IEEE TRANSACTIONS ON CYBERNETICS 1. Improving Metaheuristic Algorithms With Information Feedback Models

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1 Ths artcle has been accepted for ncluson n a future ssue of ths journal. Content s fnal as presented, wth the excepton of pagnaton. IEEE TRANSACTIONS ON CYBERNETICS 1 Improvng Metaheurstc Algorthms Wth Informaton Feedback Models Ga-Ge Wang, Member, IEEE, and Yng Tan, Senor Member, IEEE Abstract In most metaheurstc algorthms, the updatng process fals to make use of nformaton avalable from ndvduals n prevous teratons. If ths useful nformaton could be exploted fully and used n the later optmzaton process, the qualty of the succeedng solutons would be mproved sgnfcantly. Ths paper presents our method for reusng the valuable nformaton avalable from prevous ndvduals to gude later search. In our approach, prevous useful nformaton was fed back to the updatng process. We proposed sx nformaton feedback models. In these models, ndvduals from prevous teratons were selected n ether a fxed or random manner. Ther useful nformaton was ncorporated nto the updatng process. Accordngly, an ndvdual at the current teraton was updated based on the basc algorthm plus some selected prevous ndvduals by usng a smple ftness weghtng method. By ncorporatng sx dfferent nformaton feedback models nto ten metaheurstc algorthms, ths approach provded a number of varants of the basc algorthms. We demonstrated expermentally that the varants outperformed the basc algorthms sgnfcantly on 14 standard test functons and 10 CEC 011 real world problems, thereby, establshng the value of the nformaton feedback models. Index Terms Benchmark, evolutonary algorthms (EAs), evolutonary computaton, nformaton feedback, metaheurstc algorthms, optmzaton algorthms, swarm ntellgence. I. INTRODUCTION IN VARIOUS aspects of daly lfe, people try ther best to maxmze ther benefts and mnmze ther costs. Ths type of reasonng s modeled mathematcally by optmzaton problems. In mathematcs, computer scence, decson-makng, and Manuscrpt receved September 0, 017; revsed November 0, 017; accepted December, 017. Ths work was supported n part by the Natonal Natural Scence Foundaton of Chna under Grant , Grant , Grant , and Grant , n part by the Natural Scence Foundaton of Jangsu Provnce under Grant BK01509, n part by the Bejng Natural Scence Foundaton under Grant 41609, and n part by the Natonal Key Basc Research Development Plan (97 Plan) Project of Chna under Grant 015CB50. Ths paper was recommended by Y. S. Ong. (Correspondng author: Yng Tan.) G.-G. Wang s wth the Department of Computer Scence and Technology, Ocean Unversty of Chna, Qngdao 66100, Chna, also wth the School of Computer, Jangsu Normal Unversty, Xuzhou 1116, Chna, and also wth the Insttute of Algorthm and Bg Data Analyss and School of Computer Scence and Informaton Technology, Northeast Normal Unversty, Changchun 10117, Chna (e-mal: gagewang@gmal.com). Y. Tan s wth the Key Laboratory of Machne Percepton and the Department of Machne Intellgence, School of Electroncs Engneerng and Computer Scence, Pekng Unversty, Bejng , Chna (e-mal: ytan@pku.edu.cn). Color versons of one or more of the fgures n ths paper are avalable onlne at Dgtal Object Identfer /TCYB other felds, optmzaton problems seek the maxmum or mnmum value of a gven objectve functon. These problems are often approached usng optmzaton algorthms. Optmzaton algorthms can be dvded loosely nto two categores: 1) the tradtonal determnstc methods and ) modern metaheurstc algorthms. The former wll generate the same results for dfferent runs under the same condtons. For the latter, dfferent runs wll generate dfferent solutons n most cases, even under the same condtons. Because metaheurstc algorthms can solve many complcated problems successfully, they have receved ncreased attenton n many felds, rangng from academc research to engneerng practce. Inspred by nature, a varety of metaheurstc algorthms have been proposed recently to deal wth complcated optmzaton problems [1] [5]. Many of them have solved complex, challengng problems that are dffcult to approach usng tradtonal mathematcal optmzaton technques. These nature-nspred algorthms nclude ant colony optmzaton (ACO) [6], [7], artfcal bee colony [8], [9], dfferental evoluton (DE) [10] [1], evolutonary strategy (ES) [1], cuckoo search (CS) [14], [15], freworks algorthm (FWA) [16], bran storm optmzaton [17], [18], earthworm optmzaton algorthm [19], elephant herdng optmzaton [0], krll herd (KH) [1] [8], bogeography-based optmzaton (BBO) [9], genetc algorthm (GA) [0] [], harmony search (HS) [] [5], monarch butterfly optmzaton (MBO) [6], probabltybased ncremental learnng (PBIL) [7], moth search algorthm [8], partcle swarm optmzaton (PSO) [9] [46], and bat algorthm (BA) [47], [48]. However, these basc metaheurstc algorthms have faled to make full use of valuable nformaton avalable from the ndvduals n prevous teratons to gude ther current and later search. Some of them, such as ABC [8], ACO [6], [49], BA [47], and BBO [9], [50], abandon prevous nstances drectly. Others, such as CS [14], FWA [16], [51], PSO [9] [4], KH [1], [], and MBO [6], use only the best prevous ndvduals. In practce, any of the prevous ndvduals could contan a varety of useful nformaton. If such nformaton could be fully exploted and utlzed n the later optmzaton process, the performance of these metaheurstc algorthms surely would be sgnfcantly mproved. Accordngly, many researchers enhanced these metaheurstc algorthms, and some useful nformaton obtaned from the surrogate, an ndvdual, the whole populaton/swarm, dynamcal envronments, and/or neghbors has been extracted c 017 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

2 Ths artcle has been accepted for ncluson n a future ssue of ths journal. Content s fnal as presented, wth the excepton of pagnaton. IEEE TRANSACTIONS ON CYBERNETICS and reused to a certan degree. Few of these mprovements were based on a ftness functon, wth the excepton of Bngul [5]. Bngul transformed the multobjectve problem (MOP) nto sngle-objectve problems by usng a ftness functon. In addton, the square-based ftness functon was used n Bngul [5]. In contrast, most of the prevous studes amed to mprove the performance of a partcular metaheurstc algorthm by reusng the exploted nformaton. However, they faled to form a general framework for reusng the obtaned nformaton. In ths paper, we present our research, based on a ftness functon, n whch we constructed a systematc nformaton feedback model that reused the nformaton from ndvduals n prevous teratons. Ths proposed nformaton feedback model was demonstrated to provde a general framework that could be used to mprove the performance of most metaheurstc algorthms. In ths paper, we studed how to make the best use of the nformaton avalable from prevous ndvduals by usng the followng technques. Frst, a certan number of ndvduals n prevous teratons were selected n ether a fxed or random manner. For ths paper, we selected one, two, or three ndvduals from prevous teratons. Second, the prevous ndvduals selected as feedback nformaton were gven to the updatng process. In ths way, the nformaton from prevous ndvduals could be reused fully. Last, each ndvdual of the current teraton was updated accordng to the ndvdual generated by the basc algorthm and some selected prevous ndvduals through a weghted sum method. It should be noted that there were many dfferent ways to determne ther weghts. Ths paper used ther ftness to do so. An ndvdual wth better ftness had a greater weght. Combnng nformaton feedback models wth metaheurstc algorthms led to mproved methods. They were then benchmarked through 14 test cases and ten CEC 011 real world problems. The expermental results demonstrated that the nformaton feedback from prevous ndvduals sgnfcantly outperformed all the basc algorthms. The organzaton of ths paper s as follows. Secton II provdes a revew of the related lterature regardng reusng nformaton n metaheurstc algorthms. In Secton III, we ntroduce the optmzaton process for metaheurstc algorthms. Ths secton then explans how we ncorporated the useful nformaton n prevous ndvduals nto the basc methods, and demonstrates how to mprove PSO wth nformaton feedback models. Secton IV provdes the mathematcal analyses. In Secton V, we explore varous expermental models and provde the smulaton results. Further dscusson s gven n Secton VI. Secton VII concludes ths paper. II. RELATED WORK Recently, n order to mprove the performance of the metaheurstc algorthms, many scholars have extracted and reused the nformaton from varous sources, such as the surrogate, an ndvdual, the whole populaton/swarm, and/or a neghbor. They have also used nformaton from dynamcal envronments, drectonal nformaton, mutual nformaton (MI), and other forms of nformaton. Ther work regardng varous types of nformaton reuse s revewed brefly below. A. Surrogate Informaton Surrogate nformaton s found to be very effectve n reducng user effort. Therefore, many researchers have mproved varous metaheurstc algorthms through the use of surrogate nformaton, as n GA and PSO. Sun et al. [1] proposed a new surrogate-asssted nteractve genetc algorthm (IGA), where the uncertanty n subjectve ftness evaluatons was exploted both n tranng the surrogates and n managng surrogates. Moreover, uncertanty n the nterval-based ftness values was also consdered n model management, so that not only the best ndvduals but also the most uncertan ndvduals would be chosen to be re-evaluated by the human user. The expermental results ndcated that the new surrogate-asssted IGA could allevate user fatgue effectvely and was more lkely to fnd acceptable solutons n solvng complex desgn problems. Gong et al. [5] proposed a computatonally cheap surrogate model-based multoperator search strategy for evolutonary optmzaton. In ths strategy, a set of canddate offsprng solutons were generated by usng the multple offsprng reproducton operators. The best one accordng to the surrogate model was chosen as the offsprng soluton. The proposed strategy was used to mplement a multoperator ensemble n two popular evolutonary algorthms (EAs), DE, and PSO. Amng to solve medum-scale problems (.e., 0 50 decson varables), Lu et al. [54] proposed a Gaussan process surrogate model-asssted EA for medum-scale computatonally expensve optmzaton problems (GPEME). A new framework was developed and used n GPEME that carefully coordnated the surrogate modelng and the evolutonary search. In ths way, the search could focus on a small promsng area and was supported by the constructed surrogate model. Sammon mappng was also ntroduced to transform the decson varables from tens of dmensons to a few dmensons, n order to take advantage of Gaussan process surrogate modelng n a low-dmensonal space. Wang et al. [55] dvded data-drven optmzaton problems nto two categores: 1) offlne and ) onlne data-drven optmzaton. An EA was then presented to optmze the desgn of a trauma system, whch s a typcal offlne datadrven multobjectve optmzaton problem. As each sngle functon evaluaton nvolved a large amount of patent data, Wang et al. [55] developed a multfdelty surrogate management strategy to reduce the computaton tme of the evolutonary optmzaton. Mendes et al. [56] proposed the use of genetc programmng to obtan hgh-qualty surrogate functons that were evaluated quckly. Such functons could be used to compute the values of the optmzaton functons n place of the burdensome methods. The proposal was tested successfully on a verson of the TEAM benchmark problem wth uncertantes n decson parameters. Kattan and Ong [57] proposed a surrogate genetc programmng (or sgp for short) to retan the appeal of the

3 Ths artcle has been accepted for ncluson n a future ssue of ths journal. Content s fnal as presented, wth the excepton of pagnaton. WANG AND TAN: IMPROVING METAHEURISTIC ALGORITHMS WITH INFORMATION FEEDBACK MODELS semantc-based evolutonary search for handlng challengng problems wth enhanced effcency. The proposed sgp dvded the populaton nto two parts, then t evolved the populaton usng standard GP search operators and meta-models that served as a surrogate to the orgnal objectve functon evaluaton. In contrast to prevous works, two forms of metamodels were ntroduced n ths paper to make the dea of usng a surrogate n GP search feasble and successful. Rosales-Pérez et al. [58] ntroduced an approach for addressng model selecton for support vector machnes used n classfcaton tasks. The model selecton problem was transferred mathematcally as a multobjectve one, amng to mnmze smultaneously two components closely related to the error of a model. A surrogate-asssted evolutonary multobjectve optmzaton approach was adopted to explore the hyper-parameters space. The surrogate-asssted optmzaton was used to reduce the number of solutons evaluated by the ftness functons so that the computatonal cost would be reduced as well. Hldebrandt and Branke [59] presented a new way to use surrogate models wth GP. Rather than usng the genotype drectly as nput to the surrogate model, they used a phenotypc characterzaton n ther method. Ths phenotypc characterzaton could be computed effcently, whch allowed them to defne approxmate measures of equvalence and smlarty. Usng a stochastc, dynamc job shop scenaro as an example of smulaton-based GP wth an expensve ftness evaluaton, they demonstrated that these deas can be used to construct surrogate models and mprove the convergence speed and soluton qualty of GP. PSO s one of the most excellent swarm ntellgencebased metaheurstc algorthms [9], n whch partcles are updated accordng to the best ndvduals n the populaton and the best poston for each partcle so far. Ln et al. [60] proposed a bnary PSO based on surrogate nformaton wth proportonal acceleraton coeffcents (BPSOSIPAC) for the 0-1 multdmensonal knapsack problem (MKP). The BPSOSIPAC was based on the surrogate nformaton concept to repar an nfeasble partcle and make the nfeasble soluton become a feasble one. B. Indvdual Informaton ABC s a relatvely new swarm ntellgence-based metaheurstc algorthm [8]. In the basc ABC, prevous ndvduals were not reused at all. In addton, Gao et al. [61] proposed a bare bones ABC called BABC that used parameter adaptaton and ftness-based neghborhood. In BABC, the useful nformaton n the best ndvdual and a Gaussan search equaton were used to generate a new canddate ndvdual at the onlooker phase [61]. On other hand, at the employed bee phase, the nformaton from the prevous search and from the better ndvduals was ncorporated nto the parameter adaptaton strategy and a ftness-based neghborhood mechansm n order to mprove the search ablty [61]. GA has been appled successfully to address all knds of engneerng problems, especally n dscrete optmzaton [0], [1]. Bngul [5] frst used nformaton feedback n adaptve GAs for dynamc MOPs. Bngul transformed the multobjectve optmzaton problem nto a sngleobjectve problem by usng a statc ftness functon and rule-based weght ftness functon. Bngul [5] also used a square-based ftness functon because t generated the best solutons among varous types of ftness functons. Gong et al. [6] combned the advantages of the GA and PSO, and proposed a generalzed learnng PSO paradgm, the *L-PSO. In *L-PSO, genetc operators were used to generate exemplars accordng to the hstorcal search nformaton of partcles. By performng crossover, mutaton, and selecton on the hstorcal nformaton of partcles, the constructed exemplars were not only well dversfed but also hghly qualfed. Ly and Lpson [6] proposed a strategy to select the most nformatve ndvduals n a teacher-learner type coevoluton by usng the surprsal of the mean, based on Shannon nformaton theory. Ths selecton strategy was verfed by an teratve coevolutonary framework, whch conssted of symbolc regresson for model nference, and a GA for optmal experment desgn. In order to explot fully both global statstcal nformaton and ndvdual locaton nformaton, Zhou et al. [64] combned an estmaton of dstrbuton algorthm wth computatonally cheap and expensve local search (LS) methods. Xong et al. [65] combned stochastc elements nto a resource nvestment project schedulng problem (RIPSP), and proposed a stochastc extended RIPSPs. A knowledgebased multobjectve EA (K-MOEA) was proposed to solve the problem. In K-MOEA, the useful nformaton n the obtaned nondomnated solutons (ndvduals) was extracted and then used to update the populaton perodcally to gude subsequent search. C. Populaton/Swarm Informaton Gao et al. [66] proposed a novel ABC algorthm based on nformaton learnng, called ILABC. In ILABC, at each generaton, the whole populaton was dvded dynamcally nto several subpopulatons by the clusterng partton based on the prevous search experence. Furthermore, the dfferent ndvduals n one subpopulaton and n dfferent subpopulatons exchanged nformaton after all the ndvduals were updated. In ths way, all the ndvduals would fnd the best soluton cooperatvely. In addton to ILABC, Gao et al. [67] proposed another mproved ABC algorthm usng more nformatonbased search equatons. Inspred by the echo locaton behavor of bats n nature, BA was proposed for global optmzaton problems [47]. The poston of the bats was updated by the bats frequency, velocty, and dstance to food. Therefore, ther poston had no relatonshp wth any knd of nformaton reuse. Wang et al. [68] proposed a multswarm BA (MBA) for global optmzaton problems. In MBA, the nformaton between dfferent swarms was exchanged by an mmgraton operator wth dfferent parameter settngs. Thus, ths confguraton was able to make a good tradeoff between global and LS.

4 Ths artcle has been accepted for ncluson n a future ssue of ths journal. Content s fnal as presented, wth the excepton of pagnaton. 4 IEEE TRANSACTIONS ON CYBERNETICS Wth regard to DE, t s well accepted that two control parameters: 1) scale factor (F) and ) crossover rate (Cr), have great nfluence on the performance of DE. Based on nformaton from the populaton, Ghosh et al. [69] proposed a smple yet useful adaptaton technque for tunng F and Cr. In order to boost the populaton dversty when addressng large-scale global problems, Al et al. [70] proposed a new, mproved DE called mde-bes. Ths verson was a multpopulaton algorthm, and the populaton was dvded nto ndependent subgroups, each wth dfferent mutaton and update strateges. The nformaton of the best ndvdual was used to generate a novel mutaton strategy that produced qualty solutons wth a balance between exploraton and explotaton. At each generaton, the ndvduals exchanged ther nformaton between the subgroups. Cu et al. [71] desgned a novel adaptve multple subpopulatons-based DE named MPDE, n whch the parent populaton was splt nto three subpopulatons based on ther ftness values. In MPDE, the useful nformaton from the tral vectors and target vectors was exploted fully to form a replacement strategy that amed to mprove the search ablty. Inspred by team cooperaton n the real world, Gao et al. [7] proposed a dual-populaton DE (DPDE) wth coevoluton for constraned optmzaton problems (COPs). The COP was dvded nto two objectves that were solved by two subpopulatons at each generaton, respectvely. In DPDE, an nformaton-sharng strategy was used to exchange search nformaton between the dfferent subpopulatons. Wang et al. [7] proposed a cooperatve multobjectve DE (CMODE) wth multple populatons for multobjectve optmzaton problems (MOPs), whch ncluded M sngleobjectve optmzaton subpopulatons and an archve populaton for an M-objectve optmzaton problem. These (M + 1) populatons cooperated to optmze all objectves of MOPs by usng adaptve DEs. The addtonal dfference term was added to the proposed method wth the am of sharng nformaton from the archve. In ths way, an ndvdual could use the search nformaton not only from ts own subpopulaton but also from other populatons. The ndvdual was expected to search along the whole Pareto front (PF) by usng the nformaton of all the populatons nstead of beng attracted to the margn or extreme pont only by the search nformaton of ts own subpopulaton. Hence, CMODE could approxmate the whole PF quckly wth the help of the archved nformaton. Dhal et al. [74] proposed two varants of FA: 1) FA va Lévy flghts and ) FA va chaotc sequence. In these two algorthms, the nformaton of populaton dversty was fully extracted to generate the ndvduals at each generaton. Pan et al. [75] proposed a local-best HS algorthm wth dynamc subpopulatons (DLHS) for global optmzaton problems. In DLHS, the whole harmony memory (HM) was dvded nto a certan number of small-szed sub-hms that exchanged nformaton wth each other by usng a perodc regroupng schedule. Furthermore, the useful nformaton n the local best harmony vector was used to generate a novel harmony mprovsaton scheme [75]. D. Informaton From Dynamcal Envronments Though many versons of multobjectve PSO (MOPSO) have been desgned, few MOPSOs have been desgned to adjust the balance between exploraton and explotaton dynamcally accordng to the feedback nformaton detected from the evolutonary envronment. Hu and Yen [76] proposed a new algorthm, the parallel cell coordnate system (PCCS), accordng to the nformaton about the evolutonary envronment, ncludng densty, rank, and dversty ndcators. PCCS was then ncorporated nto a self-adaptve MOPSO, and a new MOPSO was proposed: the pccsamopso. Foss nvestgated how a vable system, the honey bee swarm, gathered meanngful nformaton about potental new nest stes n ts problematc envronment [77]. Ths nvestgaton used a cybernetc model of a self-organzng nformaton network to analyze the fndngs from the last 60 years of publshed research about swarm behavor. Informaton gatherng by a honey bee swarm was frst modeled as a self-organzng nformaton network. E. Neghborhood and Drecton Informaton In the basc DE, the base and dfference vectors are always selected randomly from the whole populaton for the mutaton operators, but the neghborhood and drecton nformaton fals to be used effectvely [10], [11], [78], [79]. In order to address ths problem, several scholars have put forward mproved strateges. Peng et al. [80] proposed a novel DE framework wth dstrbuted drecton nformaton-based mutaton operators (DE-DDI) for dealng wth complex problems n bg data. In DE-DDI, the dstrbuted topology was used to generate a neghborhood for each ndvdual frst. Then the drecton nformaton derved from the neghbors was ntroduced nto the mutaton operator of DE. Consequently, the neghborhood and drecton nformaton fully exploted the regons of better ndvduals, and guded the search to the promsng area. Lao et al. [81] proposed another DE framework wth a drectonal mutaton based on cellular topology, called cellular drecton nformaton-based DE (DE-CDI). For each ndvdual n DE-CDI, the cellular topology was formed to defne a neghborhood. Next, the drecton nformaton based on the neghborhood was ncorporated nto the mutaton operator. In ths way, DE-CDI not only extracted the neghborhood nformaton to explot the regons of better ndvduals and accelerate convergence but also ntroduced the drecton nformaton to gude the search to the promsng area. In order to use the neghborhood and drecton nformaton fully, Ca et al. [8] proposed a new DE framework wth neghborhood and drecton nformaton (ND-DE). Though ND-DE had better performance than most of the DEs, ts performance reled manly on the selecton of drecton nformaton. To overcome ths dsadvantage, the adaptve operator selecton mechansm was ncorporated nto the ND-DE, whch was able to select adaptvely the drecton nformaton for the specfc DE mutaton strategy. Accordngly, an mproved ND-DE called adaptve drecton nformaton-based ND-DE

5 Ths artcle has been accepted for ncluson n a future ssue of ths journal. Content s fnal as presented, wth the excepton of pagnaton. WANG AND TAN: IMPROVING METAHEURISTIC ALGORITHMS WITH INFORMATION FEEDBACK MODELS 5 (and-de) was proposed by Ca et al. [8], whch performed much better than ND-DE. Fang et al. [8] proposed a decentralzed quantum-nspred PSO (QPSO) wth cellular structured populaton called cqpso. In cqpso, the partcles were located n a -D grd and allowed to get nformaton only from ther neghbors. The overlappng partcles exchange the nformaton among the nearest neghborhoods. Wang et al. [84] proposed an mproved verson of BA namely varable neghborhood bat algorthm (VNBA), s thus proposed. In VNBA, the bat ndvdual can get useful nformaton from ther neghbors. F. Mutual Informaton He et al. [85] ntroduced the multresoluton analyss, MI, and PSO nto artfcal neural network models. They proposed a hybrd wavelet neural network model for forecastng monthly ranfall from antecedent monthly ranfall and clmate ndces. G. Other Informaton ACO s one of the most representatve metaheurstc algorthms for global optmzaton problems, especally, for dscrete optmzaton [6], [49]. Because the ants are updated accordng to the pheromone, the prevous nformaton fals to be used n ACO. Shang et al. [86] ntroduced heurstc nformaton nto antdecson rules, and then proposed a new verson of ACO named AntMner for epstass detecton. In AntMner, the heurstc nformaton was used to gude ants durng the search process wth the am of enhancng the computatonal effcency and soluton accuracy. Wang and Tang [87] proposed an adaptve DE based on analyss of search data for the MOPs. In ths algorthm, frst the useful nformaton was derved from the search data durng the evoluton process by usng clusterng and statstcal methods. Then the derved nformaton was used to gude the generaton of new populaton and the LS. Park and Lee [88] proposed a novel opposton-based learnng method by usng a beta dstrbuton wth partal dmensonal change and selecton swtchng. They combned ths approach wth DE to enhance the convergence speed and search ablty. In the proposed method, the partal dmensonal changng scheme was used to preserve useful nformaton. Smulated annealng (SA) s one of the oldest classcal metaheurstc algorthms [89] that s a trajectory-based optmzaton algorthm. Yang and Kumar [90] proposed an nformaton guded framework for SA. Informaton gathered from the exploraton stage was used as feedback to drve the optmzaton procedure, leadng to the rse of the annealng temperature durng the optmzaton process. The resultng algorthm had two phases: phase I performed nearly unrestrcted exploraton, and phase II re-heated the annealng procedure and exploted nformaton gathered durng phase I. Muñoz et al. [91] proposed a robust nformaton contentbased method for contnuous ftness landscapes that generated four measures related to the landscape features. In addton, t could overcome the dsadvantage of samplng the ftness landscape usng random walks wth varable step sze. From the descrptons above, we can see that for most metaheurstc algorthms, some useful nformaton obtaned from a surrogate, an ndvdual, the whole populaton/swarm, dynamcal envronments, neghbor and drecton, and/or mutual relatonshp s extracted and reused to a certan degree. However, few of them are based on a ftness functon (except [5]). Bngul [5] transferred the MOP nto some sngle-objectve problems by usng a ftness functon as explaned prevously. Furthermore, whle most of the studes above amed to mprove the performance of a certan metaheurstc algorthm by reusng the exploted nformaton, they faled to form a general framework for reusng the obtaned nformaton. In ths paper, we present our research, based on a ftness functon, n whch we constructed a systematc nformaton feedback model that reused the nformaton from ndvduals n prevous teratons. Ths proposed nformaton feedback model was demonstrated to provde a general framework that could be used to mprove the performance of most metaheurstc algorthms. III. IMPROVING METAHEURISTIC ALGORITHMS WITH INFORMATION FEEDBACK MODELS In ths secton, we explan how metaheurstc algorthms have been mproved based on nformaton feedback models. Frst, we provde a bref outlne of the basc optmzaton process, and then we gve a descrpton of the nformaton feedback models. Fnally, usng PSO as an example, we demonstrate how to mprove the algorthm usng nformaton feedback models. A. Optmzaton Process Despte the fact that dfferent metaheurstc algorthms have dfferent updatng strateges, ther optmzaton processes can be summarzed brefly by the followng general steps. 1) Intalzaton: Intalzaton can be dvded nto populaton ntalzaton and parameter ntalzaton. The runnng envronments for later search are set durng ths process. ) Search: In general, metaheurstc algorthms frst mplement global search and then LS,.e., exploraton and then explotaton. These two searches perform n parallel, beng adjusted by certan parameters. The search process s repeated untl some termnaton condton s satsfed. ) Output: Output the fnal best solutons. B. Informaton Feedback Models In theory, for our model k(k 1) prevous ndvduals can be selected, but usng a substantal number of ndvduals mght complcate the method. Therefore, n ths paper, k {1,, }. As mentoned above, we wll take PSO as an example to llustrate the framework of our proposed method. Some symbols are gven before the nformaton feedback models are descrbed. Suppose that x t s the th ndvdual at teraton t, and x and f t are ts poston and ftness value, respectvely. Here,

6 Ths artcle has been accepted for ncluson n a future ssue of ths journal. Content s fnal as presented, wth the excepton of pagnaton. 6 IEEE TRANSACTIONS ON CYBERNETICS t s the current teraton, 1 N P s an nteger number, and N P s the populaton sze. y t+1 s the ndvdual generated by the basc PSO, and f t+1 s ts ftness. The framework of the proposed method s gven through the ndvduals at the (t )th, (t 1)th, tth, and (t + 1)th teratons. 1) Model F1 and Model R1: Ths s the smplest case. The th ndvdual x t+1 can be generated as follows: x t+1 = αy t+1 + βxj t (1) where xj t s the poston for ndvdual j(j {1,,...,N P}) at teraton t, and fj t s ts ftness. α and β are weghtng factors satsfyng α + β = 1. They can be gven as α = f t j j,β = f t+1 j. () Here, ndvdual j can be determned n the followng ways. Defnton 1: The model n (1) s called model F1 when j =. The ndvduals n prevous and current generatons are used to generate the ndvdual for the next generaton. Defnton : The model n (1) s called model R1 when j = r 1, where r 1 s an nteger randomly selected between 1 and N P. The ndvdual generated by Defnton has a hgher populaton dversty than the one generated by Defnton 1. We can see that f r 1 =, the model R1 wll be F1 wth the probablty of 1/N P. Ther ncorporaton nto the basc PSO results n PSOF1 and PSOR1, respectvely. ) Model F and Model R: Two ndvduals at two prevous teratons are collected and used to generate ndvdual. For ths case, the th ndvdual x t+1 can be generated as follows: x t+1 = αy t+1 + β 1 xj t 1 + β xj t 1 () where xj t 1 and xj t 1 are the poston for ndvduals j 1 and j (j 1, j {1,,...,N P }) at teraton t and t 1, and fj t 1 and fj t 1 are ther ftness values, respectvely. α, β 1, and β are weghtng factors satsfyng α +β 1 +β = 1. They can be provded as follows: α = 1 fj t 1 j 1 1 j 1 β 1 = 1 fj t 1 + f t+1 β = 1 f t+1 1 j j 1 1 j j 1 j 1. (4) Indvduals j 1 and j n () can be determned n several dfferent ways. For ths model, ths paper focused on Defntons and 4. Defnton : The model n () s called model F when j 1 = j =. The ndvduals at two prevous and current generatons are used to generate the ndvdual for the next generaton. Defnton 4: The model n () s called model R when j 1 = r 1, and j = r, where r 1 and r are ntegers that are randomly selected between 1 and N P. Smlarly, the ndvdual generated by Defnton 4 has more dversty of populaton than the ndvdual generated by Defnton. Here, we can see, f r 1 = r =, the model R wll be F wth the probablty of 1/N P. Ther ncorporaton nto the basc PSO results n PSOF and PSOR, respectvely. ) Model F and Model R: Three ndvduals at three prevous teratons are collected and used to generate ndvdual. For ths case, the th ndvdual x t+1 can be generated as follows: x t+1 = αy t+1 + β 1 xj t 1 + β xj t 1 + β xj t (5) where xj t 1, xj t 1, and xj t are the poston of ndvduals j 1, j, and j (j 1, j, j {1,,...,N P }) at teraton t, t 1, and t, and fj t 1, fj t 1, and fj t are ther ftness values, respectvely. Ther weghtng factors are α, β 1,β, and β wth α + β 1 + β + β = 1, whch can be gven as α = 1 fj t 1 + fj t 1 + fj t β 1 = 1 f t+1 β = 1 f t+1 β = 1 f t+1 + fj t 1 + fj t 1 + fj t 1 j j + fj t 1 + fj t 1 + fj t j 1 j + fj t 1 + fj t 1 + fj t j 1 1 j + fj t 1 + fj t 1 + fj t. (6) Though j 1 j can be determned n many dfferent ways, we adopted Defntons 5 and 6 for ths model. Defnton 5: The model n (5) s called model F when j 1 = j = j =. The ndvduals at two prevous and current generatons are used to generate the ndvdual for the next generaton. Defnton 6: The model n (5) s called model R when j 1 = r 1, j = r, and j = r, where r 1 r are nteger numbers that are selected randomly between 1 and N P. Smlarly, the ndvdual generated by Defnton 6 has more populaton dversty. Here, we can see that f r 1 = r = r =, model R wll be F wth the probablty of 1/N P. Ther ncorporaton nto the basc PSO results n PSOFand PSOR, respectvely. By ncorporatng the nformaton feedback model nto the basc optmzaton process, we have a new updatng optmzaton process as shown n Fg. 1. C. PSO Usng Model F1 We now take PSO and model F1 as an example to explan how to ntroduce nformaton feedback nto a metaheurstc algorthm. PSO [9] s one of the most representatve swarm ntellgence paradgms. The solutons (called partcles) are located ntally n the whole search regon at random. Subsequently,

7 Ths artcle has been accepted for ncluson n a future ssue of ths journal. Content s fnal as presented, wth the excepton of pagnaton. WANG AND TAN: IMPROVING METAHEURISTIC ALGORITHMS WITH INFORMATION FEEDBACK MODELS 7 Fg. 1. Schematc flowchart of updatng optmzaton process. Fg.. Improvng PSO wth nformaton feedback models (k = 1). the velocty and poston of the partcles are updated as (7) and (8), respectvely. v t+1 = ωv t + c ( ) ( ) 1r 1 p,best x + c r g,best x (7) = x t + vt+1 (8) x t+1 where x and v are the poston and velocty of partcle, respectvely; p,best and g,best are the poston wth the optmal objectve value searched untl now by partcle and the whole populaton, respectvely; w s an nerta parameter controllng the dynamcs of flyng; r 1 and r are random real numbers n [0, 1]; and c 1 and c are factors controllng the related weghtng of correspondng terms. After updatng velocty and poston for partcle, p,best and g,best wll be updated. Ths process wll be repeated untl a certan stop condton s met. Next, lookng at the general outlne of the optmzaton process, we can see the man steps for mprovng PSO by usng the nformaton feedback model (k = 1). 1) Intalzaton: The parameters used n PSO are set, and the partcle populaton s ntalzed randomly wth the predefned regons. Ths process s the same as performed n the basc PSO. ) Search: Ths s the crtcal part for mprovng PSO. Frst, the velocty and poston of partcle are updated accordng to (7) and (8). The updated partcle can be called y. If the generaton count t s bgger than 1, partcle wll be further updated by (1), and the newly generated partcle wll be consdered as the fnal partcle for the next generaton. The search process s repeated untl some termnaton condton s satsfed. ) Output: PSO returns the values of g best and f (g best ) as ts fnal soluton. The detaled steps of the combnaton of PSO and the nformaton feedback model (k = 1) can be seen n Fg..InFg., G max s the maxmum of the generaton. Smlarly, the other fve models (R1, F ) can be ncorporated nto the basc PSO. Gven the lmts on the length of ths paper, we wll not descrbe them n detal. IV. MATHEMATICAL ANALYSES In ths secton, we provde a mathematcal analyss to prove the convergence of the proposed method. We frst prove the algorthm under model F and R. Here, the followng lemmas are provded, and they are true for any algorthm dscussed n ths paper. Lemma 1: An algorthm A can reach ts fnal soluton x best all of the tme. Here, algorthm A can be any of the algorthms dscussed n ths paper, such as ACO [6], BA [47], BBO [9], CS [14], DE [10], ES [1], KH [1], MBO [6], PBIL [7], and PSO [9]. x best s the best soluton for algorthm A, and ts lower bound and upper bound are x mn and x max, respectvely. Lemma 1 ndcates that algorthm A s able to fnd the fnal soluton all of the tme, f t can search for the gven doman wth enough tme.

8 Ths artcle has been accepted for ncluson n a future ssue of ths journal. Content s fnal as presented, wth the excepton of pagnaton. 8 IEEE TRANSACTIONS ON CYBERNETICS TABLE I BENCHMARK FUNCTIONS In sum, for each nformaton feedback model, where the model s Model F1 F or R1 R, an algorthm A under these models can reach ts fnal soluton x best every tme. Lemma : The soluton x t+1 s one of the feasble solutons for algorthm A. Proof: Here, we only prove that the lower bound and upper bound of x t+1 for algorthm A are x mn and x max, respectvely. For ease of descrpton, (5) can be descrbed n the followng form: x t+1 = αy + β 1 x 1 + β x + β x. (9) It s clear that for algorthm A, the lower bound and upper bound of the solutons x 1, x, and x are x mn and x max, respectvely. In other words, x mn y x max, x mn x 1 x max, x mn x x max, and x mn x x max. Next, we can get α x mn α y α x max, β 1 x mn β 1 x 1 β 1 x max, β x mn β x β x max, and β x mn β x β x max. Therefore, we get (α + β 1 + β + β ) x mn x t+1 = αy + β 1 x 1 + β x + β x (α + β 1 + β + β ) x max. (10) Accordng to the defnton of α, β 1, β, and β n (5), we know α +β 1 +β +β = 1. Therefore, (10) can be updated as x mn x t+1 = αy + β 1 x 1 + β x + β x x max. (11) We observe clearly that x mn x t+1 x max. In other words, the newly generated soluton x t+1 va our proposed method s a feasble soluton for algorthm A. Theorem : A proposed algorthm A can reach ts fnal soluton x best all the tme. Proof: Here, A represents the proposed algorthm dscussed n the prevous secton. Therefore, accordng to Lemmas 1 and, the proposed algorthm A s able to fnd the fnal soluton x best f t can search for the gven doman wth enough tme. For Models F1 and R1, t s obvous that these models are specal cases of Models F and R. Therefore, any proposed algorthm A s smlarly proven. We do not gve them n detal n ths paper. V. SIMULATION RESULTS Secton III gves sx nformaton feedback models,.e., F1 F and R1 R, each of whch can be ncorporated nto a basc metaheurstc algorthm, thereby, yeldng sx varants of each basc method. For example, gven PSO, we have PSOF1- and PSOR1-. The basc PSO can be named as PSOF0. Smply, we can call them F0 F, and R1 R for short. We must pont out that n order to nvestgate fully the superorty of dfferent nformaton feedback models, sx varants were compared wth each other only and wth the correspondng basc algorthm. Through ths comparson, we were able to look at the performance of sx nformaton feedback models and determne whether these models could mprove the performance of the basc algorthm. Sx nformaton feedback models were combned wth the basc metaheurstc algorthms, and these newly combned methods were further benchmarked by 14 standard test functons as shown n Table I [9]. Each functon had 0 ndependent varables, that s, the dmenson of each problem was 0. Some of functons were multmodal, whch means that they had multple local mnma. Some were nonseparable, whch means that they could not be wrtten as a sum of functons of ndvdual varables. The benchmarks were compared by mplementng the nteger versons of all the metaheurstc algorthms n MATLAB [9]. The granularty or precson of each benchmark functon was 0.1, except for the Quartc functon. Snce the doman of each dmenson of the Quartc functon was only ±1.8, t was mplemented wth a granularty of 0.01 [9]. More nformaton about these functons can be seen by referrng to [9]. Frst, we nvestgated the performance of PSO under Models F1 F and R1 R, and then these sx models were extended to be ncorporated nto more metaheurstc algorthms. A. Performance of PSO Wth Models F1 F and R1 R In ths secton, we wll look at the performance of PSO under Models F1 F and R1 R on 14 benchmarks n Table I. In order to get ther representatve statstcal results, 50 ndependent runs were done for PSO. In addton, PSO had a populaton sze of 50, an eltsm parameter of, and was run for 50 generatons. The results were recorded n Table II. In more detal, the best, average, and worst performances of each method were collected, as shown n Table II. The results were hghlghted n bold f PSO performed the best on a benchmark. The total numbers of the bold results were collected, as shown n the last row n Table II. In order to nvestgate the nfluence of F1 and R1, the number of functons on whch PSO performed the best was calculated, as shown n the last two columns of Table II. From Table II, we see that R1 was the best nformaton feedback model, havng the greatest mpact on PSO. F was nferor only to R1. In addton, for sx nformaton feedback

9 Ths artcle has been accepted for ncluson n a future ssue of ths journal. Content s fnal as presented, wth the excepton of pagnaton. WANG AND TAN: IMPROVING METAHEURISTIC ALGORITHMS WITH INFORMATION FEEDBACK MODELS 9 TABLE II FUNCTION FITNESS OBTAINED BY PSO WITH SIX MODELS models and F0, ther average rankng from good to bad was as follows: R1 > F > R > F > F0 > R1 > F1 = R. Models R1 have slghtly greater mpact than F1 for the PSO algorthm on 14 benchmarks (1 versus 18). From Table II, we can see that our sx proposed models, especally R1 and F, were able to mprove sgnfcantly the performance of PSO by balancng the exploraton and explotaton. Let us gve the detaled analyses as follows. In PSO, partcle learned manly from the nformaton of the global search and ts own best poston. On one hand, ths stuaton meant that most partcles would fly toward the promsng area, and the PSO would have a fast convergence. That s to say, PSO would have a good exploraton ablty. On other hand,

10 Ths artcle has been accepted for ncluson n a future ssue of ths journal. Content s fnal as presented, wth the excepton of pagnaton. 10 IEEE TRANSACTIONS ON CYBERNETICS TABLE III PARAMETER SETTINGS TABLE IV FUNCTION FITNESS OBTAINED BY TEN METAHEURISTIC ALGORITHMS WITH SIX INFORMATION FEEDBACK MODELS TABLE V CPU TIME USED BY TEN METAHEURISTIC ALGORITHMS WITH SIX INFORMATION FEEDBACK MODELS f the optmal were local, t would be hard to escape from t. R1 ntroduced dversty nto the optmzaton process of PSO, whch would enable the trapped partcles to escape from the local postons. If the partcles were not trapped nto local postons, the addton of populaton dversty dd no harm to PSO, because the global best partcle was always memorzed durng the whole optmzaton process. Ths s why F performed better than other models except R1. In sum, the PSO combned wth sx proposed models (especally Models R1 and F) performed better than or equally to the basc PSO. B. Performance of Sx Informaton Feedback Models In ths secton, we explan how sx nformaton feedback models were combned wth other nne metaheurstc algorthms,.e., ACO [6], BA [47], BBO [9], CS [14], DE [10], ES [1], KH [1], MBO [6], and PBIL [7]. These newly combned methods were further benchmarked by 14 standard test functons, as shown n Table I [9]. For an algorthm, dfferent parameter settngs have a great mpact on ts performance. In order to compare farly, ther parameters were set as shown n Table III. For ACO, BBO, DE, ES, PBIL, and PSO, ther parameters were the same as n [9]. For most algorthms, dfferent runs may generate dfferent results. In order to get ther representatve statstcal results, 50 ndependent runs were done for each method. In addton, each method had a populaton sze of 50, an eltsm parameter of, and were run for 50 generatons. The best, average, and worst performances of each method were collected and summarzed n Table IV. The results were hghlghted n bold f the algorthms performed the best for a benchmark. In order to nvestgate the nfluence of F1 and R1, the number of functons on whch the metaheurstc algorthms performed the best was calculated, as shown n the last two columns of Table IV. Table V shows the average CPU tme for each method on each benchmark. We must pont out that PSO was also ncluded n Tables IV and V n order to get more accurate statstcal results. From Table IV, we see that F was the best nformaton feedback model, and had the greatest mpact on the three algorthms: 1) BA; ) CS; and ) MBO. R1 s nferor only to F and had the greatest mpact on three algorthms: 1) ES; ) KH; and ) PSO. F1 ranked thrd and had the greatest mpact on two algorthms: 1) BBO and ) DE. For R and R, except ACO, they had the best mpact on MBO and PBIL, respectvely. Lookng carefully at Table IV, for ACO, R, and R had the same mpact; for MBO, F, and R had the same mpact. In addton, for sx nformaton feedback models and F0, ther average rankng from good to bad was as follows: F > R1 > F1 > F = R > F0 > R. Models F1 had a greater mpact than R1 for ten metaheurstc algorthms on 14 benchmarks (0 versus 16). From Table V, we observed that, except BA, all the varants consumed more tme than ther respectve basc algorthms. Ths result falls fully under the adage, there s no free lunch [9]. The addtonal tme was used manly to evaluate the ftness values, and that acton can be tme consumng. C. Comparsons Usng t-test Based on the fnal results of 50 ndependent runs on 14 functons, Table VI presents the t values on every functon of the two-taled test, wth the 5% level of sgnfcance between the basc method and mproved methods wth sx nformaton

11 Ths artcle has been accepted for ncluson n a future ssue of ths journal. Content s fnal as presented, wth the excepton of pagnaton. WANG AND TAN: IMPROVING METAHEURISTIC ALGORITHMS WITH INFORMATION FEEDBACK MODELS 11 TABLE VI COMPARISONS BETWEEN THE BASIC METHOD AND SIX IMPROVED METHODS WITH INFORMATION FEEDBACK MODELS AT α=0.05 ON TWO-TAILED t-tests TABLE VII TEN REAL WORLD PROBLEMS SELECTED FROM CEC 011 TABLE VIII OPTIMIZATION RESULTS OBTAINED BY TEN METAHEURISTIC ALGORITHMS WITH SIX INFORMATION FEEDBACK MODELS FOR CEC 011 RWPS feedback models. In the table, the value of t wth 98 degrees of freedom was sgnfcant at α = 0.05 by a two-taled test. Boldface ndcates that the correspondng method performed sgnfcantly better than the basc method. The best, equal, and worst n Table VI ndcate that the correspondng method performed better than, equal to, or worse than ts basc one. In more detal, the best, equal, and worst performance of each method was collected and summarzed, as shown n Table VI. For nstance, comparng ACO and sx varants of ACO, ACOF1, and ACOR1 outperformed ACO sgnfcantly on ten, twelve, eleven, twelve, ten, and eleven functons, respectvely, and performed as well as ACO on two, one, one, zero, two, and one functons, respectvely. These results ndcate that sx varants of ACO generally performed better than ACO n terms of the soluton accuracy. Though the performance of ACOF1 and ACOR1 was slghtly weaker on some functons, Table VI also reveals that they outperformed ACO on most functons. Smlarly, Table VI shows that most methods (ACO, BA, CS, DE, ES, MBO, PBIL, and PSO) had absolute advantage over ther basc algorthms. The performance of BBO and KH was better than or equal to ther basc ones on most benchmarks. In addton, as seen from the last three rows of Table VI, R1was the best nformaton model; F1, R1, and F were the three best models among the sx dfferent nformaton feedback models. Ths concluson concdes wth results n Table IV. D. Real World Problems In addton to the standard functons dscussed n the secton above, ten more real world problems (RWPs) (see Table VII) were also used to valdate the sx nformaton feedback models. More detaled nformaton about ten RWPs can be found n [9]. Here, the parameters used n the ten approaches were the same as the above. The populaton sze and generatons were set to 50 and 50, respectvely. The results obtaned by 50 ndependent runs on ten RWPs were recorded n Table VIII. The results were hghlghted n bold f an algorthm performed the best on a benchmark. For each model, the total numbers of the bold results were collected, as shown n the last row. From Table VIII, we see that F1 was the best nformaton feedback model, and had the greatest mpact on the three metaheurstc algorthms: 1) BBO; ) MBO; and ) PSO. R1 was only slghtly nferor to F1 and had the greatest mpact on fve metaheurstc algorthms: 1) ACO; ) BA; ) ES; 4) KH; and 5) PBIL. F ranked the thrd and had the greatest mpact on three metaheurstc algorthms: 1) BA; ) CS; and ) KH. For the other three nformaton feedback models (R, F, and R), F had a greater nfluence on ten metaheurstc algorthms than R and R. For KH, we can see, R1 and F had equal nfluence. Moreover, for BBO, F1 had the same performance as F0 (the basc BBO). For PBIL, R1 had the same performance as F0 (the basc PBIL). In addton, for sx nformaton feedback

12 Ths artcle has been accepted for ncluson n a future ssue of ths journal. Content s fnal as presented, wth the excepton of pagnaton. 1 IEEE TRANSACTIONS ON CYBERNETICS models and F0, ther average rankng from good to bad was as follows: F1 > R1 > F0 > F > F > R > R. Models F1 had a greater mpact than R1 for ten metaheurstc algorthms on ten RWPs (155 versus 89). From the results on 14 standard functons and ten RWPs, F1, R1, and F performed the best among sx nformaton models. Except RWPs studed here, there are many dffcult ssues deservng to be extensvely studed, lke cloud data [94], encrypted outsourced data [95], [96], and mage copy detecton [97]. Shen et al. [94] desgned a new effcent and effectve publc audtng protocol wth novel dynamc structure for cloud data wth the am of decreasng the computatonal and communcaton overheads. Devsng a searchable and desrable encrypton scheme cannot only support personalzed search but also mprove user search experence. For ths purpose, Fu et al. [96] handled the ssue of personalzed multkeyword ranked search over encrypted data whle preservng prvacy n cloud computng. Fu et al. [95] presented a contentaware search scheme, whch can make semantc search more smart. In addton, they verfed the prvacy and effcency of ther schemes n the experments. Zhou et al. [97] desgned a global context verfcaton scheme to flter false matches for copy detecton. Concretely, the overlappng regon-based global context descrptor was desgned to verfy these matches to flter false matches. Gu and Sheng [98] proposed an equvalent dual formulaton for v-svc and a robust v-svcpath based on lower upper decomposton wth partal pvotng. Also, ther robust regularzaton path algorthm can avod the exceptons completely, and handlng the sngulartes n the key matrx. VI. DISCUSSION From the experments conducted n the prevous secton, each of the ten algorthms was mproved by a partcular nformaton feedback model. Here, KH s taken as an example to analyze why the nformaton feedback model can mprove the performance of all of the algorthms on 14 functons. KH s a relatvely new and promsng algorthm proposed by Gandom and Alav n 01 [1]. R1 had the greatest mpact on KH among sx nformaton feedback models,.e., k = 1, and j = r 1 n (1). For krll, the frst and second movements are based manly on the best krll [1]. Ths wll surely make most krll move toward the promsng area. However, at the later search stage, the KH algorthm mght be trapped nto the local optmum. R1 added more dversty to the populaton for the optmzaton process at the later search stage. Meanwhle, the generated krll had a smaller possblty of surpassng the gven range. So, the performance of KH was mproved sgnfcantly. In addton, dfferent models were able to create a good balance between exploraton and explotaton. When k was small, few of the prevous ndvduals were used. In ths way, the ablty of exploraton could be mproved. Conversely, when k was bg, as many of the prevous ndvduals were used as possble. In ths way, the ablty of explotaton could be greatly mproved. On other hand, the algorthms under models R1 had more populaton dversty and exploratve ablty than models F1. Ths could mprove sgnfcantly the performance of the metaheurstc algorthms at the late stage. After fully nvestgatng the performance of the proposed methods, the followng ponts should be hghlghted n future. Frst, the varants of a basc method (except BA) consume more CPU tme than the basc one because of ncreased ftness evaluaton. Methods to reduce CPU tme are worthy of further study. Second, KH and PSO were taken as examples to explan the prncple of our models. Further analyss usng theores should be performed to ascertan the reasons why the models can mprove the performance of ther basc algorthms. VII. CONCLUSION In the study of optmzaton, few metaheurstc algorthms reuse the prevous nformaton to gude the later updatng process. 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Dr. Wang served as an Assocate Edtor of the Internatonal Journal of Computer Informaton Systems and Industral Management Applcatons and an Edtoral Board Member of Internatonal Journal of Bo-Inspred Computaton from 016. He served as a Guest-Edtor for many journals ncludng the Internatonal Journal of Bo-Inspred Computaton, Operatonal Research, Memetc Computng, andfuture Generaton Computer Systems. Yng Tan (SM 0) receved the Ph.D. degree from Southeast Unversty, Nanjng, Chna, n He s a Full Professor and the Ph.D. Advsor wth the School of Electroncs Engneerng and Computer Scence, Pekng Unversty, Bejng, Chna. He s the Inventor of freworks algorthm. Hs current research nterests nclude swarm ntellgence, machne learnng, and data mnng and ther applcatons. Dr. Tan was a recpent of the nd-class Natural Scence Award of Chna n 009. He serves as the Edtor-n-Chef of the Internatonal Journal of Computatonal Intellgence and Pattern Recognton and the Assocate Edtor of the IEEE TRANSACTIONS ON CYBERNETICS, the IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, andtheinternatonal Journal of Swarm Intellgence Research. He served as an Edtor of Sprnger s LNCS for over 0 volumes and a Guest Edtor of several referred journals, ncludng Informaton Scences, Soft Computng, Neurocomputng, Natural Computng, and the IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS. He has been the Foundng General Char of the seres Internatonal Conference on Swarm Intellgence snce 010, and the Jont General Char of the frst and second BRICS CCI, and the 014 IEEE WCCI Program Commttee Co-Char.