Research Article An Enhanced Differential Evolution with Elite Chaotic Local Search

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1 Computatonal Intellgence and Neuroscence Volume 215, Artcle ID , 11 pages Research Artcle An Enhanced Dfferental Evoluton wth Elte Chaotc Local Search Zhaolu Guo, 1 Haxa Huang, 2 Changshou Deng, 3 Xuezh Yue, 1 and Zhjan Wu 4 1 Insttute of Medcal Informatcs and Engneerng, School of Scence, Jangx Unversty of Scence and Technology, Ganzhou 341, Chna 2 School of Lterature and Law, Jangx Unversty of Scence and Technology, Ganzhou 341, Chna 3 School of Informaton Scence and Technology, Jujang Unversty, Jujang 3325, Chna 4 State Key Laboratory of Software Engneerng, Wuhan Unversty, Wuhan 4372, Chna Correspondence should be addressed to Zhaolu Guo; gzl@whu.edu.cn Receved 8 October 214; Accepted 27 Aprl 215 AcademcEdtor:RafkAlyev Copyrght 215 Zhaolu Guo et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Dfferental evoluton (DE) s a smple yet effcent evolutonary algorthm for real-world engneerng problems. However, ts search ablty should be further enhanced to obtan better solutons when DE s appled to solve complex optmzaton problems. Ths paper presents an enhanced dfferental evoluton wth elte chaotc local search (DEECL). In DEECL, t utlzes a chaotc search strategy based on the heurstc nformaton from the elte ndvduals to promote the explotaton power. Moreover, DEECL employs a smple and effectve parameter adaptaton mechansm to enhance the robustness. Experments are conducted on a set of classcal test functons. The expermental results show that DEECL s very compettve on the majorty of the test functons. 1. Introducton Numerous problems n scence and engneerng can be converted nto optmzaton problems. Therefore, t s of sgnfcance both n theory and n engneerng applcatons to develop effectve and effcent optmzaton algorthms for solvng complex problems of scence and engneerng. Dfferental evoluton (DE), proposed by Storn and Prce n 1997 [1], s a smple yet effectve global optmzaton algorthm. Accordng to frequently reported theoretcal and expermental studes, DE has exhbted compettve performancethanmanyotherevolutonaryalgorthmsntermsof both convergence speed and soluton precson over several benchmark functons and real-lfe problems [2 4]. Due to ts smplcty, easy mplementaton, and effcency, DE has stmulated many researchers nterests snce ts development. Therefore, t has become a hot research topc n evolutonary computaton over the past decades [5 7]. However, ts search ablty should be further enhanced to obtan better solutons when DE s used to solve varous real-lfe optmzaton problems [2, 8, 9]. Partcularly,DE may suffer from premature convergence and/or slow convergence when solvng complex multmodal optmzaton problems. In order to mprove the performance of the conventonal DE, a number of DE varants have been proposed n recent decades [2, 6, 1]. Recognzng that the performance of DE depends on the control parameters, Brest et al. [11]presented a self-adaptve DE (jde), n whch both F and CR are created ndependently for each ndvdual by an adaptve mechansm. Specfcally, the new F s created by a random value from.1 to.9 wth a probablty.1 durng the search process. Meanwhle, the new CR obtans a random value from. to 1. wth a probablty.1. Unlke jde, JADE, proposed by Zhang and Sanderson [12], utlzes a dstnct parameter adaptaton mechansm, n whch the new F and CR are created for each ndvdual by a normal dstrbuton and a Cauchy dstrbuton, respectvely. In addton, JADE learns knowledge from the recent successful F and CR and apples the learned knowledge for creatng new F and CR. Identfyng that both the mutaton strateges and ther assocated control parameters can drectly nfluence the performance of DE, Qn et al. [7] proposed a novel self-adaptve DE, SaDE, whch adaptvely tunes the tral vector generaton strateges and ther assocated control parameter values by extractng

2 2 Computatonal Intellgence and Neuroscence knowledge from the prevous search process n generatng promsng solutons. Mallpedd et al. [13] ntroduced an mproved DE wth ensemble of parameters and mutaton strateges (EPSDE), whch employs a pool of dverse tral vector generaton strateges and a pool of values for the control parameters F and CR. By ncorporatng an oppostonbased learnng strategy nto the tradtonal DE for populaton ntalzaton and generatng new solutons, Rahnamayan et al. [14] proposed an opposton-based DE (ODE). The expermental results confrmed that the opposton-based learnng strategy can mprove the convergence speed and the soluton accuracy of DE. Further, Wang et al. [15] mproved the opposton-based learnng strategy, proposed a generalzed opposton-based learnng strategy, and presented an enhanced DE wth generalzed opposton-based learnng strategy(gode).jaetal.[16] presented an effectve memetc DE algorthm, DECLS, whch utlzes a chaotc local search wth a shrnkng strategy to mprove the search ablty. Expermental results ndcated that the performance of the canoncal DE s sgnfcantly mproved by the chaotc local search. Recently, Wang et al. [17] proposed a composte DE, called CoDE, the man dea of whch s to randomly combne several well studed tral vector generaton strateges wth a number of control parameter settngs hghly recommended by other researchers at each generaton to create new tral vectors. Expermental results on all the CEC25 contest test nstancesshowthatcodesverycompettve. Although there already exst many DE varants for solvng complex optmzaton problems, accordng to the no free lunch (NFL) theory [18], the performance of DE for some benchmark functons and real-lfe problems should be further enhanced to obtan better solutons. Moreover, many studes have revealed that embeddng local search strategy can greatly enhance the search ablty of DE [14, 16, 19]. Motvated by these consderatons, n order to promote the performance of DE on complex optmzaton problems, ths study proposes an enhanced dfferental evoluton wth elte chaotc local search, called DEECL. In DEECL, we utlze a chaotc search strategy based on the heurstc nformaton from the elte ndvduals to promote the explotaton power. Further, we also desgn a smple and effectve parameter adaptaton mechansm to enhance the robustness. The rest of the paper s organzed as follows. The conventonal DE s ntroduced n Secton 2. Secton 3 presents the enhanced DE. Numercal experments are presented n Secton 4 for the comparson and analyss. Fnally, the paper s concluded n Secton Dfferental Evoluton Wthout loss of generalty, only mnmzaton problems are consdered n ths study. We suppose that the objectve functon to be mnmzed s Mn f(x), X=[X 1,X 2,...,X D ],and the search space s Ω= D j=1 [LB j, UB j ], (1) where D sthenumberofdmensonsoftheproblem,lb j and UB j denote the lower and upper boundares of the search space, respectvely. Smlar to other evolutonary algorthms, DE also has a smple structure, only ncludng three smple operators, namely, mutaton, crossover, and selecton operators [2]. In the ntal phase, DE creates an ntal populaton P(t) = X t }, whch s randomly generated from the search space, where X t =[X t,1,xt,2,...,xt,d ], =1, 2,...,NP; NP s the populaton sze and t s the generaton. After ntalzaton, the mutaton and crossover operators are performed to create thetralvectors,andthentheselectonoperatorsutlzedto select the better one between the offsprng ndvdual and the parent ndvdual for the next generaton. DE performs these steps repeatedly to converge toward the global optma untl the termnatng crteron s reached [2]. In the followng subsectons, the evolutonary operators of DE wll be ntroducedndetal Mutaton Operator. In the mutaton operator, a mutant vector V t s created by usng a predetermned mutaton strategy for each ndvdual X t,namely,targetvector,nthe current populaton [17]. DE has many mutaton strateges used n ts mplementatons, such as DE/rand/1, DE/best/1, DE/rand-to-best/1, DE/best/2 and DE/rand/2 [2]. Among these mutaton strateges, DE/rand/1 s the most frequently used mutaton strategy, whch s expressed as follows [1]: V t =X t r1 +F (Xt r2 Xt r3 ), (2) where r1, r2, and r3 are randomly selected from the set 1, 2,...,NP}\}, and they are mutually dfferent from each other. F s called as scalng factor, amplfyng the dfference vector X t r2 Xt r Crossover Operator. Followng mutaton, a tral vector U t s generated by executng the crossover operator for each par of target vector X t and ts correspondng mutant vector V t [2]. Bnomal crossover s the most commonly used crossover operators n current popular DE. The bnomal crossover s descrbed as follows [1]: U t,j =,j, f rand (, 1) < CR or j==jrand X t,j, otherwse, (3) V t where rand(, 1) s generated for each j and takes a value from. to 1. n a unformly random manner, and CR [, 1] s the crossover probablty, whch lmts the number of parameters nherted from the mutant vector V t.thenteger jrand s randomly chosen from the range [1,D], whch guarantees that at least one parameter of the tral vector U t s nherted from the mutant vector V t [7] Selecton Operator. Lke the genetc algorthm, the selectonprocessofdesalsobasedonthedarwnanlawof survval of the fttest. The selecton process s performed n order to choose the more excellent ndvduals for the

3 Computatonal Intellgence and Neuroscence 3 t=; = ; / Intalze the populaton / for =1to NP do for j=1to D do X t,j = LB j +rand(, 1) (UB j LB j ); end for Evaluate ndvdual X t ; = + 1; end for whle < MAX do for =1to NP do Choose three mutually dfferent ntegers r1, r2, r3from the set 1, 2,...,NP}\}n a random manner; jrand = randnt(1, D); for j=1to D do f rand(, 1) < CRor j==jrand then U t,j = Xt r1,j + F (Xt r2,j Xt r3,j ); else U t,j = Xt,j ; end f end for / Selecton step / f f(u t ) f(xt ) then X t+1 =U t ; f f(u t )<f(xt Best ) then X t Best =Ut ; end f else X t+1 =X t ; end f = +1; end for t=t+1; end whle Algorthm 1: DE algorthm. next generaton. For mnmzaton problems, the selecton operator can be defned n the followng form [1]: = U t, f f(ut ) f(xt ) X t, otherwse, (4) X t+1 where f(x t ) and f(ut ) ndcate the ftness values of the target vector X t and ts correspondng tral vector Ut,respectvely Algorthmc Framework of DE. Based on the above elaborate ntroducton of the DE s operators, we present the framework of DE wth DE/rand/1/bn strategy n Algorthm 1, where s the number of ftness evaluatons, Max s the maxmum number of evaluatons, rand(,1) ndcates arandomrealnumberntherange[, 1], randnt(1, D) represents a random nteger n the range [1, D], andx t Best s the global best ndvdual found so far. 3. Proposed Approach 3.1. Motvatons. DE has been demonstrated to yeld superor performance for solvng varous real-world optmzaton problems [21 23]. However, t tends to suffer from premature convergence and/or slow convergence when solvng complex optmzaton problems [6, 24]. To enhance the performance of DE, many researchers have proposed varous mproved DE algorthms durng the past decade [25 27]. Among the DE varatons, memetc method s a promsng approach to mprove the performance of the tradtonal DE, whch utlzes varous local search strateges, such as chaotc search strategy [16], smplex crossover search strategy [19], and orthogonal search strategy [28], to strengthen the explotaton ablty of the tradtonal DE and consequently accelerate the convergence speed. Among the local search strateges commonly used n memetc DE, chaotc search strategy s nspred by the chaos phenomenon n nature. Chaos s a classc nonlnear dynamcal system, whch s wdely known as a system wth the propertes of ergodcty, randomcty, and senstvty to ts ntal condtons [16, 29, 3]. Due to ts ergodcty and randomcty, a chaotc system can randomly generate a longtme sequence whch s able to traverse through every state of the system and every state s generated only once f gven a long enough tme perod [16, 31]. Takng advantage of the well-known characterstcs of the chaotc systems, researchers have proposed many chaotc search strateges for optmzng varous problems [16, 32 34]. However, to the best of our knowledge, among many chaotc search strateges, they pay more attenton to the characterstcs of the ergodcty and randomcty of the chaotc system. Therefore, the exploraton capacty can be ndeed mproved. However, n order to mantan a balance between exploraton and explotaton, the explotaton ablty of the chaotc search strategy should be further enhanced. Thus, when desgnng a relatvely comprehensve chaotc search strategy, we should further ntegrate more heurstc nformaton nto the chaotc search strategy to promote ts explotaton power. Generally, the elte ndvduals n the current populaton known as a promsng search drecton toward the optmum are the favorable source that can be employed to enhance the explotaton ablty. Based on these consderatons, we present an elte chaotc search strategy, whch not only utlzes the characterstcs of the ergodcty and randomcty of the chaotc system, but also merges the superor nformaton of the current populaton nto the chaotc search process Elte Chaotc Search. In many chaotc search strateges, the Logstc chaotc functon s utlzed to generate a chaotc sequence, whch s formulated as follows [16]: K = rand (, 1), K =.25,.5,.75, K n = 4. K n 1 (1 K n 1 ), n=1, 2,...,N, where K sthentalvalueofthechaotcsystem,whch s randomly generated from the range [, 1], but cannot be equal to.25,.5, or.75. K n s the nth state of the chaotc system. As known, the ntal state K of the chaotc system (5)

4 4 Computatonal Intellgence and Neuroscence n=; N=D/5; Randomly choose an ndvdual X t I from the current populaton; K = rand(, 1), K =.25,.5,.75; p=rand(2./np,.1); whle n<ndo Randomly choose an ndvdual X t pbest from the top 1p% ndvduals n the current populaton; for j =1toDdo E n I,j =Xt I,j +Kn (X t pbest,j Xt I,j ); f E n I,j > UB j or E n I,j < LB j then E n I,j = rand(lb j, UB j ); end f end for f f(e n I )<f(xt I ) then X t I =En I ; break; end f n=n+1; K n = 4. K n 1 (1 K n 1 ); = + 1; end whle Algorthm 2: Elte chaotc search operator. s randomly produced. Due to ts ergodcty and senstvty to the ntal state K, K n s a random long-tme sequence, whch can traverse through every state of the system and every state s generated only once f N s large enough. In order to enhance the explotaton ablty of the tradtonal chaotc search strategy, we ntegrate the heurstc nformaton learned from the elte ndvduals nto the chaotc search strategy to promote the explotaton power. The proposed elte chaotc search strategy s defned by E n I =Xt I +Kn (X t pbest Xt I ), (6) where X t I s an ndvdual to be performed the elte chaotc search, whch s randomly chosen from the current populaton. K n sthechaotcsequence,wheren = 1, 2,...,N, N = D/5, and X t pbest s an elte ndvdual, whch s randomly chosen from the top 1p% ndvduals n the current populaton wth p=rand(2./np,.1). In the proposed elte chaotc search operator, an ndvdual X t I s randomly selected from the current populaton to undergo the elte chaotc search strategy. After that, the ntal value of the chaotc system takes a value from range [, 1.] n a unformly random manner. Then, an elte chaotc search procedure for ndvdual X t I s repeatedly performed untl fndng a better soluton than ndvdual X t I or the number of teratons n s equal to N.Theframeworkoftheeltechaotc search operator s descrbed n Algorthm Parameter Adaptaton. Snce the settng of control parameters can sgnfcantly nfluence the performance of DE, parameter adaptaton mechansm s essental for an effcent DE [7, 11, 12]. To ths end, we desgn a smple and effectve parameter adaptaton mechansm nspred by [11] nto DEECL. In DEECL, each ndvdual s ndependently assocated wth ts own mutaton factor F t and crossover probablty CR t.forndvdual, ts control parameters Ft and CR t are ntalzed to.5 and.9, respectvely. Generally, a normal dstrbuton wth mean value.5 and standard devaton.3 s a promsng adaptve approach for the mutaton factor of DE [7], whereas Cauchy dstrbuton s more favorable to dversfy the mutaton factors and thus avod premature convergence [12]. Based on these consderatons, at each generaton, the new mutaton factor NF t assocated wth ndvdual s generated by a Cauchy dstrbuton random real number wth locaton parameter.5 and scale parameter.3 wth probablty.1. Addtonally, followng the suggestons n [11], the new crossover probablty NCR t assocated wth ndvdual acqures a random value from. to 1. wth probablty.1. Mathematcally, the new control parameters NF t and NCR t assocated wth ndvdual for generatng ts correspondng tral vector U t are obtaned by NF t = randc (.5,.3), f rand (, 1) <.1 F t, otherwse, NCR t = rand (, 1), f rand (, 1) <.1 CR t, otherwse, where randc(.5,.3) s a Cauchy dstrbuton random real number wth locaton parameter.5 and scale parameter.3 and rand(, 1) s a unformly random number wthn the range [, 1]. After obtanng the new control parameters NF t and NCRt, the correspondng tral vector Ut are created (7)

5 Computatonal Intellgence and Neuroscence 5 Table 1: The 13 classcal test functons. Functon Name Intal range f mn f1 Sphere Problem [ 1, 1] D f2 Schwefel sproblem2.22 [ 1, 1] D f3 Schwefel sproblem1.2 [ 1, 1] D f4 Schwefel sproblem2.21 [ 1, 1] D f5 Rosenbrock s Functon [ 3, 3] D f6 Step Functon [ 1, 1] D f7 Quartc Functon wth Nose [ 1.28, 1.28] D f8 Schwefel s Problem 2.26 [ 5, 5] D f9 Rastrgn s Functon [ 5.12, 5.12] D f1 Ackley s Functon [ 32, 32] D f11 Grewank Functon [ 6, 6] D f12 Penalzed Functon 1 [ 5, 5] D f13 Penalzed Functon 2 [ 5, 5] D by usng the new control parameters NF t and NCR t.its wdely acknowledged that better control parameter values tend to produce better ndvduals that have a greater chance to survve and thus these values should be propagated to the next generatons [12]. Therefore, n the selecton step, the control parameters F t+1 and CR t+1 assocated wth ndvdual for the next generaton are updated by = NF t, f f(ut )<f(xt ) F t, otherwse, F t+1 = NCR t, f f(ut )<f(xt ) CR t, otherwse. CR t+1 From the above desgned parameter adaptaton mechansm, we can nfer that the better control parameters of DEECL can be propagated to the next generatons. Therefore, the control parameters of DEECL can be adaptvely tuned accordng to the feedback from the search process. 4. Numercal Experments 4.1. Expermental Setup. In order to assess the performance oftheproposeddeecl,weuse13classcaltestfunctons (f1 f13)thatarewdelyusedntheevolutonarycomputaton communty [8, 12, 35] to verfy the effectveness of the proposed DEECL. We descrbe these test functons n Table 1. Among these test functons [35], f1 f4 are contnuous unmodal functons. f5 s the Rosenbrock functonwhch s unmodal for D = 2 and 3;however,t may have multple mnma n hgh dmenson cases [36]. f6 sadscontnuous step functon, and f7 s a nosy functon. f8 f13 are multmodal functons and they exst many local mnma [35]. In all experments, we set the number of dmensons D to 3 for all these test functons. We carry out 3 ndependent runs for each algorthm and each test functon wth 15, functon evaluatons () as the termnaton crteron. Moreover, we record the average and standard devaton of the functon error value (f(x) f(x )) for estmatng the (8) performance of the algorthms, as recommended by [17], where x s the best soluton ganed by the algorthm n a run and x s the global optmum of the test functon Beneft of the Two Components. There are two mportant components n the proposed DEECL: the proposed elte chaotc search strategy and the desgned parameter adaptaton mechansm. Accordngly, t s nterestng to recognze the beneft of the two components of the proposed DEECL. For ths purpose, we conduct experments to compare the proposed DEECL wth the tradtonal DE wth DE/rand/1 strategy and two varants of DEECL, namely, DE wth the proposed elte chaotc search strategy (DEwEC) and DE wth the desgned parameter adaptaton mechansm (DEwPA). In the experments, we set the populaton sze of all the algorthms to 1. For the other parameters of DE and DEwEC, we set F =.5 and CR =.9, followng the suggestons n [11]. We present the expermental results of the above mentoned algorthms n Table 2. The best results among the four algorthms are hghlghted n boldface. MeanError and Std Dev ndcate the mean and standard devaton of the functon error values acheved n 3 ndependent runs, respectvely. From the results of comparson between DE and DEwEC, DEwEC performs better than DE on all test functons wth the excepton of f6. On test functon f6, both DE and DEwEC exhbt smlar performance. In total, DEwEC s better than DE on twelve test functons. The results of comparson between DE and DEwEC ndcate that our ntroduced elte chaotc search strategy s effectve to enhance the performance of the tradtonal DE. From the comparson of DE wth DEwPA, DEwPA surpasses DE on all test functons except for f5 and f6. On test functon f6, both DE and DEwPA demonstrate smlar performance, whereas DE s better than DEwPA on test functon f5. In summary, DEwPA outperforms DE on eleven test functons. The comparson of DE wth DEwPA reveals that our desgned parameter adaptaton mechansm scapableofmprovngtheeffcencyofthetradtonalde. By ncorporaton of both the proposed elte chaotc search strategy and the desgned parameter adaptaton mechansm, DEECL acheves promsng performance, whch s better than other three DE algorthms on the majorty of the test functons. To be specfc, DEECL s better than DE, DEwEC, anddewpaoneleven,nne,andtentestfunctons,respectvely. DE, DEwEC, and DEwPA can outperform DEECL only on one test functon. Comparson results suggest that both the ntroduced elte chaotc search strategy and the desgned parameter adaptaton mechansm demonstrate postve effect on the performance of DEECL. In addton, the comparson results confrm that the ntroduced elte chaotc search strategy and the desgned parameter adaptaton mechansm can help DE wth both outperform DE wth ether or nether one on the majorty of the test functons. Moreover, the ntroduced elte chaotc search strategy and the desgned parameter adaptaton mechansm work together to mprove the performance of the tradtonal DE rather than contradct each other. The evoluton of the average functon error values derved from DE, DEwEC, DEwPA, and DEECL versus

6 6 Computatonal Intellgence and Neuroscence Table 2: Expermental results of DE, DEwEC, DEwPA, and DEECL over 3 ndependent runs for the 13 test functons. Functon DE DEwEC DEwPA DEECL Mean ± Std Dev Mean ± Std Dev Mean ± Std Dev Mean ± Std Dev f1 2.23E 16 ± 2.5E E 24 ± 3.95E E 33 ± 4.76E E 38 ± 6.6E 38 f2 2.86E 8 ± 1.26E E 12 ± 6.75E E 2 ± 2.8E E 22 ± 1.21E 22 f3 1.88E 1 ± 6.12E E 2 ± 1.3E E 2 ± 5.94E E 2 ± 3.44E 2 f4 1.7E 1 ± 2.13E E 2 ± 2.1E E 4 ± 1.55E 4 4.6E 5 ± 3.5E 5 f5 1.39E+ 1 ± 8.74E E + 1 ± 5.12E E+ 1 ± 1.58E E+ 1 ± 2.21E+ 1 f6.e + ±.E +.E + ±.E +.E + ±.E +.E + ±.E + f7 8.82E 3 ± 2.61E E 3 ± 8.72E E 3 ± 1.59E E 3 ± 6.52E 4 f8 7.31E+ 3 ± 3.75E E+ 3 ± 3.25E E 2 ± 1.82E E 2 ± 1.19E 12 f9 1.77E+ 2 ± 1.1E+ 1.E + ±.E +.E + ±.E +.E + ±.E + f1 5.93E 9 ± 3.1E E 13 ± 1.76E E 15 ± 1.7E 15 4.E 15 ±.E + f E 16 ± 1.16E 15.E + ±.E +.E + ±.E +.E + ±.E + f12 2.2E 17 ± 1.81E E 25 ± 1.38E E 3 ± 7.69E E 32 ± 2.74E 48 f E 17 ± 3.59E E 24 ± 6.14E E 32 ± 2.2E E 32 ± 3.7E 34 the number of s plotted n Fgure1 for some typcal test functons. As can be seen from Fgure 1, DEECL converges fasterthande,dewec,anddewpa Comparson wth Other DE Varants. In order to verfy the effectveness of the proposed DEECL algorthm, we compare DEECL wth the tradtonal DE and three other DE varants, namely, jde [11], ODE [14], and DECLS [16]. In addton, jde s a self-adaptve DE, n whch both parameters F and CR are generated ndependently for each ndvdual by an adaptve mechansm [11]. ODE s proposed by Rahnamayan et al. [14], whch ncorporates the opposton-based learnng strategy nto the tradtonal DE for populaton ntalzaton and creatng new solutons. DECLS s an effectve memetc DE algorthm [16], whch utlzes the chaotc local search strategy and an adaptve parameter control approaches smlar to jde [11] to mprove the search ablty. In the experments, n order to have a far comparson, we set thepopulatonszeofallthealgorthmsto1.theother parametersettngsofthesethreedevarantsarethesameas n ther orgnal papers. The mean and standard devaton of the functon error values acheved by each algorthm for the 13 classcal test functons are presented n Table 3. For convenence of analyss, the best results among the four DE algorthms are hghlghted n boldface. In order to gan statstcally sgnfcant conclusons, we conduct two-taled t-tests at the sgnfcance level of.5 [28, 35] on the expermental results. The summary comparson results are descrbed n the last three rows of Table 3. +,, and suggest that DEECL s better than, worse than, and smlar to the correspondng algorthm n terms of the two-taled t-tests at the sgnfcance level of.5, respectvely. From Table 3, we can nfer that DEECL acheves the better results than all the other four algorthms on the majorty of the 13 classcal test functons. Specfcally, DEECL s sgnfcantly better than DE, jde, ODE, and DECLS on eleven, seven, nne, and sx test functons accordng to the two-taled t-test, respectvely. In addton, DEECL s smlar to DE, jde, ODE, and DECLS on one, fve, two, and fve test functons, respectvely. DE and jde surpasses DEECL only on one test functon. Addtonally, ODE and DECLS perform betterthandeeclonlyontwotestfunctons. Overall, DEECL performs better than the tradtonal DE, jde, ODE, and DECLS on the majorty of the test functons. Ths can be because the proposed elte chaotc search strategy learnng the heurstc nformaton from the elte ndvduals can promote the explotaton power, and the desgned parameter adaptaton mechansm can enhance the robustness. The evoluton of the average functon error values derved from DE, jde, ODE, DECLS, and DEECL versus the number of s plotted n Fgure 2 for some typcal test functons. It can be known from Fgure 2 that DEECL converges faster than DE, jde, ODE, and DECLS. In order to compare the total performance of the fve DE algorthms on the all 13 classcal test functons, we carry out the average rankng of Fredman test on the expermental results followng the suggestons n [37 39]. Table 4 presents the average rankng of the fve DE algorthms on the all 13 classcal test functons. We can sort these fve DE algorthms by the average rankng nto the followng order: DEECL, DECLS, jde, ODE, and DE. Therefore, DEECL obtans the best average rankng, and ts total performance s better than that of the other four algorthms on the all 13 test nstances. 5. Conclusons DE s a popular evolutonary algorthm for the contnuous global optmzaton problems, whch has a smple structure yet exhbts effcent performance on varous real-world engneerng problems. However, accordng to the no free lunch (NFL) theory, the performance of DE should be further enhanced to obtan better solutons n some cases. Inthspaper,weproposeanenhanceddfferentalevoluton wth elte chaotc local search, called DEECL, whch uses a chaotc search strategy based on the heurstc nformaton

7 Computatonal Intellgence and Neuroscence f1 f f f f12 DE DEwEC DEwPA DEECL DE DEwEC f f f13 DEwPA DEECL Fgure 1: Evoluton of the average functon error values derved from DE, DEwEC, DEwPA, and DEECL versus the number of.

8 8 Computatonal Intellgence and Neuroscence f1 f f f f12 DE jde ODE DECLS DEECL DE jde ODE f f f13 DECLS DEECL Fgure 2: Evoluton of the average functon error values derved from DE, jde, ODE, DECLS, and DEECL versus the number of.

9 Computatonal Intellgence and Neuroscence 9 Table 3: Expermental results of DE, jde, ODE, DECLS, and DEECL over 3 ndependent runs for the 13 test functons. Functon DE jde ODE DECLS DEECL Mean ± Std Dev Mean ± Std Dev Mean ± Std Dev Mean ± Std Dev Mean ± Std Dev f1 2.23E 16 ± 2.5E E 31 ± 1.82E E 29 ± 3.92E E 31 ± 2.22E E 38 ± 6.6E 38 f2 2.86E 8 ± 1.26E E 19 ± 3.7E E 9 ± 2.61E E 18 ± 9.36E E 22 ± 1.21E 22 f3 1.88E 1 ± 6.12E E 2 ± 6.45E E 1 ± 1.17E E 4 ± 5.78E E 2 ± 3.44E 2 f4 1.7E 1 ± 2.13E E 4 ± 1.23E E 7 ± 3.43E E 5 ± 2.E 5 4.6E 5 ± 3.5E 5 f5 1.39E+ 1 ± 8.74E E+ 1 ± 5.47E E+ 1 ± 1.28E+ 5.5E 5 ± 1.45E E+ 1 ± 2.21E+ 1 f6.e + ±.E +.E + ±.E +.E + ±.E +.E + ±.E +.E + ±.E + f7 8.82E 3 ± 2.61E E 3 ± 1.45E E 3 ± 6.21E E 3 ± 2.36E E 3 ± 6.52E 4 f8 7.31E+ 3 ± 3.75E E 2 ± 1.82E E+ 3 ± 2.36E E 2 ± 1.82E E 2 ± 1.19E 12 f9 1.77E+ 2 ± 1.1E+ 1+.E + ±.E E+ 1 ± 2.21E+ 1+.E + ±.E +.E + ±.E + f1 5.93E 9 ± 3.1E E 15 ± 1.74E E 15 ± 1.74E E 15 ± 1.63E E 15 ±.E + f E 16 ± 1.16E 15+.E + ±.E +.E + ±.E +.E + ±.E +.E + ±.E + f12 2.2E 17 ± 1.81E E 32 ± 8.15E E 29 ± 2.32E E 32 ± 1.55E E 32 ± 2.74E 48 f E 17 ± 3.59E E 31 ± 2.93E E 29 ± 3.7E E 32 ± 2.37E E 32 ± 3.7E

10 1 Computatonal Intellgence and Neuroscence Table 4: Average rankngs of the fve algorthms for the 13 test functons acheved by Fredman test. Algorthm Rankng DEECL 2.4 DECLS 2.27 jde 2.65 ODE 3.5 DE 4.54 from the elte ndvduals to promote the explotaton power and employs a smple and effectve parameter adaptaton mechansm to enhance the robustness. In the experments, we use 13 classcal test functons that are wdely used n the evolutonary computaton communty to evaluate the performance of DEECL. The expermental results show that DEECL can outperform the conventonal DE, jde, ODE, and DECLSonthemajortyofthetestfunctons. In the future, we wll apply DEECL to handle more complex optmzaton problems, such as hgh-dmensonal optmzaton problems and multobjectve optmzaton problems. Conflct of Interests The authors declare that there s no conflct of nterests regardng the publcaton of ths paper. 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