Research Article Simulated Annealing-Based Krill Herd Algorithm for Global Optimization

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1 Hndaw Publshng Corporaton Abstract and Appled Analyss Volume 2013, Artcle ID , 11 pages Research Artcle Smulated Annealng-Based Krll Herd Algorthm for Global Optmzaton Ga-Ge Wang, 1,2 Lhong Guo, 1 Amr Hossen Gandom, 3 Amr Hossen Alav, 4 and Hong Duan 5 1 Changchun Insttute of Optcs, Fne Mechancs and Physcs, Chnese Academy of Scences, Changchun , Chna 2 Graduate School of Chnese Academy of Scences, Bejng , Chna 3 DepartmentofCvlEngneerng,UnverstyofAkron,Akron,OH ,U 4 Department of Cvl and Envronmental Engneerng, Engneerng Buldng, Mchgan State Unversty, East Lansng, MI 48824, U 5 School of Computer Scence and Informaton Technology, Northeast Normal Unversty, Changchun , Chna Correspondence should be addressed to Lhong Guo; guolh@comp.ac.cn Receved 27 December 2012; Accepted 1 Aprl 2013 Academc Edtor: Mohamed Tawhd Copyrght 2013 Ga-Ge Wang et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Recently, Gandom and Alav proposed a novel swarm ntellgent method, called krll herd (), for global optmzaton. To enhance the performance of the method, n ths paper, a new mproved meta-heurstc smulated annealng-based krll herd () method s proposed for optmzaton tasks. A new krll selectng (KS) operator s used to refne krll behavor when updatng krll s poston so as to enhance ts relablty and robustness dealng wth optmzaton problems. The ntroduced KS operator nvolves greedy strategy and acceptng few not-so-good solutons wth a low probablty orgnally used n smulated annealng (). In addton, a knd of eltsm scheme s used to save the best ndvduals n the populaton n the process of the krll updatng. The merts of these mprovements are verfed by fourteen standard benchmarkng functons and expermental results show that, n most cases, the performance of ths mproved meta-heurstc method s superor to, or at least hghly compettve wth, the standard and other optmzaton methods. 1. Introducton In management scence, mathematcs, and economcs, the process of optmzaton s the selecton of the best soluton from some set of feasble alternatves. More generally, optmzaton conssts of fndng the optmal values of some objectve functon wthn a gven doman. In general, a great many optmzaton technques have been developed and appled to solve optmzaton problems [1]. A general classfcaton way for these optmzaton technques s consderng the nature ofthesetechnques,andtheseoptmzatontechnquescan be categorzed nto two man groups: determnstc methods and modern ntellgent algorthms. Determnstc methods usng gradent such as hll clmbng follow a rgorous step and wll repeat the process of optmzaton f the teratons start wth the same ntal startng pont. Eventually, they wll reach the same set of solutons. On the other hand, modern ntellgent algorthms wthout adoptng gradent always have some randomness, and the process of optmzaton cannot be repeatable even wth the same ntal value. However, generally, the fnal solutons, though slghtly dfferent, wll arrve at the same optmal values wthn a gven accuracy [2]. The growth of stochastc optmzaton methods as a blessng from the mathematcal and computng theorem has opened up a new facet to complete the optmzaton of a functon. Recently, nature-nspred metaheurstc methods perform effcently and effectvely n solvng modern nonlnear numercal global optmzaton problems. To some extent, all metaheurstc methods make an attempt at makng balance between dversfcaton (global search) and ntensfcaton (local search) [2, 3]. Inspred by nature, these strong metaheurstc methods have ever been appled to solve a varety of complcated problems, such as task-resource assgnment [4], constraned

2 2 Abstract and Appled Analyss optmzaton [5], test-sheet composton [6], and water, geotechncal, and transport engneerng [7, 8]. All the problems that need to fnd extreme value could be solved by these optmzaton methods. These types of metaheurstc methods use a populaton of solutons and always obtan optmal or suboptmal solutons. Durng the 1960s and 1970s, computer researchers nvestgated the possblty of formulatng evoluton as an optmzaton method and ths eventually generated a subset of gradent-free methods, namely genetc algorthms (s) [2, 9].Inthelasttwentyyears,agreatnumberofclasscal optmzaton technques have been developed on functon optmzaton, such as bat algorthm () [10], bacteral foragng optmzaton [11], bogeography-based optmzaton (BBO) [12 14], modfed Lagrangan method [15], artfcal plant optmzaton algorthm (APOA) [16], artfcal physcs optmzaton [17], dfferental evoluton () [18, 19], genetc programmng [20], partcle swarm optmzaton () [21 25], cuckoo search () [26], and, more recently, the krll herd () method [27] that s nspred by the herdng behavor of krll ndvduals n nature [2]. Frstly presented by Gandom and Alav n 2012, based on the smulaton of the herdng behavor of krll ndvduals, algorthm s a novel metaheurstc search approach for optmzng possbly nondfferentable and nonlnear functons n contnuous space [2, 27]. In, the objectve functon for the krll moton s manly nfluenced by the mnmum dstances of each krll ndvdual from food and from hghest densty of the herd. The poston of the krll conssts of three man motons: () foragng moton, () movement led by other ndvduals, and () random physcal dffuson. algorthm does not need dervatve nformaton, because t uses a stochastc search nstead of a gradent search used n determnstc methods. Furthermore, comparng wth other metaheurstc approaches, ths new method requres few control parameters, n effect, only a sngle parameter Δt (tme nterval), to adjust, whch makes mplement smply and easly, and s well ftted for parallel computaton [2]. s an effcent algorthm n exploraton but at tmes t may trap nto some local best values so that t cannot search globally well [2]. For, the search only reles on random walks, so there s no guarantee for a fast convergence [2]. To mprove for solvng complcated optmzaton problems, a few technques have been ntroduced nto the standard method [2, 28],whchncreasethedverstyof populaton and greatly enhance the performance of the basc method. On the other hand, nspred by the annealng process, smulated annealng () algorthm [29] saprobablstc meta-heurstc search method. Dfferently form most ntellgent algorthm, s a trajectory-based optmzaton technque [30]. Frstly presented here, a novel metaheurstc method based on and s orgnally proposed n ths paper, wth the am of acceleratng convergence speed, thus makng the approach more feasble for a wder range of practcal applcatons wthout losng the attractve merts of the canoncal approach.in,frstly,weuseastandardmethod to select a good canddate soluton set. And then, a new krll selectng (KS) operator s ntroduced nto the basc method. The ntroduced KS operator nvolves greedy strategy and acceptng a not-so-good soluton wth a low probablty orgnally used n smulated annealng () [3]. Ths greedy strategy s appled to decde whether a good canddate soluton s accepted so as to mprove ts effcency and relablty for solvng global numercal optmzaton problem. Furthermore, to mprove the exploraton of and evade premature convergence, the concept of acceptance probabltes n KS operator s ntroduced nto the method through acceptng a not-so-good soluton wth a low probablty n place of prevous soluton n order to enhance the adversty of the populaton.fourteenstandardbenchmarkfunctons,whch have ever been appled to verfy optmzaton algorthms n contnuous optmzaton problems, are used to evaluate our proposed method. Expermental results show that the performs more effcently and accurately than basc,,,,,,,,,,, and. The structure of ths paper s organzed as follows. Secton 2 gves a bref descrpton of basc and algorthms. Our proposed method s descrbed n detal n Secton 3. Subsequently,comparngwth,,,,,,,,,, and, the merts of our method are verfed by 14 benchmark functons n Secton4. Fnally, Secton 5 provdes the concluson and proposals for future work. 2. Prelmnary At frst, n ths secton we wll provde a background on the krllherdandsmulatedannealngalgorthmnbref Krll Herd Algorthm. Krll herd () [2, 27] sanovel metaheurstc search method for optmzaton tasks, whch mmcs the herdng of the krll swarms n response to specfc bologcal and envronmental processes. The poston of an ndvdual krll n search space s manly nfluenced by three motons descrbed as follows [2]: () foragng acton, () movement nfluenced by other krll ndvduals, () physcal dffuson. In method, the followng Lagrangan model n a d-dmensonal decson space s used as shown n the followng: dx =F dt +N +D, (1) where F, N,andD are the foragng moton, the moton nduced by other krll ndvduals, and the random physcal dffuson of the krll [27]. The foragng moton ncludes two components that are the current food locaton and the prevous experence about the food locaton. For the krll, thsmotoncanbeformulated as follows [2]: F =V f β +ω f F old, (2) where β =β food +β best (3)

3 Abstract and Appled Analyss 3 and V f s the foragng speed, ω f s the nerta weght of the foragng moton n (0, 1), and F old s the last foragng moton. β food s the food attractve and β best s the effect of the best ftness of the th krll so far [2, 27]. In movement nduced by other krll, the drecton of moton nduced, α,sapproxmatelycalculatedbythefollowng three factors: local effect (a local swarm densty), target effect (the target swarm densty), and repulsve effect (a repulsve swarm densty). For a krll, ths movement can be expressed as follows [2, 27]: N new =N max α +ω n N old (4) and N max s the maxmum nduced speed, ω n s the nerta weght of the moton nduced n [0, 1], andn old s the last moton nduced [2]. For the krll ndvduals, the physcal dffuson can be looked on as a random process. Ths moton can be decded by a maxmum dffuson speed and a random drectonal vector. It can be provded as follows [2]: D =D max δ, (5) where D max s the maxmum dffuson speed, and δ s the random drectonal vector whose values are random numbers n [ 1, 1] [2]. Based on the three prevously mentoned movements, usng dfferent parameters of the moton durng the tme, the poston of the thkrlldurngthentervalt to t+δts expressed by [2]: X (t+δt) =X (t) +Δt dx dt. (6) Note that Δt s one of the most sgnfcant constants and should be fne-tuned n terms of the gven real-world optmzatonproblem [2]. Ths s because ths parameter can be consdered as a scale factor of the speed vector. More detals about the three actons and approach can be found n [2, 27] Smulated Annealng. Smulated annealng () algorthm s a stochastc search technque that orgnated from statstcal mechancs. The method s nspred by the annealng process of metals. In the annealng process, a metal s heated to a hgh temperature and gradually cooled to a low temperature that can crystallze. As the heatng process lets the atoms travel randomly, f the coolng s done slowly enough,sotheatomshaveenoughtmetoadjustthemselves so as to reach a mnmum energy state. Ths analogy can be appled n functon optmzaton wth the state of metal correspondng to the possble and the mnmum energy state beng the fnal best soluton [30]. The method repeats a neghbor generaton procedure and follows search paths that mnmze the objectve functon value. When explorng search space, the provdes the possbltyofacceptngworsegeneratngsolutonsnaspecal manner n order to avod trappng nto local mnma. More precsely, n each generaton, for a current soluton X whose value s f(x),aneghborx s chosen from the neghborhood of X denoted by N(X). For each step, the objectve dfference Δ=f(X ) f(x). X could be accepted wth a probablty calculated by [30] P s = exp ( Δ ). (7) T And then, ths acceptance probablty s compared to a random number r (0, 1) and X s accepted whenever p>r. T s temperature controlled by a coolng scheme [30]. The method ncludes specfc factors: a neghbor generaton move, objectve functon calculaton, a method for assgnng the ntal temperature, a procedure to update the temperature, and a coolng scheme ncludng stoppng crtera [30]. 3. Our Approach: For the orgnal approach, as the search only reles on random walks, a rapd convergence cannot always be guaranteed. To mprove ts performance, genetc reproducton mechansms have added to the approach [27]. It has been demonstrated that, comparng other approaches, II ( wth crossover operator) performed the best. In effect, makes full use of the three motons n the populaton and t s shown that the performs well n both convergence speed and fnal accuracy on unmodal problems and most smple multmodal problems. However, once n a whle, n a rough regon of the ftness landscape, cannot always succeed n fndng better solutons on some complex problems. In our present study, n order to further mprove the performance of, a modfed greedy strategy and mutaton scheme, called krll selectng (KS) operator, s ntroduced nto the method to desgn a novel smulated annealng-based krll herd () algorthm. The ntroduced KS operator s nspred from the classcal smulated annealng algorthm. That s to say, n our work, the physcal property of metal s added to the krll to produce a type of super krll that s able to perform the KS operator. The dfference between and s that the KS operator s appled to only accept the basc generatng new better soluton for each krll nstead of acceptng all the krll updatng adopted n. Ths s rather greedy. The standard s very effcent and powerful, but the solutons have slght changes as the optma are approachng n the later run phase of the search. Therefore, to evade premature convergence and further mprove the exploraton abltyofthe,ksoperatoralsoacceptsfewnot-so-good krll wth a low acceptance probablty p as new soluton. Ths probablty p s also called transton probablty. Ths acceptance probablty technque can ncrease dversty of thepopulatonnanefforttoavodprematureconvergence and explore a large promsng regon n the pror run phase to search the whole space extensvely. The man step of KS operator adopted n method s gven n Algorthm 1. In Algorthm 1,tobegnwth,thetemperaturesupdated accordng to T=α T. (8)

4 4 Abstract and Appled Analyss Begn End. T=α T; Δf = f(x ) f(x ); % Accept f mproved If ( Δf > f n ) then do X +1 =X ; end f % Accept wth a low probablty f not mproved If (Δf <= f n &exp( Δf/(k T)) >r) then do X +1 =X ; end f Algorthm 1: Krll selectng (KS) operator. Here, T s the temperature for controllng the acceptance probablty p. And then,the change of the objectve functon value Δf s computed by (9) Δf = f (X ) f(x ). (9) Here, X s new generatng krll for krll by three motons n basc. Whether or not we accept a change, we usually use aconstantnumberf n as a threshold. If Δf > f n,thenewly generatng krll X s accepted as ts latest poston for krll. Otherwse, when (10)strue,thenewlygeneratngkrllX s also accepted as ts latest poston for krll p=exp ( Δf )>r. (10) k T Here, r s random number drawn from unform dstrbuton n (0, 1). k s Boltzmann s constant. For smplcty wthout losng generalty, we use k=1n our present study. In, the crtcal operator s the KS operator that comes from algorthm, whch s smlar to the LLF operator used n L [2]. The core dea of the proposed KS operator s based on two consderatons. Frstly, good solutons can make themethodconvergefaster.secondly,theksoperatorcan sgnfcantly mprove the exploraton of the new search space. In, at frst, standard method wth hgh convergence speed s used to shrnk the search regon to a more promsng area. And then, KS operator wth great greedy ablty s appled to only accept better solutons to mprove the qualty of the whole populaton. In ths way, method can explore the new search space wth and extract optmal populaton nformaton by KS operator. In addton, transton probablty p n KS operator s appled to accept few nonmproved krll wth a low acceptance probablty p n an effort to ncrease dversty of the populaton and evade premature convergence. In addton, except krll selectng operator, another vtal mprovement s the addton of eltsm strategy nto the approach. Undoubtedly, both and have some basc eltsm. However, t can be further enhanced. As wth other optmzaton approaches, we ntroduce some sort of eltsm wth the am of holdng some optmal krll n the populaton. Here, a more ntensve eltsm on the optmal krll s appled, whch can stop the optmal krll from beng spoled by three motons and krll selectng operator. In, at frst, the KEEP optmal krll are memorzed n a vector KEEPKRILL.In the end, the KEEP worst krll are replaced by the KEEP stored optmal krll. Ths eltsm strategy always has a guarantee that the whole populaton cannot declne to the populaton wth worseftness.notethat,n,aneltsmstrategysappled to keep some excellent krll that have the optmal ftness, so even f three motons and krll selectng operator corrupts ts correspondng krll, we have memorzed t and can recover to ts prevous good status f needed. Byntegratngprevouslymentonedkrllselectngoperator and ntensve eltsm strategy nto basc approach, the has been desgned that can be presented n Algorthm 2. Here, NP s the sze of the parent populaton P. 4. Smulaton Experments In ths secton, the effectveness of our proposed method s tested to global numercal optmzaton through a number of experments mplemented n standard benchmark functons. To allow an unbased comparson of tme requrements, all the experments were conducted n the same hardware and software envronments wth [2]. Well-defned problem sets are ftted for verfyng the performance of optmzaton approaches proposed n ths paper. Based on mathematcal expressons [31], benchmark functons can be consdered as objectve functons to mplement such tests. In our present work, fourteen dfferent benchmark functons are used to evaluate our proposed metaheurstc method. The formulaton of these benchmark functons and ther propertes can be found n [12, 32]. It s worth pontng out that, n [32], Yao et al. have ever used 23 benchmarks to test optmzaton approaches. However, for the other low-dmensonal benchmark functons (d = 2, 4, and 6), all the methods perform slghtly dfferently wth each other [33], because these low-dmensonal benchmarks aresosmplethattheycannotdstngushtheperformance dfference among dfferent approaches. Thus, n our study, only fourteen hgh-dmensonal benchmarks are appled to verfy our proposed method [2].

5 Abstract and Appled Analyss 5 Begn Step 1: Intalzaton. Set the generaton counter t, the populaton P of NP krll, the foragng speed V f, the dffuson speed D max, and the maxmum speed N max, temperature T 0,Boltzmannconstantk,coolngfactorα, and an acceptance threshold number f n, and eltsm parameter KEEP. Step 2: Ftness calculaton. Calculate the ftness for each krll based on ther ntal poston. Step 3: Whle t<maxgeneraton do Sort all the krll accordng to ther ftness. Store the KEEP best krll. for =1:NP(all krll) do Implement three moton as descrbed n Secton 2.1. Update poston for krll by krll selectng operator n Algorthm 1. Calculate the ftness for each krll based on ts new poston X +1. end for Substtute the KEEP best krll for thekeep worst krll. Sort all the krll accordng to ther ftness and fnd the current best. t=t+1; Step 4: end whle Step 5: Output the best soluton. End. Algorthm 2: Smulated annealng-based krll herd algorthm General Performance of. In ths secton, the performance of approach on global numerc optmzaton problem wth eleven optmzaton methods was compared so as to look at the merts of. The eleven optmzaton methods are (artfcal bee colony) [34], (bat algorthm) [10], (cuckoo search) [35], (dfferental evoluton) [18], (evolutonary strategy) [36], (genetc algorthm) [9], (harmony search) [1, 37], [27], (probablty-based ncremental learnng) [38], (partcle swarm optmzaton) [21, 39, 40], and [29]. More nformaton about these comparatve methods can be found n [2]. Besdes, we must pont out that, n [27], comparng all the algorthms, the expermental results show that the performance of II was the best whch proves the superorty of the approach. Consequently, n our present study, we use II as basc method. In the followng experments, the same parameters for, and are adopted, whch are the foragng speed V f = 0.02, the maxmum dffuson speed D max = 0.005, the maxmum nduced speed N max = 0.01, ntaltemperature T 0 = 1.0, maxmum number of accept Accept max = 15, Boltzmann constant k = 1,coolngfactorα = 0.95, and an acceptance threshold number f n = 0.01 (only for ). For other methods used here, ther parameters are selected as [2, 12, 41]. We set populaton sze NP = 50 and maxmum generaton Maxgen = 50 for each approach. We ran 100 Monte Carlo smulatons of each approach on each benchmark functon to get typcal performances [1]. The results of the experments are recorded n Tables 1 and 2. Table 1 llustrates theaveragemnmafoundbyeachapproach,averagedover 100 Monte Carlo runs. Table 2 llustrates the absolute best mnma found by each approach over 100 Monte Carlo runs. That s to say, Table 1 represents the average performance of each approach, whle Table 2 represents the best performance of each approach. The best value obtaned for each test problem s shown n bold. Note that the normalzatons n the tables are based on dfferent scales, so values are not comparatve between the two tables. For each functon used nourwork,therdmensons20. From Table 1, we see that, on average, s the most effectve at fndng objectve functon mnmum on eleven of the fourteen benchmarks (F01 F06, F08, F10, and F12 F14).,, and are the second most effectve, performng thebestonthebenchmark,f07,f11,andf09whenmultple runs are made, respectvely. Table 2 shows that performs the best on thrteen of the fourteen benchmarks whch are F01 F08 and F10 F14. s the second most effectve, performng the best on the benchmark F09 when multple runs are made. Moreover, the runnng tme of the twelve optmzaton approaches was slghtly dfferent. We collected the average CPU tme of the optmzaton methods as appled to the 14 benchmarks consdered n our work. The results are marked n Table 1. was the quckest optmzaton method. was the nnth fastest of the twelve approaches. However, t should be noted that n most real-world applcatons, t s the ftness functon evaluaton that s by far the most tmeconsumng part of an optmzaton approach. In addton, to prove the superorty of the proposed method n the future, convergence graphs of,,,,,,,,,,, and are also provded n ths secton. However, lmted by the length of paper, here only some most typcal benchmarks are llustrated n Fgures 1 7 whch mean the process of optmzaton. The functon values shown n Fgures 1 7 are the average objectve functon mnmum obtaned from 100 Monte Carlo smulatons, whch are the true objectve functon soluton, notnormalzed.bytheway,notethattheglobaloptmaof the benchmarks (F04, F05, F08, F10, and F14) are llustrated

6 6 Abstract and Appled Analyss Table 1: Mean normalzed optmzaton results n fourteen benchmark functons. F F F F04 1.2E6 5.8E7 3.3E3 1.9E5 2.8E7 3.8E5 4.2E7 1.3E4 6.0E7 4.1E F05 3.0E7 8.3E8 1.2E6 1.0E7 4.2E8 1.5E7 6.4E8 1.1E6 8.2E8 9.6E F06 3.6E6 6.3E7 4.5E5 1.3E6 4.5E7 3.5E6 4.2E7 4.2E5 5.3E7 1.0E7 5.4E F F F F F F F13 2.7E3 1.4E E3 1.5E4 5.0E3 1.3E E4 5.4E F E E E Tme Total The values are normalzed so that the mnmum n each row s These are not the absolute mnma found by each algorthm, but the average mnma found by each algorthm. Table 2: Best normalzed optmzaton results n fourteen benchmark functons. F F F F E E4 9.1E E E8 3.9E F05 5.4E6 5.6E8 4.5E4 8.5E6 7.9E8 1.5E5 1.1E9 7.4E5 2.0E9 9.2E F06 7.5E6 5.3E8 3.3E6 1.1E7 9.1E8 8.8E6 5.0E8 3.7E6 1.1E9 9.7E7 2.2E F F F F E E3 1.1E E F F F13 6.3E3 3.9E4 2.6E3 5.1E3 6.9E4 1.4E4 5.5E E4 1.3E F E E E E3 2.1E Total The values are normalzed so that the mnmum n each row s These are the absolute best mnma found by each algorthm. n the form of the semlogarthmc convergence plots. We use short for II n the legend of Fgures 1 7 and next texts. Fgure 1 showstheresultsobtanedforthetwelvemethods when the F01 Ackley functon s appled. From Fgure 1, clearly, we can draw the concluson that s sgnfcantly superor to all the other algorthms durng the process of optmzaton. For other algorthms, although slower, eventually fnds the global mnmum close to, whle,,,,,,,, and fal to search the global mnmum wthn the lmted teratons. Here, all the algorthms show the almost same startng pont; however outperforms them wth fast and stable convergence rate. Fgure 2 showstheresultsforf04penalty#1functon. From Fgure 2, although slower later, shows a fastest convergence rate ntally among twelve methods; however, t s outperformed by after 6 generatons. Furthermore, outperforms all other methods durng the whole progress of optmzaton n ths multmodal benchmark functon. Eventually, performs the second best at fndng the global mnmum. and perform the thrd and the fourth best at fndng the global mnmum wth a relatvely slow and stable convergence rate. Fgure 3 shows the performance acheved for F05 Penalty #2 functon. For ths multmodal functon, smlar to the F04 Penalty #1 functon as shown n Fgure 2,ssgnfcantly

7 Abstract and Appled Analyss Benchmark functon value Benchmark functon value Number of generatons Fgure 1: Comparson of the performance of the dfferent methods for the F01 Ackley functon Number of generatons Fgure 3: Comparson of the performance of the dfferent methods for the F05 Penalty Benchmark functon value Benchmark functon value Number of generatons Fgure 2: Comparson of the performance of the dfferent methods for the F04. superor to all the other algorthms durng the process of optmzaton. performs the second best at fndng the global mnmum. For other algorthms, the fgure shows that there s lttle dfference between the performance of and. However, carefully studyng Table 1 and Fgure 3,wecan conclude that performs slghtly better than n ths multmodal functon Number of generatons Fgure 4: Comparson of the performance of the dfferent methods for the F08 Rosenbrock functon. Fgure 4 shows the results for F08 Rosenbrock functon. From Fgure 4, very clearly, has the fastest convergence rate at fndng the global mnmum and sgnfcantly outperforms all other approaches. Lookng carefully at Fgure 4, s only nferor to and performs the second best n ths unmodal functon. In addton, and perform very

8 8 Abstract and Appled Analyss Benchmark functon value Benchmark functon value Number of generatons 10 1 Number of generatons Fgure 5: Comparson of the performance of the dfferent methods for the F10. Fgure 7: Comparson of the performance of the dfferent methods for the F14 Step functon. Benchmark functon value Number of generatons Fgure 6: Comparson of the performance of the dfferent methods for the F12 Schwefel. well and have ranks of 3 and 4, respectvely. Furthermore, hasafastconvergencentallytowardstheknown mnmum; however, t s outperformed by after 7 generatons.,,,,,,, and do not manage to succeed n ths benchmark functon wthn maxmum number of generatons, showng a wde range of obtaned results. Fgure 5 shows the results for F10 Schwefel 1.2 functon. From Fgure 5, we can see that performs far better than other algorthms durng the optmzatonprocess n ths relatve smple unmodal benchmark functon. shows a faster convergence rate ntally than ; however, t s outperformed by after 5 generatons. and perform very well and rank 3 and 4, respectvely. Fgure 6 showstheresultsforf12schwefel2.21functon. Veryclearly,hasthefastestconvergencerateatfndng the global mnmum and sgnfcantly outperforms all other approaches.forotheralgorthms,,,andperform very well and have ranks of 2, 3 and 4, respectvely. Partcularly, s only nferor to and eventually converges to the value that s very close to C. Fgure 7 showstheresultsforf14stepfuncton.apparently, shows the fastest convergence rate at fndng the global mnmum and sgnfcantly outperforms all other approaches. Here, all the algorthms show the almost same startng pont; however outperforms the other algorthms wth fast and stable convergence rate. For other algorthms, performs the second best among 12 methods. Furthermore,,,,,,,,,,and do not manage to succeed n ths benchmark functon wthn maxmum number of generatons, showng a wde range of obtaned results. From prevous analyses about Fgures 1 7, we can arrve at a concluson that our proposed metaheurstc method sgnfcantly outperforms the other eleven approaches. In general, s only nferor to and performs the second best among twelve methods. and perform the thrd

9 Abstract and Appled Analyss 9 best only nferor to the and. Furthermore, the llustraton of benchmarks F04, F05, F08, and F10 shows that has a faster convergence rate ntally, whle later t converges slower and slower to the true objectve functon value Dscusson. For all the benchmark functons consdered here, t has been demonstrated that the proposed method s superor to, or at least hghly compettve wth, the standard and other eleven great state-of-the-art populaton-based methods. The advantages of nvolve mplementng smply and easly and have few parameters to regulate.theworkcarredouthereprovesthetobe effectve, robust, and effcent over a varety of benchmark functons. Benchmark evaluaton s a good way for testng the effectveness of the metaheurstc methods, but t also has some lmtatons [2]. Frst, we made lttle effort to carefully adjust the optmzaton methods used n ths paper. In most cases, dfferent adjustng parameters n the optmzaton methods mght generate great dfferences n ther performance. Second, real-world optmzaton problems may not have much of a relatonshp wth benchmark functons. Thrd, benchmark evaluaton may reach sgnfcantly dfferent conclusons f the gradng crtera or problem setup change. In our work, we securtzed the average and best results acheved wth some populaton sze and after some number of teratons. However, we mght arrve at dfferent conclusons f we change the populaton sze, or consder how many populaton szestneedstoreachacertanfunctonvalueorfwemodfy the teraton. Regardless of these caveats, the expermental results obtaned here are promsng for and show that thsnovelmethodmghtbeabletofndancheamongthe plethora of optmzaton approaches [2]. It s worth pontng out that tme requrement s a lmtaton to the mplementaton of most optmzaton methods. If an approach does not has a fast converge, t wll be nfeasble, snce t would take much tme to get an optmal or suboptmal soluton. seems not to requre an mpractcal amount of CPU tme; of the twelve optmzaton approaches dscussed n ths paper, was the nnth fastest. How to speed up the s convergence s stll worthy of further study. In our study, 14 benchmark functons are appled to prove the merts of our method; we wll verfy our proposed method on more optmzaton problems, such as the hghdmensonal (d 20) CEC 2010 test sut [42] andthe real-world engneerng problems. Moreover, we wll compare wth other optmzaton methods. In addton, we only look at the unconstraned functon optmzaton here. Our future work ncludes addng more optmzaton technques nto for constraned optmzaton problems, such as constraned real-parameter optmzaton CEC 2010 test sut [43]. 5. Concluson and Future Work Due to the lmted performance of on complex problems [2], KS operator has been ntroduced to the standard to develop a novel mproved metaheurstc optmzaton method based on and, called smulated annealngbased krll herd () algorthm, for optmzaton problem. In, frstly, we use a standard algorthm to select a good canddate soluton set. And then, a new krll selectng (KS) operator s ntroduced nto the basc method. The ntroduced KS operator ncludes greedy strategy and acceptng a not-so-good soluton wth a small probablty. Ths greedy strategy s appled to accept a good canddate soluton so as to mprove ts effcency and relablty for solvng global numercal optmzaton problem. Furthermore, theksoperatornotonlyacceptschangesthatmprovethe objectve functon, but also keeps some changes n not-sogood solutons wth a low probablty that are not deal. Ths can enhance the adversty of the populaton, mprove the exploraton of, and evade premature convergence. Furthermore, we can see that ths new method can speed up the global convergence rate wthout losng the strong robustness of the basc. From the analyses of the expermental results, we observe that the greatly mproves the relablty of the global optmalty and they also mprove the qualty of the solutons on unmodal and most multmodal problems. In addton, the s smple and mplements easly. In the feld of numercal optmzaton, there are numerous ssues that deserve further scrutny. Our future work wll emphasze the two ssues. On the one hand, we would employ our proposed method to solve real-world engneerng optmzaton problems, and, obvously, can be a great method for real-world engneerng optmzaton problems. On the other hand, we would develop more new metaheurstc methods to solve optmzaton problems more accurately and effcently [2]. Acknowledgments Ths work was supported by State Key Laboratory of Laser Interacton wth Materal Research Fund under Grant no. 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