Internatonal Journal of Mathematcs and Computers n Smulaton, Volume 5, Issue 2, 2011, pp [9] Da C., Chen W., Zhu Y., Jang Z., You Z., Seeker optmzaton algorthm for tunng the structure and parameters of neural networks, Neurocomputng, Volume 74, Issue 6, 2011, pp [10] Da C., Chen W., Zhu Y., Seeker optmzaton algorthm for dgtal IIR flter desgn, IEEE Transactons on Industral Electroncs, Volume 57, Issue 5, 2010, pp [11] Da C., Chen W., Cheng Z., L Q., Jang Z., Ja J.. Seeker optmzaton algorthm for global optmzaton: A case study on optmal modellng of proton echange membrane fuel cell (PEMFC), Internatonal Journal of Electrcal Power and Energy Systems, 2011, Volume 33, Issue 3, pp [12] B. Shaw, S. Ghoshal, V. Mukherjee, S. P. Ghoshal, Soluton of Economc Load Dspatch Problems by a Novel Seeker Optmzaton Algorthm, Internatonal Journal on Electrcal Engneerng and Informatcs, Vol. 3, No. 1, 2011, pp ISBN: