Utilizing cumulative population distribution information in differential evolution

Transcription

1 Utlzng cumulatve populaton dstrbuton nformaton n dfferental evoluton Yong Wang a,b, Zh-Zhong Lu a, Janbn L c, Han-Xong L d, e, Gary G. Yen f a School of Informaton Scence and Engneerng, Central South Unversty, Changsha , PR Chna b School of Computer Scence and Informatcs, De Montfort Unversty, Lecester LE1 9BH, UK c Insttute of Informaton Securty and Bg Data, Central South Unversty, Changsha , Chna d Department of Systems Engneerng and Engneerng Management, Cty Unversty of Hong Kong, Hong Kong e State Key Laboratory of Hgh Performance Complex Manufacturng, Central South Unversty, Changsha , PR Chna f School of Electrcal and Computer Engneerng, Oklahoma State Unversty, Stllwater OK 74078, USA 1 Abstract: Dfferental evoluton (DE) s one of the most popular paradgms of evolutonary algorthms. In general, DE does not explot dstrbuton nformaton provded by the populaton and, as a result, ts search performance s lmted. In ths paper, cumulatve populaton dstrbuton nformaton of DE has been utlzed to establsh an Egen coordnate system by makng use of covarance matrx adaptaton. The crossover operator of DE mplemented n the Egen coordnate system has the capablty to dentfy the features of the ftness landscape. Furthermore, we propose a cumulatve populaton dstrbuton nformaton based DE framework called CPI-DE. In CPI-DE, for each target vector, two tral vectors are generated based on both the orgnal coordnate system and the Egen coordnate system. Then, the target vector s compared wth these two tral vectors and the best one wll survve nto the next generaton. CPI-DE has been appled to two classc versons of DE and three state-of-the-art varants of DE for solvng two sets of benchmark test functons, namely, 28 test functons wth 30 and 50 dmensons at the 2013 IEEE Congress on Evolutonary Computaton, and 30 test functons wth 30 and 50 dmensons at the 2014 IEEE Congress on Evolutonary Computaton. The expermental results suggest that CPI-DE s an effectve framework to enhance the performance of DE. Keywords: Cumulatve populaton dstrbuton nformaton, dfferental evoluton, Egen coordnate system, evolutonary algorthms 1. Introducton Dfferental Evoluton (DE), proposed by Storn and Prce [1] [2] n 1995, s a very popular evolutonary algorthm (EA) paradgm. Durng the past two decades, DE has attracted a lot of attenton and has been successfully appled to solve a varety of numercal and real-world optmzaton problems [3] [4] [5]. The remarkable advantages of DE are ts smple structure and ease of mplementaton. In DE, each ndvdual n the populaton s called a target vector. DE contans three basc operators: mutaton, crossover and selecton. Durng the evoluton, DE generates a tral vector for each target vector through the mutant and crossover operators. Afterward, the tral vector competes wth ts target vector for survval accordng to ther ftness. DE also nvolves three control parameters: the populaton sze, the scalng factor, and the crossover control parameter. The performance of DE s dependent manly on these three operators and three control Emal: ywang@csu.edu.cn (Y. Wang) parameters. In order to further mprove the performance of DE, a lot of DE varants have been desgned, such as JADE [6], jde [7], SaDE [8], EPSDE [9], CoDE [10], and so on. DE s a populaton-based optmzaton algorthm; however, populaton dstrbuton nformaton has not yet been wdely utlzed n the DE communty, whch makes DE neffcent especally when solvng some optmzaton problems wth complex characterstcs. Very recently, two attempts have been made along ths lne [11] [12]. However, the methods proposed n [11] and [12] only utlze the dstrbuton nformaton from a sngle populaton of one generaton, and the cumulatve dstrbuton nformaton of the populaton over the course of evoluton has been gnored. Moreover, these methods ntroduce some extra parameters. Therefore, new nsghts nto the usage of the populaton dstrbuton nformaton n DE are qute necessary. In 2001, Hansen and Ostermeer [13] proposed the well-known covarance matrx adaptaton evoluton strategy, called CMA-ES. CMA-ES generates offsprng by samplng a multvarate normal dstrbuton, whch ncludes three man elements: mean vector of the search dstrbuton, covarance matrx, and step-sze. Indeed, covarance matrx reflects the populaton dstrbuton nformaton to a certan degree [12]. In CMA-ES, the covarance matrx s self-adaptvely updated accordng to the nformaton from the prevous and current generatons. In ths paper, we make use of the cumulatve dstrbuton nformaton of the populaton to establsh an Egen coordnate system n DE, by consderng CMA as an effectve tool. Furthermore, we suggest a cumulatve populaton dstrbuton nformaton based DE framework called CPI-DE. In CPI-DE, for each target vector, the crossover operator of DE s mplemented n both the orgnal coordnate system and the Egen coordnate system and, as a result, two tral vectors are generated. Subsequently, the target vector s compared wth these two tral vectors and the best one wll enter the next populaton. CPI-DE s appled to two classc DE versons as well as three state-of-the-art DE varants. Extensve experments across two benchmark test sets from the 2013 IEEE Congress on Evolutonary Computaton (IEEE CEC2013) [14] and the 2014 IEEE Congress on Evolutonary Computaton (IEEE CEC2014) [15] have been mplemented to verfy the effectveness of CPI-DE. The man contrbutons of ths paper can be summarzed as

2 2 follows: Due to the fact that sngle populaton fals to contan enough nformaton to estmate the covarance matrx relably, ths paper updates the covarance matrx n DE by an adaptaton procedure, whch makes use of the cumulatve dstrbuton nformaton of the populaton. CPI-DE provdes a smple yet effcent synergy of two knds of crossover: the crossover n the Egen coordnate system and the crossover n the orgnal coordnate system. The former ams at dentfyng the propertes of the ftness landscape and mprovng the effcency and effectveness of DE by producng the offsprng toward the promsng drectons. In addton, the purpose of the latter s to mantan the superorty of the orgnal DE. Moreover, no extra parameters are requred n CPI-DE. Our expermental studes have shown that CPI-DE s capable of enhancng the performance of several classc DE versons and advanced DE varants. The rest of ths paper s organzed as follows. Secton 2 descrbes the basc procedure of DE. Secton 3 brefly revews the recent developments of DE n the last fve years. The proposed CPI-DE s presented n Secton 4. The expermental results and the performance comparson are gven n Secton 5. Fnally, Secton 6 concludes ths paper. 2. Dfferental evoluton (DE) Smlar to other EA paradgms, DE starts wth a populaton ( g) ( g) ( g) ( g) of ndvduals,.e., P = { x = ( x,1,..., xd, ), = 1,..., }, where g s the generaton number, D s the dmenson of the ( g decson space, and s the populaton sze. In P ), each ndvdual s also called a target vector. At g=0, the jth decson varable of the th target vector s ntalzed as follows: (0) x, j = Lj + rand(0,1)*( U j Lj), = 1,...,, j = 1,..., D (1) where rand(0,1) represents a unformly dstrbuted random varable between 0 and 1, and L j and U j are the lower and upper bounds of the jth decson varable, respectvely. After the ntalzaton, DE repeatedly mplements three basc operators,.e., mutaton, crossover, and selecton, to search for the optmal soluton of an optmzaton problem. Note that n DE, a combnaton of the mutaton operator and the crossover operator s called a tral vector generaton strategy Mutaton Operator At each generaton, a mutant vector s generated for each target vector by the mutaton operator. The followng are four commonly used mutaton operators n the DE communty: DE/rand/1 ( g) ( g) ( g) ( g) v = x r1 + F*( x r2 x r3 ) (2) DE/rand/2 ( g) ( g) ( g) ( g) ( g) ( g) v = x + F*( x x ) + F*( x x ) (3) r1 r2 r3 r4 r5 DE/current-to-best/1 ( g) ( g) ( g) ( g) ( g) ( g) v = x + F*( x best x ) + F*( x r1 x r 2 ) DE/current-to-rand/1 ( g) ( g) ( g) ( g) ( g) ( g) v = x + F*( x x ) + F*( x x ) r1 r2 r3 In the above equatons, the ndces r 1, r, r, r, and r are dstnct ntegers randomly selected from [1,..., ] and are also dfferent from, x best s the best target vector n the current populaton, F s the scalng factor, and v s the mutant vector Crossover Operator After mutaton, the crossover operaton s appled to each par of x and v ( g) ( g) to generate a tral vector u = ( u,1,..., u ). The bnomal crossover can be expressed as follows: D, ( ),, f (0,1) rand g v rand CR or j = j j u, j =, j = 1,, D (6) x otherwse, j, where j rand s a random nteger between 1 and D, rand(0,1) s a unformly dstrbuted random number between 0 and 1, and CR s the crossover control parameter. The condton j = j rand makes the tral vector dfferent from the correspondng target vector by at least one dmenson Selecton Operator The selecton operator of DE adopts a one-to-one competton between the target vector and ts tral vector. For a mnmzaton problem, f the objectve functon value of the tral vector s less than or equal to that of the target vector, then the tral vector wll survve nto the next generaton; otherwse, the target vector wll enter the next generaton: ( g) ( g) ( g) ( g + 1) u, f f( u ) f( x ) x = (7) x, otherwse where f () s the objectve functon. It s evdent that, F, and CR are three man control parameters of DE. The settng of s related to the dmenson of the decson space. In general, the hgher the dmenson of the decson space, the larger the value of. In addton, F s always chosen from the range [0.4, 1.0] and CR s usually set to a value close to 0.1 or 1.0 dependng on the characterstcs of an optmzaton problem [10]. 3. The related work Recent two decades have wtnessed sgnfcant progress n the developments of DE. In 2011, Das and Suganthan [16] presented a comprehensve survey on DE, ncludng the basc concepts and major varants of DE, as well as the applcatons and theoretcal studes of DE. Next, we wll brefly ntroduce the recent developments of DE n the last fve years Introducton of New Tral Vector Generaton Strateges Zhou et al. [17] proposed an ntersect mutaton operator, n whch the ndvduals n the populaton are dvded nto two (4) (5)

3 3 parts accordng to ther ftness: the worse part and the better part. Zhang and Yuen [18] presented a drectonal mutaton operator, n whch a dfferental vector pool s establshed once the best-so-far ftness of the populaton has been mproved at one generaton. Subsequently, ths dfferental vector pool s utlzed to create the tral vectors n the next generaton. Hu et al. [19] ntroduced a subspace clusterng mutaton operator whch selects an elte ndvdual as the base vector and employs the dfference between two randomly generated boundary ndvduals as the perturbaton vector. Gong and Ca [20] proposed the rankng-based mutaton operators. In ths knd of mutaton operators, the ndvduals are selected based on ther rankngs, whch means the ndvduals wth better rankng have more opportunty to be selected. Ca and Wang [21] ncorporated the neghborhood and drecton nformaton nto the mutaton operator. They also proposed two strateges. In the frst strategy the neghborhood nformaton s used to select the base and dfferental vectors, and n the second strategy the drecton nformaton s ncorporated nto the mutaton operator. Wang et al. [22] proposed a multobjectve sortng-based mutaton operator. In ths operator, the ftness and dversty nformaton are smultaneously consdered as two objectve functons n DE, wth the am of selectng those ndvduals wth both hgh ftness and better dversty for mutaton. Guo et al. [23] proposed a successful-parent-selectng framework to select ndvduals for mutaton and crossover. In ths framework, successful solutons are stored nto an archve and some ndvduals n the archve are chosen to mplement mutaton and crossover when stagnaton s happenng. Yu et al. [24] desgned an adaptve greedy mutaton strategy, n whch one of the vectors n mutaton s randomly selected from the top k ndvduals n the current populaton. Moreover, n order to adjust the greedness degree, the parameter k s set by an adaptve scheme. Wang et al. [25] proposed Gaussan bare-bones DE, the core technque of whch s a Gaussan mutaton operator. Zhao and Suganthan [26] emprcally nvestgated the performance of the exponental crossover operator of DE, and suggested a lnearly scalable exponental crossover operator. In classc DE, the tral vector generated by the crossover operator s usually a vertex of the hyper-rectangle defned by the mutant and target vectors. In order to allevate ths drawback, Wang et al. [27] exploted orthogonal crossover to probe the hyper-rectangle defned by the mutant and target vectors, thus enhancng the search ablty of DE Adaptng the Control Parameter Settngs Gong et al. [28] analyzed the behavor of the crossover operator and proposed a crossover rate repar technque for adaptve DE varants. Tanabe and Fukunaga [29] suggested a success-hstory based parameter adaptaton scheme to revse the settngs of both the scalng factor F and the crossover control parameter CR n JADE [6]. Under the framework n [29], Tanabe and Fukunaga [30] further ncorporated lnear populaton sze reducton. By usng the correlaton coeffcent, Takahama and Saka [31] mproved the settngs of F and CR n JADE [6]. In [32], F and CR are determned by a mechansm based on exponentally weghtng movng average. Sarker et al. [33] defned three sets for F, CR, and the populaton sze, respectvely. Durng the evoluton, dynamc selecton s executed for these three control parameters of DE. He and Yang [34] controlled F by takng advantage of Lévy dstrbuton. Yu and Zhang [35] ntroduced an adaptve parameter control scheme based on optmzaton state estmaton. Zhu et al. [36] proposed an adaptve populaton tunng scheme to dynamcally adjust accordng to a status montor. Zamuda et al. [37] proposed a populaton reducton DE wth multple mutaton operators, n whch s reduced wth the ncrease of the generaton number. In adaptve or self-adaptve DE varants, t s always expected that the crossover control parameter CR whch nduces a larger amount of replacements can generate solutons of hgher qualty. Segura et al. [38] carred out an emprcal nvestgaton on ths ssue by analyzng the correlaton between the qualty of the obtaned solutons and the probablty of replacement nduced by dfferent CR values. In [39], Segura et al. studed on the effectveness of ncorporatng feedback nformaton from the search process to gude the adaptaton of F, and ponted out that further research s requred to successfully adapt F Integratng Multple Tral Vector Generaton Strateges wth Multply Control Parameter Settngs n a Sngle Populaton Tang et al. [40] made use of the ftness nformaton to tune the control parameters and choose the mutaton operators. Moreover, they desgned an ndvdual-dependent parameter settng and an ndvdual-dependent mutaton operator whch are assocated wth each ndvdual n the populaton. Y et al. [41] proposed a novel DE, called HSDE. HSDE combnes two mutaton operators to balance the exploraton and explotaton abltes of DE and ntegrates them wth a self-adaptve parameter control strategy ntroduced n [7]. Fan and Yan [42] presented a self-adaptve DE n whch each ndvdual has ts own F, CR, and mutaton operator. Moreover, fve mutaton operators have been adopted and F and CR are automatcally adjusted. Very recently, Fan and Yan [43] proposed zonng evoluton of control parameters, n whch sutable combnatons of F and CR can be generated by zonng evoluton. Moreover, adaptve mutaton operator s also employed. In [44], two mutaton operators are combned by a lnear ncrement rule. In addton, F s generated accordng to two Gaussan dstrbutons and CR s produced by two unform dstrbutons based on the success rato. Takahama and Saka [45] proposed a novel method to detect the modalty of landscape beng searched,.e., unmodalty or multmodalty. Afterward, a mutaton operator s selected accordng to the modalty of landscape. Moreover, F and CR are tuned dynamcally n [45]. Zhou et al. [46] dynamcally dvded the populaton nto three groups by consderng the poston and ftness nformaton of each ndvdual. Moreover, these three groups are assgned wth dfferent roles and equpped wth dfferent mutaton operators and control

4 4 parameter values Mult-populated DE Bujok and Tvrdk [47] developed a parallel DE, whch contans a parallel mgraton model employng varous adaptve DE varants. Huo et al. [48] proposed a mult-swarm DE wth swarm sharng management. In ths method, each swarm explores the search space ndependently and the sharng management s appled to adjust swarm sze. Kushda et al. [49] desgned an sland-based DE. Ths method allocates dfferent control parameters to each sland, performs mgraton among slands, and dynamcally vares subpopulaton sze by ndvdual transfer. Zhou et al. [50] proposed a two-layer herarchcal DE, n whch the populaton n the top layer conssts of the best ndvduals obtaned from the several populatons n the bottom layer. In [51], an sland based dstrbuted DE framework has been proposed. Cheng et al. [52] presented a dstrbuted DE wth multcultural mgraton, whch makes use of two mgraton selecton approaches to mantan the dversty n the subpopulatons and an affnty based replacement strategy to control the dversty among the ndvduals. Peng and Wu [53] presented a heterozygous DE. In ths method, the populaton s frstly dvded nto four sub-swarms, and then each sub-swarm corresponds to a parameter adjustment scheme Combnng DE wth Other Technques Durng the past fve years, combnng DE wth other technques has attracted consderable attenton. For example, DE has been combned wth opposton-based learnng [54] [55], restart technque [56], adaptve dsturbance mechansm [57], and Taguch local search [53]. Recently, Yang et al. [58] proposed an auto-enhanced populaton dversty mechansm, whch frstly dentfes whether the populaton s convergng or stagnant, and then redversfes the populaton at the dmensonal level. L et al. [59] presented a novel dea,.e., the cumulatvely learned evoluton path. In ths method, after a tral vector has been created, an addtonal dfferental vector s added to ths tral vector based on the evoluton path nformaton. In addton, DE has also been combned wth surrogate models to deal wth computatonally expensve global numercal optmzaton problems n [60], [61], and [62] Hybrdzed DE At present, DE has been hybrdzed wth a lot of meta-heurstc methods, such as artfcal bee colony algorthm [63], varable neghborhood search [64], smulated annealng [65], estmaton of dstrbuton algorthm [66], genetc programmng [67], and Cuckoo search [68]. Moreover, several DE varants have been developed under the memetc framework [69] [70]. Our work n ths paper falls n the frst category,.e., ntroducng new crossover operator by utlzng cumulatve populaton dstrbuton nformaton n DE. 4. Proposed approach 4.1. Motvaton Based on the above ntroducton, t s clear that populaton dstrbuton nformaton has seldom been nvolved n the current state-of-the-art DE. Very recently, Guo and Yang [11] and Wang et al. [12] made the frst attempt to explot the populaton dstrbuton nformaton n DE. The methods proposed n [11] and [12] share some smlar deas. More specfcally, these two methods frstly compute the covarance matrx of the populaton. Subsequently, the Egenvectors obtaned from the Egen decomposton are used to establsh an Egen coordnate system. Fnally, the crossover operator of DE s mplemented n the Egen coordnate system to generate the tral vectors. Compared wth the crossover operator n the orgnal coordnate system, the crossover operator n the Egen coordnate system makes the recombnaton process of DE rotatonally nvarant [11]. However, these two methods only make use of the dstrbuton nformaton of the current populaton and, consequently, the estmaton of the covarance matrx s unrelable due to nsuffcent nformaton. Besdes, some problem-dependent parameters have been ntroduced, such as the number of ndvduals adopted to compute the covarance matrx and the frequency of the crossover operator beng executed n the Egen coordnate system. The above dscusson motvates us to carry out an n-depth nvestgaton on the utlzaton of populaton dstrbuton nformaton n DE. Actually, the methods n [11] and [12], and ths paper are all nspred by CMA-ES [13]. However, unlke [11] and [12], ths paper adapts the covarance matrx accordng to the nformaton of the prevous and current generatons to ncrease the probablty of producng successful search dstrbuton for the subsequent evoluton, whch results n a more reasonable search behavor. Besdes, ths paper proposes a DE framework called CPI-DE, whch elmnates the problem-dependent parameters n [11] and [12]. In CMA-ES, the ndvduals n the populaton are generated by the followng equaton [13]: ( g+ 1) ( g) ( g) ( g) x = m + σ Ν ( 0, C ), = 1,, λ (8) where m s the mean vector of the search dstrbuton, σ s the step-sze, C s the covarance matrx, and Ν(, 0 C ) s a multvarate normal dstrbuton wth zero ( g mean and covarance matrx C ). The man am of CMA-ES ( g s to calculate m + 1) ( g, σ + 1) ( g + 1), and C for the next generaton (g+1). Moreover, the step-sze and the covarance matrx n CMA-ES are self-adaptvely updated as the search goes on. It s noteworthy that ES and DE have dfferent search patterns. In ES, the offsprng s produced accordng to a predefned probablstc dstrbuton. However, n DE the offsprng s generated by the arthmetc operaton of the base and dfferental vectors and by the nformaton exchange between the target vector and the mutant vector. Therefore, t s unnecessary to update the step-sze for DE and we only focus on the adaptaton of the covarance matrx (.e., CMA).

5 Rank--Update of the Covarance Matrx n DE In CMA-ES, two updatng strateges have been ntroduced to adapt the covarance matrx: rank-μ-update and rankone-update. Ths paper only employs rank-μ-update based on the followng consderatons: 1) rank-μ-update plays a prmary role when the populaton s large. Compared wth ES, DE usually adopts a relatvely larger populaton sze; and 2) rank-one-update explots correlatons between consecutve generatons and constructs an evoluton path to update the covarance matrx, whch nevtably adds computatonal complexty. By elmnatng the rank-one-update, CMA n DE becomes smpler. In rank-μ-update, μ represents the populaton sze. Snce the populaton sze s equal to n DE, rank-μ-update n CMA-ES s called rank--update n ths paper. The covarance matrx C s ntalzed as C (0) = I, D D where I R s a unty matrx. In addton, the mean vector of the search dstrbuton m s ntalzed as a randomly generated pont n the search space. At generaton ( g 1) (g+1), m + s updated as follows: ( g+ 1) ( g+ 1) m = wx :2* (9) = 1 ( g 1) where x + :2* s the th best ndvdual n the offsprng populaton (note that n CPI-DE, the offsprng populaton ( g+ 1) ( g+ 1) conssts of 2* ndvduals),.e., f( x1:2* ) f( x2:2* ) ( g 1) f( x + :2* ), w s the th postve weght coeffcent, and ( g 1) w = 1. It s evdent that m + s the weghted average of = 1 the best ndvduals n the offsprng populaton. In order to ntroduce a search bas toward the promsng area, the value of the weght coeffcent depends on the qualty of the ndvdual,.e., w1 w2 w > 0. Accordng to the suggeston n [13], w s set as follows: w w =, 1,..., (10) w ' j = 1 ' j and ' w = ln( + 0.5) ln( ), 1,..., (11) In order to update the covarance matrx, frstly an estmator of C s computed: ( g 1) ( g 1) ( g) ( g 1) ( g) T C + = w ( x + :2* m )( x + :2* m ) (12) = 1 ( g + 1) Afterward, C at generaton (g+1) s updated as follows: 2 ( 1) ( ) ( ) 1 ( 1) C g+ = (1 c ) g g C + c σ C + (13) 2 where c mn(1, / D ) s the learnng rate and = eff = ( w ) s the varance effectve selecton mass. eff It s clear from (13) that the nformaton from both the prevous and current generatons are used to update the covarance matrx, whch means that the cumulatve dstrbuton nformaton of the populaton has been utlzed to adapt the search dstrbuton. It s necessary to note that (13) ncludes the step-sze σ at generaton g. As ponted out prevously, t does not make sense to adjust the step-sze for DE. Therefore, for the sake of smplcty, σ s set to 1 n ths paper. Indeed, σ = 1 resembles the covarance matrx from the estmaton of multvarate normal algorthm [71]. Moreover, σ = 1 mples that the covarance matrx at each generaton s of equal mportance [72] Crossover n the Egen Coordnate System The man dea of the crossover n the Egen coordnate system s the followng [12]. Frstly, by mplementng the Egen decomposton on the covarance matrx, an orthonormal bass of Egenvectors can be obtaned, whch forms an Egen coordnate system. Then the target vector and ts mutant vector are transformed nto the Egen coordnate system. Afterward, the crossover of DE s executed on the transformed target and mutant vectors, and thus, a tral vector s produced n the Egen coordnate system. Fnally, ths tral vector s transformed back nto the orgnal coordnate system. Next, we wll gve the detals of the above procedure. The Egen decomposton of the covarance matrx C at generaton g can be descrbed as: 2 T C = B D B (14) where each column of the orthogonal matrx B s the ( g correspondng Egenvector of C ), and each dagonal element of the dagonal matrx D s the correspondng ( g Egenvalue of ) C. By dong ths, B ncludes an orthonormal bass of Egenvectors and forms an Egen coordnate system. T Note that B has the capablty to rotate a vector nto the Egen coordnate system. After the rotaton, the elements of the resultng vector are related to the projectons onto the Egenvectors [72]. Based on the above property, the target vector x and ts mutant vector v are transformed nto the Egen coordnate system: T '( g) ( g) ( g) x = B x (15) T '( g) ( g) ( g) v = B v (16) Afterward, the crossover operator s mplemented n the Egen coordnate system and a tral vector ' ( ' ( g u = u ),, ) s produced: ' ( ) u g D, ' ' ( ),, f (0,1) g v j rand CR or j = jrand u, j =, j = 1,, D (17) ' x, j, otherwse Snce B s able to rotate the result back nto the orgnal coordnate system [72], the fnal tral vector u n the orgnal coordnate system can be obtaned by the followng transformaton: ( g) ( g) '( g) u = B u (18) 4.4. CPI-DE The Pseudocode of CPI-DE s shown n Fg. 1. As,1

6 6 1: g = 0; // g s the generaton number 2: (0) Intalze m and C (0) ; 3: Generate an ntal populaton (0) (0) (0) P = { x 1,, x } by unformly and randomly samplng ndvduals n the decson space; (0) (0) 4: Evaluate the objectve functon values of the ntal populaton f( x1 ),, f( x ); 5: FEs = ; // FEs records the number of ftness evaluatons 6: Whle FEs < MaxFEs do // MaxFEs represents the maxmum number of ftness evaluatons 7: ( ) ( ) O = Ø, E g = Ø, and P ( 1) = Ø; 8: For = 1: do 9: Implement the mutaton and crossover operators of DE n the orgnal coordnate system to generate a tral vector u _1 for the target vector ( g x ) ( ) ( ) ( ), and O g = O g u g ; _1 10: Implement the mutaton and crossover operators of DE n the Egen coordnate system to generate another tral vector u _2 for the target vector x ( ) ( ) ( ) accordng to (14)-(18), and E g = E g u g ; _2 11: Evaluate the objectve functon values of u _1 and u ; _2 12: Implement the selecton operator of DE to select the best one from ( g x ), u, _1 and u, ( g _2 denoted as x + 1) ; 13: ( g+ 1) ( g+ 1) ( g+ P = P x 1) ; 14: End For 15: FEs = FEs + 2* ; 16: ( g) ( g Select the best ndvduals from O E ) ( g+ 1) ( g+ 1), denoted as x1:2*,, x :2*, and use these ndvduals and m to compute ( g+ 1) ( g 1) C accordng to (12) and (13). Subsequently, use these ndvduals to compute m + accordng to (9); 17: g = g+ 1; 18: End Whle Fg. 1. Pseudocode of CPI-DE. x mutaton crossover n the orgnal coordnate system u _1 selecton ( g 1) x + mutaton crossover n the Egen coordnate system u _2 Fg. 2. The mutaton, crossover, and selecton of CPI-DE. P O select the best ndvduals: ( g+ 1) ( g+ 1) x,, x 1:2* :2* updatec ( g+ 1) E (a) rank--update n CPI-DE λ ndvduals select the best μ ndvduals updatec ( g+ 1) (b) rank-μ-update n CMA-ES Fg. 3. The relatonshp between rank--update n CPI-DE and rank-μ-update n CMA-ES. ntroduced prevously, n CPI-DE the cumulatve dstrbuton nformaton of the populaton s used to update the covarance matrx by rank--update, and then the Egen coordnate system s establshed by the Egen decomposton of the covarance matrx. At each generaton, for each target vector, two tral vectors are generated by mplementng the mutaton and crossover operators n both the orgnal coordnate system and the Egen coordnate system. Thereafter, the best

7 7 Table 1 Expermental results of DE/rand/1/bn, CPI-DE/rand/1/bn, DE/current-to-best/1/bn, and CPI-DE/current-to-best/1/bn over 51 ndependent runs on 28 test functons wth 30D from IEEE CEC2013 usng 300,000 FEs. Test Functons wth 30D from IEEE CEC2013 Unmodal Functons Basc Multmodal Functons Composton Functons DE/rand/1/bn CPI-DE/rand/1/bn DE/current-to-best/1/bn CPI-DE/current-tobest/1/bn CEC E+00±0.00E E+00±0.00E E+00±0.00E E+00±0.00E+00 CEC E+08±2.54E E-03±1.21E E+07±1.78E E+00±0.00E+00 CEC E+09±5.11E E+04±6.87E E+06±4.08E E+02±5.91E+02 CEC E+04±5.65E E+00±0.00E E+04±3.50E E+00±0.00E+00 CEC E-06±8.94E E+00±0.00E E+00±0.00E E+00±0.00E+00 CEC E+01±1.28E E+00±6.27E E+01±4.69E E+00±8.59E+00 CEC E+01±7.09E E+01±4.90E E+00±4.61E E+00±4.34E+00 CEC E+01±4.77E E+01±4.31E E+01±5.56E E+01±4.17E-02 CEC E+01±1.13E E+01±1.25E E+01±3.04E E+01±1.36E+01 CEC E+02±5.34E E-04±1.93E E-02±1.00E E-03±3.84E-03 CEC E+01±9.45E E+02±8.88E E+00±2.97E E+00±2.82E+00 CEC E+02±1.22E E+02±1.05E E+02±1.16E E+02±9.83E+00 CEC E+02±1.49E E+02±1.27E E+02±1.31E E+02±1.19E+01 CEC E+03±1.77E E+03±2.18E E+03±4.99E E+03±3.89E+02 CEC E+03±2.97E E+03±2.17E E+03±2.62E E+03±2.92E+02 CEC E+00±2.91E E+00±2.52E E+00±2.97E E+00±2.94E-01 CEC E+02±1.04E E+02±1.16E E+02±8.73E E+02±1.30E+01 CEC E+02±1.20E E+02±1.21E E+02±1.15E E+02±9.46E+00 CEC E+01±1.22E E+01±1.03E E+01±1.21E E+01±1.17E+00 CEC E+01±2.13E E+01±2.67E E+01±3.11E E+01±3.09E-01 CEC E+02±5.01E E+02±8.98E E+02±7.02E E+02±8.67E+01 CEC E+03±3.91E E+03±3.53E E+03±1.06E E+03±6.29E+02 CEC E+03±2.53E E+03±2.76E E+03±2.89E E+03±3.42E+02 CEC E+02±6.56E E+02±3.67E E+02±2.01E E+02±1.29E+01 CEC E+02±4.13E E+02±2.51E E+02±8.78E E+02±3.99E+00 CEC E+02±1.94E E+02±2.59E E+02±7.12E E+02±5.63E+01 CEC E+03±2.89E E+02±2.91E E+02±2.96E E+02±1.02E+02 CEC E+02±8.82E E+02±1.49E E+02±1.44E E+02±1.36E one among the target vector and two correspondng tral vectors wll enter the next generaton. The above mutaton, crossover, and selecton are shown n Fg. 2. Meanwhle, the covarance matrx and the mean vector of the search dstrbuton are updated at the end of each generaton accordngly. In ths paper, the advantages of mplementng the crossover operator n the Egen coordnate system are twofold: At the early stage of evoluton, the populaton mantans hgh dversty. Under ths condton, by utlzng the cumulatve dstrbuton nformaton of the populaton, the covarance matrx has the capabltes to quckly provde reasonable search dstrbuton and to contnuously gude the populaton toward the promsng areas. At the mddle and later stages of evoluton, the dversty of populaton may gradually decrease and the search may concentrate on a relatvely small area. In ths case, by utlzng the covarance matrx, the modalty of the ftness landscape can be dentfed and the crossover n the Egen coordnate system s able to strengthen the explotaton ablty of DE n the area surrounded by the populaton. On the other hand, n order to keep the superorty and the search behavor of the orgnal DE, the crossover operator s also executed n the orgnal coordnate system. Based on the above dscusson, CPI-DE not only acheves a tradeoff between the exploraton and explotaton n DE, but also provdes a smple yet effcent framework to ncorporate the cumulatve dstrbuton nformaton of the populaton nto DE. Moreover, CPI-DE does not ntroduce ts own parameter. Remark 1: In rank-μ-update of CMA-ES, μ parents n the populaton are used to generate λ offsprng by a multvarate normal dstrbuton. Subsequently, the best μ ndvduals are selected from these λ offsprng and used to update the covarance matrx. In rank--update of CPI-DE, ndvduals n the populaton are used to create 2* offsprng by mplementng the mutaton and crossover operators n both the orgnal coordnate system and the Egen coordnate system. Afterward, the best ndvduals are chosen from these 2* offsprng and used to update the covarance

8 8 Table 2 Expermental results of DE/rand/1/bn, CPI-DE/rand/1/bn, DE/current-to-best/1/bn, and CPI-DE/current-to-best/1/bn over 51 ndependent runs on 28 test functons wth 50D from IEEE CEC2013 usng 500,000 FEs. Test Functons wth 50D from IEEE CEC2013 Unmodal Functons Basc Multmodal Functons Composton Functons DE/rand/1/bn CPI-DE/rand/1/bn DE/current-to-best/1/bn CPI-DE/current-tobest/1/bn CEC E+01±1.71E E-01±1.60E E+00±0.00E E+00±0.00E+00 CEC E+08±7.93E E+05±1.48E E+08±6.31E E+02±1.29E+03 CEC E+10±4.67E E+09±2.01E E+08±6.14E E+06±8.78E+06 CEC E+04±7.21E E+01±1.66E E+04±6.36E E+00±0.00E+00 CEC E+01±3.08E E+00±2.12E E+00±0.00E E+00±0.00E+00 CEC E+02±2.15E E+01±2.85E E+01±7.63E E+01±4.43E-12 CEC E+02±8.54E E+01±9.54E E+01±1.85E E+01±8.69E+00 CEC E+01±3.50E E+01±3.01E E+01±3.39E E+01±3.41E-02 CEC E+01±1.64E E+01±9.43E E+01±1.32E E+01±1.88E+00 CEC E+03±2.50E E+00±3.09E E+00±1.92E E-04±2.87E-03 CEC E+02±1.79E E+02±1.81E E+01±2.66E E+01±2.15E+01 CEC E+02±1.92E E+02±1.48E E+02±1.65E E+02±1.61E+01 CEC E+02±2.25E E+02±1.62E E+02±1.51E E+02±1.45E+01 CEC E+04±2.29E E+04±3.70E E+03±3.39E E+04±3.41E+02 CEC E+04±3.79E E+04±3.79E E+04±3.59E E+04±4.03E+02 CEC E+00±3.12E E+00±3.64E E+00±2.67E E+00±2.66E-01 CEC E+02±1.86E E+02±1.85E E+02±1.45E E+02±1.42E+01 CEC E+02±1.98E E+02±1.55E E+02±1.19E E+02±1.60E+01 CEC E+02±2.35E E+01±1.66E E+01±1.48E E+01±1.12E+00 CEC E+01±2.09E E+01±2.17E E+01±2.56E E+01±2.22E-01 CEC E+03±5.45E E+02±2.29E E+02±4.38E E+02±4.24E+02 CEC E+04±2.80E E+04±4.16E E+03±6.39E E+03±3.97E+02 CEC E+04±3.28E E+04±3.43E E+04±3.83E E+04±4.36E+02 CEC E+02±4.40E E+02±1.24E E+02±3.74E E+02±1.58E+01 CEC E+02±4.42E E+02±7.06E E+02±3.61E E+02±2.03E+01 CEC E+02±8.47E E+02±1.35E E+02±1.04E E+02±9.32E+01 CEC E+03±3.59E E+03±6.56E E+03±2.16E E+02±3.95E+02 CEC E+02±8.86E E+02±7.82E E+02±7.54E E+02±5.84E matrx. Hence, rank--update n CPI-DE s a natural extenson of rank-μ-update n CMA-ES. The relatonshp between rank--update n CPI-DE and rank-μ-update n CMA-ES s shown n Fg Expermental Study In ths paper, two sets of benchmark test functons are employed to demonstrate the effectveness of CPI-DE,.e., 28 test functons wth 30 dmensons (30D) and 50 dmensons (50D) at IEEE CEC2013 [14], and 30 test functons wth 30 dmensons (30D) and 50 dmensons (50D) at IEEE CEC2014 [15]. The 28 test functons n the frst set are denoted as CEC CEC , and the 30 test functons n the second set are denoted as CEC CEC best In our experments, the functon error value ( f( x ) * f( x )) of each run s recorded, where x * s the optmal best soluton and x s the best soluton found at the end of a run. The average and standard devaton of the functon error values n all runs (denoted as Mean Error and Std Dev ) are consdered as two performance metrcs to assess the performance of the algorthms. Moreover, Wlcoxon s rank sum test at a 0.05 sgnfcance level s used to test the statstcal sgnfcance between parwse algorthms. Accordng to the suggestons n [14] and [15], the maxmum number of ftness evaluatons (FEs) MaxFEs was set to 10000* D and the average functon error value smaller than 10-8 was taken as zero. Note that when our framework s appled to a specfed DE algorthm, the name of ths DE algorthm wll be changed by addng four letters CPI-. For example, DE/rand/1/bn under our framework s called CPI-DE/rand/1/bn CPI-DE for Two Classc DE Versons The proposed DE framework,.e., CPI-DE, s frstly appled to two classc versons of DE ntroduced n Secton 2,.e., DE/rand/1/bn and DE/current-to-best/1/bn. DE/rand/1/ bn selects ndvduals for mutaton n a random manner and does not add any search bas; therefore t s an unbased DE.

9 9 Table 3 Expermental results of DE/rand/1/bn, CPI-DE/rand/1/bn, DE/current-to-best/1/bn, and CPI-DE/current-to-best/1/bn over 51 ndependent runs on 30 test functons wth 30D from IEEE CEC2014 usng 300,000 FEs. Test Functons wth 30D from IEEE CEC2014 Unmodal Functons Smple Multmodal Functons Hybrd Functons Composton Functons DE/rand/1/bn CPI-DE/rand/1/bn DE/current-to-best/1/bn CPI-DE/current-tobest/1/bn CEC E+07±1.74E E-05±1.73E E+07±9.06E E+00±0.00E+00 CEC E+02±2.62E E-06±3.12E E+00±0.00E E+00±0.00E+00 CEC E+01±5.01E E+00±0.00E E+00±0.00E E+00±0.00E+00 CEC E+02±1.32E E+00±8.87E E+01±3.42E E+00±1.24E+01 CEC E+01±5.39E E+01±5.71E E+01±6.28E E+01±5.15E-02 CEC E+01±1.16E E+01±7.11E E+00±1.08E E-01±9.08E-01 CEC E-01±1.82E E-06±1.11E E-04±2.00E E-04±2.10E-03 CEC E+01±8.22E E+01±8.87E E+00±2.69E E+00±2.73E+00 CEC E+02±1.28E E+02±1.06E E+02±1.47E E+02±1.26E+01 CEC E+03±2.65E E+03±3.12E E+02±3.50E E+03±8.31E+02 CEC E+03±2.94E E+03±2.35E E+03±4.13E E+03±2.88E+02 CEC E+00±2.75E E+00±2.71E E+00±2.30E E+00±2.84E-01 CEC E-01±6.60E E-01±5.60E E-01±5.33E E-01±5.06E-02 CEC E-01±6.13E E-01±3.40E E-01±1.63E E-01±6.72E-02 CEC E+01±1.33E E+01±8.33E E+01±1.03E E+01±8.25E-01 CEC E+01±2.21E E+01±1.91E E+01±2.44E E+01±2.86E-01 CEC E+06±5.36E E+03±1.81E E+05±2.99E E+02±3.23E+02 CEC E+04±6.31E E+01±6.65E E+02±2.75E E+01±2.03E+01 CEC E+01±1.14E E+00±1.36E E+00±9.95E E+00±8.02E-01 CEC E+02±3.37E E+01±4.23E E+01±8.81E E+01±1.17E+01 CEC E+05±4.06E E+02±1.35E E+03±7.01E E+02±2.40E+02 CEC E+02±5.76E E+02±6.86E E+02±1.00E E+02±1.19E+02 CEC E+02±4.28E E+02±4.36E E+02±4.02E E+02±4.01E-13 CEC E+02±3.60E E+02±1.59E E+02±4.64E E+02±5.87E+00 CEC E+02±2.95E E+02±2.73E E+02±2.73E E+02±2.67E-01 CEC E+02±5.97E E+02±4.79E E+02±5.09E E+02±5.31E-02 CEC E+02±1.29E E+02±4.17E E+02±4.42E E+02±4.44E+01 CEC E+03±2.41E E+02±2.90E E+02±5.24E E+02±5.06E+01 CEC E+04±3.42E E+02±1.19E E+03±8.89E E+02±1.16E+02 CEC E+03±9.06E E+03±1.70E E+03±4.17E E+02±4.25E In contrast, DE/current-to-best/1/bn s a relatvely greedy DE, snce the nformaton of the best ndvdual n the populaton s exploted to produce the tral vectors. The am here s to nvestgate how CPI-DE nfluences the performance of unbased and greedy DE. The parameter settngs of these two classc versons of DE were: =D, F=0.9, and CR=0.5. For each test functon, 51 ndependent runs were mplemented. The expermental results of CEC CEC wth 30D and 50D have been reported n Tables 1 and 2, and the expermental results of CEC CEC wth 30D and 50D have been reported n Tables 3 and 4, where +, -, and denote that the performance of a classc DE verson s better than, worse than, and smlar to that of ts augmented algorthm, respectvely. One of the frst observatons from Tables 1-4 s that our framework s able to enhance the performance of these two classc DE versons on the majorty of test functons. The detaled performance comparsons from Tables 1-4 are summarzed as follows: In the case of CEC CEC wth D=30, CPI-DE/rand/1/bn and CPI-DE/current-to-best/1/bn exhbt better performance than ther orgnal algorthms on 17 and 15 test functons, respectvely. When D=50, they surpass ther orgnal algorthms on 16 test functons. In terms of CEC CEC wth D=30, CPI-DE/rand/1/bn and CPI-DE/current-to -best/1/bn perform better than ther orgnal algorthms on 21 and 15 test functons, respectvely. Wth respect to D=50, they have an edge over ther orgnal algorthms on 20 and 19 test functons, respectvely. However, t can be seen from Tables 1-4 that the number of test functons that DE/rand/1/bn and DE/current-to-best/1/bn beat ther augmented algorthms s less than fve. For CEC CEC , our framework fals to

10 10 Table 4 Expermental results of DE/rand/1/bn, CPI-DE/rand/1/bn, DE/current-to-best/1/bn, and CPI-DE/current-to-best/1/bn over 51 ndependent runs on 30 test functons wth 50D from IEEE CEC2014 usng 500,000 FEs. Test Functons wth 50D from IEEE CEC2014 Unmodal Functons Smple Multmodal Functons Hybrd Functons Composton Functons DE/rand/1/bn CPI-DE/rand/1/bn DE/current-to-best/1/bn CPI-DE/current-tobest/1/bn CEC E+08±6.65E E+05±1.43E E+08±4.54E E+03±6.10E+03 CEC E+09±2.91E E+06±2.65E E+02±1.10E E-06±7.24E-06 CEC E+04±7.65E E+00±1.53E E+04±4.72E E+00±0.00E+00 CEC E+02±4.35E E+01±2.42E E+01±3.65E E+01±4.48E+01 CEC E+01±3.43E E+01±4.52E E+01±3.95E E+01±3.43E-02 CEC E+01±1.50E E+01±2.27E E+00±9.12E E+00±1.27E+00 CEC E+00±4.17E E-01±6.84E E-04±1.38E E+00±0.00E+00 CEC E+02±1.25E E+02±1.21E E+01±3.01E E+01±2.66E+01 CEC E+02±1.67E E+02±1.53E E+02±1.38E E+02±1.61E+01 CEC E+03±3.25E E+03±3.19E E+03±1.61E E+03±6.91E+02 CEC E+04±3.98E E+04±3.31E E+04±3.34E E+04±3.83E+02 CEC E+00±2.94E E+00±2.60E E+00±2.63E E+00±2.48E-01 CEC E-01±7.75E E-01±6.78E E-01±6.55E E-01±5.65E-02 CEC E-01±2.24E E-01±4.85E E-01±2.53E E-01±1.61E-01 CEC E+02±1.69E E+01±1.55E E+01±1.38E E+01±1.78E+00 CEC E+01±1.93E E+01±1.96E E+01±2.71E E+01±2.58E-01 CEC E+07±5.69E E+03±2.90E E+07±3.17E E+03±3.05E+02 CEC E+05±4.23E E+02±9.90E E+03±1.26E E+02±2.66E+01 CEC E+01±1.60E E+01±1.91E E+01±9.16E E+01±8.88E-01 CEC E+04±6.63E E+02±9.87E E+04±3.09E E+02±3.94E+01 CEC E+06±2.18E E+03±1.89E E+06±1.27E E+03±2.91E+02 CEC E+03±1.25E E+03±1.49E E+03±2.13E E+03±1.74E+02 CEC E+02±6.23E E+02±3.03E E+02±4.50E E+02±4.59E-13 CEC E+02±2.49E E+02±2.62E E+02±2.70E E+02±2.84E+00 CEC E+02±9.85E E+02±4.36E E+02±8.99E E+02±3.26E-01 CEC E+02±6.79E E+02±5.80E E+02±6.12E E+02±5.49E-02 CEC E+03±3.77E E+03±1.47E E+02±5.46E E+02±5.07E+01 CEC E+03±1.00E E+03±4.28E E+03±4.00E E+03±5.15E+01 CEC E+05±1.36E E+03±1.47E E+04±2.06E E+02±9.30E+01 CEC E+04±1.88E E+03±4.07E E+04±1.87E E+03±6.36E Average Functon Error Value DE/rand/1/bn CPI-DE/rand/1/bn DE/current-to-best/1/bn CPI-DE/current-to-best/1/bn FEs consstently provde the results of hgher qualty on three basc multmodal functons (.e., CEC , CEC , and CEC ) and two composton functons (.e., CEC and CEC ). As far as CEC CEC are consdered, the smlar Average Functon Error Value DE/rand/1/bn CPI-DE/rand/1/bn DE/current-to-best/1/bn CPI-DE/current-to-best/1/bn FEs (a) CEC wth 30D (b) CEC wth 30D Fg. 4. Evoluton of the average functon error values derved from two classc DE versons (DE/rand/1/bn and DE/current-to-best/1/bn) and ther augmented algorthms versus the number of FE S on CEC wth 30D and CEC wth 30D. phenomenon can also be observed on three smple multmodal functons (.e., CEC2014 8, CEC , and CEC ) and one hybrd functon (.e., CEC ). The falure could be because our framework generates two tral vectors for each target

11 11 Table 5 Expermental results of JADE, CPI-JADE, jde, CPI-jDE, SaDE, and CPI-SaDE over 51 ndependent runs on 28 test functons wth 30D from IEEE CEC2013 usng 300,000 FEs. Test Functons wth 30D from IEEE CEC2013 Unmodal Functons Basc Multmodal Functons Composton Functons JADE CPI-JADE jde CPI-jDE SaDE CPI-SaDE CEC E+00±0.00E E+00±0.00E E+00±0.00E E+00±0.00E E+00±0.00E E+00±0.00E+00 CEC E+03±4.59E E+00±0.00E E+05±7.99E E-01±1.49E E+05±1.83E E+04±8.32E+03 CEC E+05±1.29E E+02±1.21E E+06±2.19E E-01±1.36E E+07±2.57E E+02± CEC E+03±1.26E E+00±0.00E E+01±2.83E E+00±0.00E E+03±1.37E E+00±0.00E+00 CEC E+00±0.00E E+00±0.00E E+00±0.00E E+00±0.00E E+00±0.00E E+00±0.00E+00 CEC E+00±6.28E E+00±5.17E E+01±4.75E E+01±5.05E E+01±2.99E E+00±6.70E+00 CEC E+00±4.40E E+00±2.22E E+00±2.42E E+00±2.23E E+01±1.32E E+01±1.46E+01 CEC E+01±9.20E E+01±4.92E E+01±4.74E E+01±5.25E E+01±4.25E E+01±5.66E-02 CEC E+01±1.54E E+01±1.44E E+01±3.83E E+01±6.29E E+01±2.82E E+01±4.94E+00 CEC E-02±2.08E E-02±1.65E E-02±2.14E E-03±7.17E E-01±1.44E E-02±4.11E-02 CEC E+00±0.00E E+00±0.00E E+00±0.00E E+00±0.00E E-01±6.77E E+00±0.00E+00 CEC E+01±5.10E E+01±3.16E E+01±9.13E E+01±7.82E E+01±1.11E E+01±1.01E+01 CEC E+01±1.35E E+01±1.18E E+01±1.61E E+01±1.36E E+01±1.88E E+01±1.82E+01 CEC E-02±2.35E E+00±8.42E E-03±6.77E E+01±1.17E E+00±1.40E E+00±1.90E+00 CEC E+03±3.13E E+03±3.27E E+03±3.92E E+03±4.51E E+03±1.03E E+03±3.99E+02 CEC E+00±6.49E E+00±4.99E E+00±3.03E E+00±2.02E E+00±3.10E E+00±3.29E-01 CEC E+01±2.65E E+01±1.44E E+01±9.42E E+01±5.22E E+01±4.25E E+01±5.58E-03 CEC E+01±7.25E E+01±6.25E E+02±1.53E E+02±1.34E E+02±4.44E E+02±1.08E+01 CEC E+00±1.01E E+00±1.30E E+00±1.30E E+00±2.06E E+00±7.04E E+00±2.85E-01 CEC E+01±5.07E E+01±4.77E E+01±3.41E E+01±3.34E E+01±6.75E E+01±5.22E-01 CEC E+02±7.22E E+02±8.64E E+02±7.28E E+02±9.43E E+02±7.70E E+02±8.87E+01 CEC E+01±2.50E E+02±2.09E E+02±1.92E E+02±6.64E E+02±2.88E E+02±8.14E+01 CEC E+03±4.74E E+03±4.23E E+03±4.45E E+03±3.62E E+03±1.11E E+03±4.69E+02 CEC E+02±1.68E E+02±1.18E E+02±8.38E E+02±4.35E E+02±6.08E E+02±5.58E+00 CEC E+02±9.97E E+02±1.37E E+02±9.63E E+02±9.52E E+02±1.06E E+02±1.25E+01 CEC E+02±5.48E E+02±3.99E E+00±4.73E E+00±1.44E E+02±2.97E E+02±2.76E+01 CEC E+02±2.30E E+02±2.16E E+02±2.25E E+02±1.19E E+02±6.93E E+02±1.14E+02 CEC E+02±0.00E E+02±0.00E E+02±0.00E E+02±0.00E E+02±0.00E E+02±0.00E vector, whch results n a smaller number of teratons. The performance of CPI-DE/rand/1/bn and CPI-DE/ current-to-best/1/bn s better than or smlar to that of ther orgnal algorthms on all the unmodal functon, regardless of the test sets and the number of decson varables. Overall, the performance mprovement provde by our framework s qute sgnfcant on four IEEE CEC2013 test functons (.e., CEC2013 2, CEC2013 3, CEC2013 4, and CEC ) and fve IEEE CEC2014 test functons (CEC2014 1, CEC , CEC , CEC , and CEC ). Compared wth the orgnal algorthms, CPI-DE/rand/ 1/bn and CPI-DE/current-to-best/1/bn have the capablty to acheve 100% successful runs for nne cases, whch have been hghlghted n boldface n Tables 1-4. It seems that under our framework, the ncrease of the dmenson (from 30 to 50) does not have a remarkable nfluence on the performance mprovement. It s also nterestng to note that the advantage of CPI-DE/ current-to-best/1/bn over DE/current-to-best/1/bn ncreases as the number of dmenson ncreases. In summary, CPI-DE s an effectve framework to mprove the performance of two classc DE versons (DE/rand/1/bn and DE/current-to-best/1/bn) n the case of two sets of benchmark test functons wth 30D and 50D from IEEE CEC2013 and IEEE CEC2014, whch ndcates that the cumulatve populaton dstrbuton nformaton does play an mportant role n DE. The convergence graphs of the average functon error values derved from two classc DE versons and ther augmented algorthms have been gven n Fg. 4 for two test functons,.e., CEC wth 30D and CEC wth 30D. 5.2 CPI-DE for Three State-of-the-Art DE Varants In order to further assess the effectveness of the proposed framework, CPI-DE s appled to three state-of-the-art DE varants,.e., JADE [6], jde [7], and SaDE [8]. In our experments, the parameter settngs of JADE, jde, and SaDE were the same as n the orgnal papers. Moreover, when applyng our framework to these three DE varants, the