A Hybrid Algorithm Based on Gravitational Search and Particle Swarm Optimization Algorithm to Solve Function Optimization Problems

Transcription

1 Engneerng Letters, 25:1, EL_25_1_04 A Hybrd Algorthm Based on Gravtatonal Search and Partcle Swarm Optmzaton Algorthm to Solve Functon Optmzaton Problems Je-Sheng Wang, and Jang-D Song Abstract Gravtatonal search algorthm (GSA) s a swarm ntellgence heurstc optmzaton algorthm based on the law of gravtaton. Amng at the dsadvantage of poor local search ablty and slow convergence speed n standard GSA, four mproved GSA-PSO hybrd algorthm are proposed by ntroducng a small constant updatng strategy n order to enhance the update ablty of velocty, acceleraton factor and the optmal ndvdual locaton, where PSO strategy was used to optmze the poston and velocty of the GSA. Through smulaton experments on typcal test functons to verfy ts performance, the smulaton results show that the optmal setup of GSA parameters can mprove the convergence rate of the algorthm and mprove the accuracy of the soluton. Index Terms gravtatonal search algorthm, partcle swarm optmzaton algorthm, functon optmzaton F I. INTRODUCTION UNCTION optmzaton problem s to fnd the optmal soluton of the obectve functon by the teratve [1]. In general, the search obectve s to optmze the functon of the obectve functon, whch s usually descrbed by the contnuous, dscrete, lnear, nonlnear, concave and convex of functon. There has been a consderable attenton pad for employng metaheurstc algorthms nspred from natural processes and/or events n order to solve functon optmzaton problems. The swarm ntellgent optmzaton algorthm [2] s a random search algorthm smulatng the evoluton of bologcal populatons. It can solve the complex anuscrpt receved June 17, 2016; revsed October 5, Ths wor was supported by the Proect by Natonal Natural Scence Foundaton of Chna (Grant No ), the Program for Laonng Excellent Talents n Unversty (Grant No. LR ), the Proect by Laonng Provncal Natural Scence Foundaton of Chna (Grant No ), the Program for Research Specal Foundaton of Unversty of Scence and Technology of Laonng (Grant No. 2015TD04) and the Openng Proect of Natonal Fnancal Securty and System Equpment Engneerng Research Center (Grant No. USTLKFGJ and USTLKEC201401). Je-Sheng Wang s wth the School of Electronc and Informaton Engneerng, Unversty of Scence and Technology Laonng, Anshan, , PR Chna; Natonal Fnancal Securty and System Equpment Engneerng Research Center, Unversty of Scence and Technology Laonng. (phone: ; fax: ; e-mal: wang_esheng@126.com). Jang-D Song s a postgraduate student n the School of Electronc and Informaton Engneerng, Unversty of Scence and Technology Laonng, Anshan, , PR Chna (e-mal: sd2011@163.com). global optmzaton problems through the cooperaton and competton among ndvduals. The representatve swarm ntellgence optmzaton algorthms nclude Ant Colony Optmzaton (ACO) algorthm [3], Genetc Algorthm (GA) [4], Bat Algorthm (BA) [5], Artfcal Bee Colony (ABC) algorthm [6], etc. But, not all metaheurstc algorthms are bo-nspred, because ther sources of nspraton often come from physcs and chemstry. For the algorthms that are not bo-nspred, most have been developed by mmcng certan physcal and/or chemcal laws, ncludng electrcal charges, gravty, rver systems, etc. The typcal physcs and chemstry nspred metaheurstc algorthms nclude Bg Bang-bg Crunch optmzaton algorthm [7], Blac hole algorthm [8], Central force optmzaton algorthm [9], Charged system search algorthm [10], Electro-magnetsm optmzaton algorthm [11], Galaxy-based search algorthm [12], Harmony search algorthm [13], Intellgent water drop algorthm [14], Rver formaton dynamcs algorthm [15], Self-propelled partcles algorthm [16], Spral optmzaton algorthm [17], Water cycle algorthm [18], etc. The gravtatonal search algorthm (GSA) was ntroduced by E. Rashed et al n 2009 [19], whch was constructed based on the law of gravty and the noton of mass nteractons. The GSA algorthm uses the theory of Newtonan physcs and ts searcher agents are the collecton of masses. It has been successfully appled n many global optmzaton problems, such as, mult-obectve optmzaton of synthess gas producton [20], forecastng of turbne heat rate [21], dynamc constraned optmzaton wth offsprng repar [22],fuzzy control system [23],grey nonlnear constraned programmng problem [24], reactve power dspatch of power systems [25], mnmum rato travelng salesman problem [26], parameter dentfcaton of AVR system [27],strategc bddng [28], etc. In ths paper, four nds of mproved GSA-PSO hybrd algorthm were proposed by ntroducng a small constant updatng mechansm, whch adopts PSO strategy to optmze the velocty and poston n the runnng process of the GSA. The smulaton analyss results show that the mproved hybrd algorthm greatly mproves the functon optmzaton convergence speed and optmzaton accuracy. (Advance onlne publcaton: 22 February 2017)

2 Engneerng Letters, 25:1, EL_25_1_04 II. GRAVITATIONAL SEARCH ALGORITH A. Physcs foundaton of GSA The law of unversal gravtaton s one of the four basc forces n nature. The gravtatonal force s proportonal to the product of the mass, and s nversely proportonal to the square of the dstance. The gravtatonal force between two obects s calculated by: F G (1) R where, F s the gravtatonal force between two obects, G s the gravtatonal constant, 1 and 2 are the masses of the obect 1 and 2 respectvely, R s the dstance between these two obects. Accordng to the nternatonal unt system, the unt of F s Newton (N), the unt of 1 and 2 s g, the unt of R s m, and the constant G s approxmately equal to N m g. The acceleraton of the partcle a s related to ts mass and of the gravtatonal force F, whch s calculated by the followed equaton. F a (2) Accordng to the Eq. (1) and (2), all of the partcles n the world are affected by gravty. The more close the dstance between two partcles, the greater the gravtatonal. Its basc prncple s shown n Fgure 1, where the mass of the partcles s represented by the mage sze. Partcle 1 s nfluenced by the gravty of the other three partcles to produce the resultant force F. Such an algorthm wll converge to the optmal soluton, and the gravtatonal force wll not be affected by the envronment, so the gravty has a strong local. obects, and then the postons of N obects are brought nto the functon, where the poston of the th obect s defned as follows. 2) Calculate the nerta mass 1 2 d X (x, x,... x,..., x ) (3) At the moment t, the mass of the partcle X s represented as t. ass t can be calculated by the followed equaton. a p (4) ft worst m best worst (5) m n m (6) where 1,2,, n, ft( ) s the ftness value of the obect, best( t ) s the optmal soluton and worst( t ) s the worst soluton. The calculaton equaton of best( t ) and worst( t ) are descrbed as follows. For solvng the maxmum problem: {1,2,..., n} best max ft (7) worst mn ft (8) {1,2,..., n} For solvng the mnmum value problem: best mn ft {1,2,..., n} (9) worst mn ft (10) {1,2,..., n} 3) Calculate gravtatonal force At the moment t, the calculaton formula for the gravtatonal force of obect to obect descrbed as follows. p a F G ( x x ) R (11) Fg. 1. Gravtatonal phenomena B. Basc prncples of gravtaton search algorthm 1) Intalze the locatons Because of no need to consder the envronmental mpact, the poston of a partcle s ntalzed as X. Frstly, randomly generate the postons x, x,..., x,... x of N 1 2 d where, s a very small constant, a ( t ) s the nertal mass of the obect tself, p ( t ) s the nertal mass of an obect. G t s the unversal gravtatonal constant at the moment t, whch s determned by the age of the unverse. The greater the age of the unverse, the smaller G t. The nner relatonshp s descrbed as follows. (t) e t T G G (12) 0 (Advance onlne publcaton: 22 February 2017)

3 Engneerng Letters, 25:1, EL_25_1_04 where G 0 s the unversal gravtatonal constant of the unverse at the ntal tme t 0, generally t s set as 100. s 20, T s the maxmum number of teratons and R ( t ) represents the Eucldean dstance between obect and obect. R X, X (13) In GSA, the sum F ( t ) of the forces actng on the X n the K th dmenson s equal to the sum of all the forces actng on ths obect: F ran F (14) 1, where ran s the random number n the range 0,1, F ( t ) s the gravty of the th obect actng on the th obect n the th dmenson space. Accordng to the Newton's Second Law, the acceleraton of the th partcle n the th dmenson at the moment t s defned as follows: 4) Change the postons F a (15) In each teraton, the obect poston can be changed by calculatng the acceleraton, whch s calculated by the followng equatons. v ( t 1) ran v a (16) x ( t 1) x v ( t 1) (17) C. Algorthm flowchart The detaled flowchart of the algorthm s shown n Fgure 2, and the optmzaton procedure s descrbed as follows. Step 1: Intalze the postons and acceleratons of all partcles, the number of teratons and the parameters of the GSA; Step 2: Accordng to the Eq. (12), calculate the ftness value of each partcle and update the gravty constant; Step 3: Accordng to the Eq. (5)-(7), calculate the qualty of the partcles based on the obtaned ftness values and the acceleraton of each partcle accordng to the Eq. (8) and Eq. (15); Step 4: Calculate the velocty of each partcle and update the poston of the partcle accordng to the Eq. (17); Step 5: If the termnaton condton s not satsfed, turn to Step 2, otherwse output the optmal soluton. III. GSA-PSO HYBRID OPTIIZATION ALGORITH A. Partcle Swarm Optmzaton Algorthm PSO algorthm adopts a large number of partcles to search the optmum n the space soluton, where the velocty of the partcle depends on the drecton and dstance of ts flght, and they have ther own ftness value. The dynamc adustment of each partcle to fnd the optmal soluton reles on the ndvdual extreme and the global extreme, untl the termnaton s met [29]. The locaton of the th partcle s represented as X x1, x 2,, xd. At each generaton, each partcle s updated by followng the two best values. The best prevous poston of the th partcle s recorded and represented as P p1, p2,, pd, whch s also called pbest. The ndex of the best pbest among all the partcles s represented by the symbol g. The locaton Pg pg1, pg2,, pgd s also called gbest. The velocty for the th partcle s represented as V v1, v2,, vd. The PSO concept conssts of, at each generaton, updatng the velocty and locaton of each partcle toward ts pbest and gbest locatons accordng to the Eq. (16) and (17), respectvely: V V c1 rand ( pbest x ) (16) c rand gbest x (17) n 1 t t d t 2 ( ) t1 t t1 X X V where w s nerta weght, c 1 and c 2 are learnng constants, and rand () s a random functon n the range [0, 1]. For Eq. (16), the frst part represents the nerta of the prevous velocty; the second part s the cognton part, whch represents the prvate thnng by tself; the thrd part s the socal part, whch represents the co-operaton among the partcles. V d s clamped to a maxmum velocty V max. B. GSA-PSO Algorthm In recent years, Seyedal. proposed BPSOGSA hybrd algorthm wth bnary parameters of the PSOGSA algorthm [29] n order to solve the bnary parameter optmzaton problem. Four hybrd algorthms (-1,, -3 and ) based on the standard GSA-PSO algorthm n the lterature [29] are proposed, whch adopts the small constant updatng strategy n three parts on the on the materal moton equaton respectvely so as to the memory functon when updatng the velocty, acceleraton, and the optmal postons. The detaled analyss are descrbed as follows. The strong exploraton ablty of GSA and the PSO algorthm are combned to obtan the better optmzaton ablty. The mproved partcle velocty updatng equatons are descrbed as follows. t ' ' V ( t 1) r V c1 r ac c2 r ( gbest x ) (18) t ' ' V ( t 1) ( r ) V c1 r ac c2 r ( gbest x ) (19) t ' ' V ( t 1) r V c1 ( r ) ac c2 r ( gbest x ) (20) t ' ' V ( t 1) r V c1 r ac c2 ( r ) ( gbest x ) (21) (Advance onlne publcaton: 22 February 2017)

4 Engneerng Letters, 25:1, EL_25_1_04 where V t s the velocty of partcle at the generaton t, c s the acceleraton coeffcent, r s a random located n the scope [0,1], s a small constant, ac ( t ) s the acceleraton of partcle at the generaton t,and gbest s the optmal soluton so far. The poston of the partcles s updated accordng to the followng equaton. X ( t 1) X V ( t 1) (22) It s worth notng that the qualty and nerta mass equatons of partcle at the generaton t are mproved n the lterature (29), whch s descrbed as follows, respectvely. Fg. 2. Flowchart of GSA then calculate the ftness value ft( t ) and the optmal value gbest of each partcle n the mproved algorthms. gbest s compared wth the gbest generated n Step 3 to obtan the optmal value gbest. Step 5: Judge whether the termnaton functon or the number of teratons s met. If t s stll not satsfed, return to Step 3. m ft 0.99 worst (23) best worst 5 m n (24) m 1 C. Algorthm Procedure The algorthm procedure of the mproved GSA-PSO algorthm shown n Fgure 3 s descrbed as follows. Step 1: Intalze the algorthm parameters, ncludng the total number of partcles N, the number of teratons t, the gravtatonal constant G 0 and the decreasng coeffcent. Step 2: Randomly generate populatons. The poston vector of the partcle s set as X ( x1, x2,, x n ), the velocty s ntalzed as v ( 1, 2,, ) T v v vs, the global optmal value s gbest and the ndvdual optmal value s pbest. Step 3: Calculate the ftness value of each ndvdual ft( t ), fnd the best ftness value best( t ) and the worst ftness value worst( t ), and record the best poston gbest. Step 4: Calculate the qualty of each ndvdual accordng to Eq. (5)-(8); then calculate the gravtatonal constant G and the Eucldean dstance between two partcles so as to calculate the ndvdual's gravty F and acceleraton a. At ths tme, by adoptng the global search ablty of PSO algorthm, four hybrd algorthms are used to update the poston and velocty of the ndvdual by usng Eq. (18)-(22); Fg. 3 The flowchart of mproved hybrd algorthm IV. SIULATION EXPERIENTS AND RESULTS ANALYSIS The parameters of the hybrd algorthm are ntalzed as follows: obect number N =30, the maxmum number of teratons max_t=1000, the gravtatonal constant G 0 =100 at tme t0, =23. Four test functons are adopted to carry out smulaton and software ATLAB s used as the smulaton platform. The ndependent operatons are carred out 50 tmes. Four test functons are shown n Table 1, whch are named as F1, F2, F3 and F4. The dmenson of F1, F2 and F3 s 30, and the dmenson of F4 s 2. On the other hand, F1 s a sngle functon, the others are polymorphc functons. Fgure 4 (a) - (b) are the compared optmzaton results of (Advance onlne publcaton: 22 February 2017)

5 Engneerng Letters, 25:1, EL_25_1_04 the above proposed algorthms. In order to mae the smulaton results more clearly, for the F3 and F4 wth fast convergence speed, the 100 generaton smulaton results are retaned wthn the optmzaton curves. The smulaton statstcal results are shown n Table 2. It can be seen from the above smulaton results that the optmzaton tme of low dmensonal functon F4 s smaller than other hgh dmensonal functon. The fast convergence rate maes each algorthm gradually tend to the optmal value and reach the steady state n the teraton number of In addton, the optmal values of the complex mult-state functon F3 and F4 are equal n each hybrd algorthm. Comparson of the optmzaton results, four nds of hybrd algorthm are better than the GSA-PSO algorthm proposed n the lterature (29). For the unmodal functon F1, the optmzaton results are compared as: < < -3 < -1. For the mult pea functon F2, the optmzaton results are compared as: -3 < < -1 <. In concluson, through the ntroducton of the small constant update strategy, the optmzaton performances of four proposed methods are mproved sgnfcantly. For sngle state functon, the ablty to update the speed and poston of the best ndvdual n the hybrd algorthm s enhanced, whch can effectvely adust the nfluence of "law of attracton" and "socal nformaton exchange" on the partcles. For the mult-pea functons, the acceleraton and velocty at the partcle movement by "law of attracton" and "speed" functon are adusted. In short, the mproved hybrd algorthms have an mportant nfluence on the law of attracton between substances. TAB. 1 SIULATION FUNCTIONS Functon Name Expresson equaton Scope Dmenson F 1 Quartc n 4 x [0,1) 1 random [-1.28,1.28] n 30 F 2 Rastrgn n F 3 Schwefel n x sn x F 4 Sheel s Foxholes n 2 x 10cos(2 x ) 10 [-5.12,5.12] n 30 1 [-600,600] n ( ) 500 ( ) x a 1 [-65.53,65.53] Best score obtaned so far Iteraton (a) Quartc (Advance onlne publcaton: 22 February 2017)

6 Engneerng Letters, 25:1, EL_25_1_04 Best score obtaned so far F Iteraton (b) Rastrgn F11 Best score obtaned so far Iteraton (c) Schwefel (Advance onlne publcaton: 22 February 2017)

7 Engneerng Letters, 25:1, EL_25_1_ Best score obtaned so far Iteraton (d) Sheel s Foxholes functon Fg. 4 Smulaton curves of fve algorthms for functon optmzaton TAB. 2 THE SIULATION RESULTS OF HYBRID ALGORITH Functon / Algorthm -1-3 Optmum F 1 Tme (s) Optmum F 2 Tme (s) Optmum F 3 Tme (s) Optmum F 4 Tme (s) V. CONCLUSION Gravtatonal search algorthm (GSA) s a swarm ntellgence heurstc optmzaton algorthm based on the law of gravtaton. In ths paper, four mproved GSA-PSO hybrd algorthm are proposed by ntroducng the optmzaton strategy of the moton equaton based on the mcro constant renewal materal. The smulaton results show that the mproved algorthm has good performance. REFERENCES [1] Y. Ren, and Y. Wu, An effcent algorthm for hgh-dmensonal functon optmzaton. Soft Computng, vol. 17, no. 6, pp , Jun [2] Z. Yuan,. A.. de Oca, and. Brattar, Contnuous optmzaton algorthms for tunng real and nteger parameters of swarm ntellgence algorthms, Swarm Intellgence, vol. 6, no. 1, pp , ar [3] Y. Ghanou, and G. Bencheh, "Archtecture Optmzaton and Tranng for the ultlayer Perceptron usng Ant System," IAENG Internatonal Journal of Computer Scence, vol. 43, no.1, pp , (Advance onlne publcaton: 22 February 2017)

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Shawd, V. uheree,and S. P. Ghoshal, Soluton of reactve power dspatch of power systems by an opposton-based gravtatonal search algorthm, Internatonal Journal of Electrcal Power & Energy Systems, vol. 55, no. 2, pp , Feb [26] Y. Lu, and L. a, Gravtatonal Search Algorthm for nmum Rato Travelng Salesman Problem, Journal of Chnese Computer Systems, vol. 34, no. 4, pp , Apr [27] C. L, H. L, and P. Kou, Pecewse functon based gravtatonal search algorthm and ts applcaton on parameter dentfcaton of AVR system, Neurocomputng, vol. 124, no. 2, pp , Jan [28] J. V. Kumar,D.. V. Kumar,and K. Eduondalu, Strategc bddng usng fuzzy adaptve gravtatonal search algorthm n a pool based electrcty maret, Appled Soft Computng, vol. 13, no. 5, pp , ay [29] S. rall, S. Z.. Hashm, New Hybrd PSOGSA Algorthm for Functon Optmzaton, n Proc Internatonal Conference on Computer and Informaton Applcaton (ICCIA), Tann, Chna, 2010, pp Je-sheng Wang receved hs B. Sc. And. Sc. degrees n control scence from Unversty of Scence and Technology Laonng, Chna n 1999 and 2002, respectvely, and hs Ph. D. degree n control scence from Dalan Unversty of Technology, Chna n He s currently a professor and aster's Supervsor n School of Electronc and Informaton Engneerng, Unversty of Scence and Technology Laonng. Hs man research nterest s modelng of complex ndustry process, ntellgent control and Computer ntegrated manufacturng. Jang-D Song s receved her B. Sc. degree from Unversty of Scence and Technology Laonng n She s currently a master student n School of Electronc and Informaton Engneerng, Unversty of Scence and Technology Laonng, Chna. Her man research nterest s modelng methods of complex process and ntellgent optmzaton algorthms. (Advance onlne publcaton: 22 February 2017)