Self-Organizing Migrating Algorithm with narrowing search space strategy for robot path planning

Applied So Compu ing 116 (2022) 108270 Con en s lis s a ailable a ScienceDi ec Applied So Compu ing jou nal homepage: www.else ie .com/loca e/asoc Sel -O ganizing Mig a ing Algo i hm wi h na owing sea ch space s a egy o obo pa h planning Quoc Bao Diep a,∗, Thanh Cong T uong b, Swaga am Das c, I an Zelinka d,a aVSB - Technical Uni e si y o Os a a, Facul y o Elec ical Enginee ing and Compu e Science, Os a a, Czech Republic bUni e si y o Finance - Ma ke ing, Ho Chi Minh Ci y, Vie Nam cIndian S a is ical Ins i u e, Elec onics and Communica ion Sciences Uni , Kolka a, India dTon Duc Thang Uni e si y, Facul y o Elec ical & Elec onics Enginee ing, Ho Chi Minh Ci y, Vie Nam a icle in o A icle his o y: Recei ed 27 June 2020 Recei ed in e ised o m 28 Oc obe 2021 Accep ed 29 No embe 2021 A ailable online 9 Decembe 2021 Keywo ds: Sel -O ganizing Mig a ing Algo i hm Op imiza ion algo i hm Swa m in elligence Nume ical op imiza ion Pa h planning D one abs ac This a icle in oduces a e sion o he Sel -O ganizing Mig a ing Algo i hm wi h a na owing sea ch space s a egy named iSOMA. Compa ed o he p e ious wo e sions, SOMA T3A and Pa e o ha anked 3 d and 5 h espec i ely in he IEEE CEC (Cong ess on E olu iona y Compu a ion) 2019 com- pe i ion, he iSOMA is equipped wi h mo e ad anced ea u es wi h no able imp o emen s including applying jumps in he o de , immedia e upda e, na owing he sea ch space ins ead o sea ching on he in e sec ing edges o hype planes, and he pa ial eplacemen o indi iduals in he popula ion when he global bes imp o ed no u he . Mo eo e , he p oposed algo i hm is o ganized in o p ocesses named ini ializa ion, sel -o ganizing, mig a ing, and eplacemen . We es ed he pe o mance o his new e sion by using h ee benchma k es sui es o IEEE CEC 2013, 2015, and 2017, which, oge he con ain a o al o 73 unc ions. No only is i supe io in pe o mance o o he SOMAs, bu iSOMA also yields p omising esul s agains he ep esen a i es o well-known algo i hmic amilies such as Di e en ial E olu ion and Pa icle Swa m Op imiza ion. Mo eo e , we demons a e he applica ion o iSOMA o pa h planning o a d one, while a oiding s a ic obs acles and ca ching he a ge . ©2021 The Au ho (s). Published by Else ie B.V. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/). 1. In oduc ion Wi h he con inuous de elopmen o science and echnology, many p ac ical p oblems a ise challenging and mos o hem can be ans o med in o op imiza ion p oblems [1]. The Swa m In elligence (SI) is one o he mos e ec i e and well-a ended me hods o ind he global op imal solu ion o such p oblems, such as Di e en ial E olu ion (DE) [2,3], A i icial Bee Colony (ABC) [4,5], Pa icle Swa m Op imiza ion (PSO) [6,7], and Sel - O ganizing Mig a ing Algo i hm (SOMA) [8,9] ha is a subjec o he epo ed esea ch he e. P oposed in he 2000s, SOMA,1a ep esen a i e o he SI, is a popula ion-based op imiza ion algo i hm, which mimics he compe i ion–coope a ion beha io among indi iduals in he pop- ula ion o c ea u es o ind he op imal solu ion. O e many mig a ion loops, he ini ial candida e solu ions a e op imized, making hese solu ions be e and be e o e ime. Wi h a non-g adien -based mechanism and lexibili y p ope y, i.e., sol - ing complex unc ions wi hou using complex ma h equa ions, ∗Co esponding au ho . E-mail add esses: [email p o ec ed] (Q.B. Diep), [email p o ec ed] (T.C. T uong), [email p o ec ed] (S. Das), [email p o ec ed], [email p o ec ed] (I. Zelinka). 1h p://somaalgo i hm.com/. SOMA demons a es i s ou s anding pe o mance and is applied in many di e en ields such as he eliabili y– edundancy alloca- ion p oblem [10], S a C a : B ood Wa — compu e games [11, 12], and d i e obo s o a oid dynamics obs acles [13,14]. On he one hand, eal-wo ld p oblems a e eme ging mo e and mo e complex, equi ing no only as eal- ime compu a ions bu also high accu acy o esul s and capable o escape om local aps. The canonical e sions o he SOMA algo i hm ha e been somewha less pe o mance agains hese issues. On he o he hand, i equi es simplici y, ease o p og amming, and ease o use o many di e en applica ion a eas. Besides, he adap a ion o he con ol pa ame e s o he algo i hm is equi ed, because no all applica ion de elope s a e expe s in he ield o he op imiza ion algo i hm. The e o e, imp o ing he algo i hm o sa is y hese illus a ion equi emen s men ioned abo e is essen ial. Many imp o ed e sions ha e been p oposed o boos up he pe o mance o he algo i hm and o e come some limi a- ions a ising du ing he applica ion p ocess, such as sel -adap ing SOMA [15], C-SOMAQI [16], he e sion o he leade selec ion in he SOMA [17], SOMA wi h non-bina y pe u ba ion [18], and sel -adap i e pa ame e s o SOMA [19]. In pa icula , he wo la es e sions, eam o eam adap i e — SOMA T3 A [20,21] and Pa e o-based SOMA [22,23], ha e made g ea s ides, hold- ing 3 d ( he same anking wi h HyDE-DF [24]) and 5 h ou o h ps://doi.o g/10.1016/j.asoc.2021.108270 1568-4946/©2021 The Au ho (s). Published by Else ie B.V. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/). Q.B. Diep, T.C. T uong, S. Das e al. Applied So Compu ing 116 (2022) 108270 38 algo i hms pa icipa ing in he 100-Digi Challenge espec- i ely, which epo ed in [25] including esul s om he 2019 Cong ess on E olu iona y Compu a ion (CEC 2019), he 2019 Gene ic and E olu iona y Compu a ion Con e ence (GECCO 2019) and he 2019 Swa m, E olu iona y and Meme ic Compu ing Con- e ence (SEMCCO 2019). F om SOMA 21 yea s his o y, i is isible ha his algo i hm belongs (based on a ious compa a i e s udies) o he mos e - icien SI and is also highly applicable o a ious p oblems o indus ial p ac ice as well as academic p oblems. Howe e , hese a e no he inal e sions o SOMA. They ha e p o en hei e ec i eness in he 100-Digi Challenge, which does no mean ha hose algo i hms will be ic o ious a all di e en es sui es. Fu he mo e, many eal-wo ld applica ions inc eas- ingly equi e algo i hms o be mo e e icien , sol e p oblems as e , and mo e accu a ely. Tha p omp ed us o de elop and p opose he nex gene a ion o SOMA T3 A, named iSOMA. And wha abou d ones? How o apply iSOMA in d ones? These ques ions will be answe ed in he nex pa o he a icle. I can be e ealed ha one o he mos impo an issues o d one applica ions is ca ching up wi h a ge s and a oiding mul iple obs acles. The es o he a icle is o ganized in o he ollowing sec ions. Sec ion 2desc ibes he p inciples o he SOMA algo i hm as well as i s s eng hs and weaknesses, which unde lie he p o- posed algo i hm. Sec ion 3p esen s he imp o ed e sion o he sel -o ganizing mig a ing algo i hm, iSOMA. Sec ion 4shows he expe imen se up. Compa ison esul s and discussion a e p e- sen ed in Sec ion 5. Applica ion o he iSOMA o pa h planning o d ones is p esen ed in Sec ion 6. Finally, he wo k is concluded in Sec ion 7. 2. The canonical SOMA 2.1. The p inciple Sel -O ganizing Mig a ing Algo i hm, a swa m-based in elli- gence op imiza ion algo i hm, wo ks based on he in e ac ion be ween indi iduals in he popula ion acco ding o a gi en ule o ind an op imal solu ion o he gi en p oblem [8,9]. The mech- anism cons i u ing he SOMA lies in how o selec indi iduals as a leade and mig an s, how mig an s mo e o he leade , as well as how o upda e be e indi iduals in o he popula- ion and elimina e he bad one. These p ocesses a e pe o med unde loops named mig a ion loops. Then, h ough many mig a- ion loops, hese solu ions a e becoming be e and be e han ini ial solu ions. This sec ion b ie ly desc ibes he p inciple o SOMA, shaping he basis o he analysis o SOMA’s s eng hs and weaknesses. A popula ion is gene a ed a he beginning o he algo i hm, con aining indi iduals as candida e solu ions o a gi en p oblem, acco ding o Eq. (1). Each a iable (dimension) o he p oblem has i s bounda y, which is also he sea ch ange o he algo i hm. They a e hen e alua ed by he gi en i ness unc ion and en e he i s mig a ion loop. P=x(lo) j+ and(x(hi) j−x(lo) j) (1) whe e: •P: he SOMA’s o iginal popula ion, •x(lo) j: he lowes limi alue, •x(hi) j: he highes limi alue, • and: a andom numbe , om 0 o 1. In each mig a ion loop, he indi idual wi h he lowes i ness alue in he popula ion is chosen as he leade , and he emaining indi iduals a e hose who a e a eling. They will jump by s ep owa d he leade using he S ep pa ame e (speci ied he g anu- la i y) be o e Pa hLeng h (a limi o dis ance) is eached. Ins ead o jumping di ec ly owa d he leade , ano he pa ame e is used o gene a e pe u ba ion mo es, named PRTVec o j, o cing he indi iduals o mo e in he N−ksubspace whe e each pai is pe pendicula o he o iginal space, as shown in Eq. (2). i andj<PRT;PRTVec o j=1;else,0.(2) The p obabili y o each mo e is de e mined by he PRT pa- ame e . A numbe is andomly gene a ed and compa ed o his h eshold. I i is less han PRT, he jump in ha dimension is pe o med, and ice e sa. This helps o main ain he di e si y o he popula ion while c ea ing be e new indi iduals. The Eq. (3) desc ibes his mo ing p ocess. xML+1 n,j=xML c,j+(xML l,j−xML c,j) PRTVec o j(3) whe e: •xML+1 n,j: posi ion in he nex mig a ion loop, •xML c,j: he mig an posi ion in he cu en mig a ion loop, •xML l,j: he leade posi ion in he cu en mig a ion loop, • : mo ing s ep, om 0, by S ep, o Pa hLeng h. A e each indi idual comple es i s mo es, he bes posi ion in he mo ing ajec o y is chosen o compa e wi h he ini ial posi ion. This one will be eplaced by he be e new posi ion, o he wise, he algo i hm will skip he new posi ion and con inue he p ocess o he emaining indi iduals. A e all indi iduals ha e comple ed he jumping, a new mi- g a ion loop is s a ed, he new leade will be chosen again, and he mig a ion p ocess will con inue un il SOMA sa is ies he speci ied e mina ion condi ion. SOMA AllToOne (SOMA ATO) is he name o ha echnique. Ra he han all indi iduals mo ing owa d he leade , unde ano he app oach, all indi iduals mo e owa d each o he , ega dless o whe he he indi idual is be - e o wo se. SOMA AllToAll (SOMA ATA) is he name o his echnique. 2.2. Weakness o SOMA S opping C i e ia: As desc ibed in he p e ious subsec ion, a e each indi idual has comple ed his mo emen , he bes posi ion in his pa h is selec ed o compa ison wi h he o iginal. I is clea ha o ind a be e posi ion, he SOMA needs o call he cos unc ion many imes (known as unc ion e alua ions - FEs, each execu ion o he cos unc ion is conside ed one FE). Fo example, wi h he s anda d se ing o SOMA: Pa hleng h = 3.0 and S ep =0.11, each a eling indi idual has 27 di e en posi ions on i s jumping pa h, which means ha he SOMA has o spend 27FEs o e alua e hese posi ions o ind he be e one. Meanwhile, o he algo i hms only use one FE o imp o e hei candida e solu ions such as DE, PSO, and ABC, o some algo i hm- speci ic pa ame e -less op imiza ion echniques like Jaya [26–28] and Rao algo i hm [29]. This cha ac e is ic causes he algo i hm o soon ace he s op condi ion o maximum unc ion e alua ions (MaxFEs) and he p ema u e con e gence scena io be o e i can ine- une he solu ion in he exploi a ion phase a he end o he op imiza ion p ocess. Mo e on he Edge: On he o he side, in each a iable o he op imiza ion p ob- lem, PRTVec o jaccep s only one o he wo alues o 0 and 1. 2 Q.B. Diep, T.C. T uong, S. Das e al. Applied So Compu ing 116 (2022) 108270 Fig. 1. All possible posi ions in 2D space. Se ing pa ame e s: S ep =0.33 and Pa hLeng h =3.0. Fig. 2. All possible posi ions in 3D space. Se ing pa ame e s: S ep =0.33 and Pa hLeng h =3.0. The e o e, his a iable will be upda ed wi h a mul iple o S ep i he alue o PRTVec o jis 1 (and ice e sa i will emain in i s posi ion). Geome ically, his means ha a eling indi iduals will mo e on he in e sec ion edges o hype planes c ea ed by pai s o sides o a iables, as shown in Figs. 1 and 2. I can be isualized as a line-sea ch s a egy. Mo ing only on he edges o hype planes wi hou mo ing in o he inne space highly limi s he sea ching abili y o he SOMA and leads o he isk o missing ou on he po en ial sea ch space. Sea ch o Nonsense: Besides, one o he majo weaknesses o SOMA is ha non- sense mo es a e aken om be e indi iduals o he wo se one, as shown in Fig. 3. This leads o a was e o compu a ional ime because he algo i hm spends a lo o FEs on hese nonsense mo es. I causes he algo i hm o ace he isk o being s opped be o e inding he op imal solu ion o he gi en p oblem. 3. The p oposed algo i hm: iSOMA The name ‘‘Sel -O ganizing Mig a ing Algo i hm’’ co e s he algo i hm’s en i e ope a ion. Consequen ly, we di ide he algo- i hmic amewo k o iSOMA in o ou p ocesses and call hem he ini ializa ion p ocess, sel -o ganizing p ocess, mig a ing p o- cess, and eplacemen (upda e) p ocess. They wo k in mig a ion loops. Indi iduals om he ini ial popula ion will mig a e o each o he in each mig a ion loop o explo e p omising subspaces and hen exploi hese spaces o ind he global op imal solu ion. As seen in Fig. 4, hese p ocedu es we e epea ed un il he speci ied s op condi ions a e me . 3.1. The ini ializa ion p ocess The iSOMA s a s wi h he ini ializa ion p ocess. Wi hin he con ol pa ame e s es ablished, an ini ial popula ion o po en- ial solu ions is andomly gene a ed using uni o mly dis ibu ed Fig. 3. The meaningless mo e om he mig an o he leade has a lowe i ness alue. In his case, he e is no posi ion wi h be e i ness alue han he mig an i sel in he capabili y sea ch space. Fig. 4. The lowcha o he iSOMA. andom numbe s o sca e ini ial indi iduals in he whole gi en sea ch space, by applying Eq. (1). This popula ion is hen e alua ed by he gi en i ness unc ion. The global bes op imal solu ion ( he indi idual wi h he small- es i ness alue) is eco ded and he algo i hm en e s he i s mig a ion loop, as desc ibed in he nex subsec ion. 3.2. The sel -o ganizing p ocess The sel -o ganizing p ocess in he p oposed algo i hm is he p ocess o de e mining which indi iduals will mo e owa d hei a ge s (named as mig an s) and which one will become he a ge (named as leade ). In he canonical e sion o ATO, all indi iduals a e mig an s and he bes indi iduals in he popula- ion become he leade o each mig a ion loop. This esul s in limi a ions as analyzed in he p e ious sec ion. On he o he hand, i all indi iduals mo e owa d each o he as he ATA e sion, SOMA no only aces he s op condi ion o FEs due o he use o a lo numbe o jumpings aken place be ween bad indi iduals bu also aces he p ema u e con e gence scena io. To o e come he men ioned sho comings, he sel -o ganizing p ocess mus bo h ensu e he elimina ion o bad indi iduals and main ain he di e si y o he popula ion by a oiding only ocusing on he global bes one. Acco dingly, he iSOMA selec s he bes indi iduals in a g oup o mo e owa d he bes indi idual in ano he g oup. To implemen his p og ess, in each mig a ion loop, he iSOMA i s andomly selec s mindi iduals in he cu en popula ion as 3 Q.B. Diep, T.C. T uong, S. Das e al. Applied So Compu ing 116 (2022) 108270 Fig. 5. The sel -o ganizing p ocess. he subpopula ion and hen selec s he bes nou o m(n⩽m), which will become mig an s. Fo each mig an , he algo i hm an- domly selec s he second subpopula ion con aining kindi iduals in he cu en popula ion (can be o e lapped wi h indi iduals). The indi idual wi h he bes i ness alue ou o kbecomes he leade o ha mig an . In case he mig an coincides wi h he leade , he algo i hm will choose he second-bes indi idual in k o be he leade . Fig. 5 desc ibes his p ocess. Fo p oblems con aining many local aps, he alues o m,n, and kshould be small. In his con ex , many mo es a e pe o med be ween andom indi iduals, boos ing he explo ing abili y o he iSOMA in he sea ch space. On he con a y, o simple p oblems, he alues o m,n, and kshould be la ge o o ce he iSOMA o ocus on be e indi iduals, inc easing he exploi ing abili y on he p omising sea ched space. These pa ame e s highly im- pac he pe o mance o he algo i hm, besides he o he con ol pa ame e s will be p esen ed in he nex subsec ion. 3.3. The mig a ing p ocess The mig a ing p ocess egula es how he mig an mo es o- wa d he leade selec ed in he p e ious subsec ion. This mo e- men ype, in he canonical e sion, is a s aigh -line-sea ch s a - egy wi h do ed-line posi ions as depic ed in Figs. 1 and 2. To enhance he algo i hm’s sea ch capabili ies and es ic he men- ioned weakness, we p opose he ollowing imp o emen s o he mig a ing way: The o de o jumps: Ins ead o jumping g adually owa d he leade as in he canonical e sion, we p opose a me hod o jumping in o de , as shown in Fig. 6. Acco dingly, he i s posi ion o he mig an is o jump ‘‘behind’’ he leade . A e ha , he mig an g adually mo es owa d he leade speci ied by he gi en S ep. In o he wo ds, he mig an s a s om he a hes s ep by s ep app oaching i s ini ial posi ion. Immedia ely upda e: Ano he aluable imp o emen de i es om e mina ing he jumping p og ess o he cu en mig an and immedia ely upda - ing i s posi ion in he popula ion i he new posi ion is be e han he ini ial posi ion. I is execu ed by he algo i hm ha will e alua e he new posi ion ound du ing he mig an ’s mig a ion and compa e i o he ini ial. I will immedia ely eplace he ini ial and s op i s mig a ion, going o he nex mig an . Fig. 6. The o de o he jumps in he iSOMA. Se ing pa ame e s: S ep =0.3, Njump =10, and Pa hLeng h =3.0. Fig. 7. All possible posi ions o he o sp ing o e mig a ion loops. Se ing pa ame e s: S ep =0.33, Pa hLeng h =3.0, wi h adap i e PRTVec o . This imp o emen , inco po a ed wi h jumping in o de , no only makes he algo i hm spend ewe FEs o ge a be e posi- ion, bu also helps he popula ion p ese e di e si y, a oiding p ema u e con e gence scena ios. Na ow he sea ch space: Fig. 7 depic s he na owing o he sea ch space. In he ea ly s ages o he op imiza ion p og ess, he algo i hm should p e e o explo e p omising subspaces a he han ocus on exploi ing hem. Thus, indi iduals mo e on he edges o hype planes c ea ed by pai s o sides o a iables (speci ied by small PRT, esul ing in mo e PRTVec o jequals ze o). Towa d he end o op imiza ion p og ess, iSOMA is mo e in- clined o exploi hese p omising subspaces. The e o e, he adap- i e PRTVec o pa ame e is p oposed so ha indi iduals can mo e inside he space c ea ed by in e sec ion hype planes, in- s ead o jus mo ing on he edges like he canonical e sion. Eq. (4) is used o enable his ea u e. i andj<PRT;PRTVec o j=1;else,PRTVec o j=FEs MaxFEs.(4) whe e: •FEs: he cu en unc ion e alua ion, •MaxFEs: he maximum o unc ion e alua ions. Besides, he adap i e PRT pa ame e is le e aged in he iSOMA, which was in oduced in [20], gi en in Eq. (5). The PRT s a s wi h a numbe close o 0 and ends wi h a numbe close o 1 o a oid he meaningless compa ison o andj<PRT in Eq. (4). In his e sion, he S ep pa ame e is ixed. PRT =0.05 +0.90 FEs MaxFEs (5) 3.4. The eplacemen p ocess The p ocess is o eplace some indi iduals in he cu en popula ion wi h new ones. This is a necessa y p og ession o be 4 Q.B. Diep, T.C. T uong, S. Das e al. Applied So Compu ing 116 (2022) 108270 aken when he algo i hm canno ind a be e global op imal solu ion a e a ce ain amoun o sea ching ime which can be measu ed by he numbe o unc ion e alua ions. Acco dingly, a e se e al FEs, i he algo i hm does no dis- co e a be e posi ion han he global bes , he iSOMA will andomly eplace a ce ain pe cen age (10% o example) o he exis ing indi iduals in he cu en popula ion (excluding he global bes indi idual) by he same numbe o andomly gene - a ed indi iduals in he whole sea ch space (acco ding o Eq. (1)). Using andom indi iduals ins ead o eco ded his o ical indi id- uals ound du ing he sea ching p og ess p e en s he algo i hm om alling in o he cu en local aps. The p oposed iSOMA is desc ibed in Algo i hm 1. Algo i hm 1 : iSOMA 1: C ea e and e alua e he ini ial popula ion P 2: S o e he bes indi idual as he global bes one 3: while s op condi ion no eached do 4: Selec andom mindi iduals om P 5: Pick he bes nou o mindi iduals as mig an s 6: o i=1 o nmig an s do 7: Selec andom kindi iduals om P 8: Picked ou he bes o kas he leade . 9: i he leade is he mig an hen 10: Change he leade o he second-bes one in k. 11: end i 12: while (njump ⩽Njump) and (no be e posi ion) do 13: Upda e PRT alues 14: The mig an mo es o he leade 15: Checking bounda y 16: Re-e alua e i ness unc ion 17: Upda ed his mig an 18: Upda ed he global bes posi ion 19: end while 20: i he global bes is no upda ed a e FEs hen 21: Randomly eplace x% o he popula ion P 22: end i 23: end o 24: end while 25: e u n Fig. 8 isually illus a es how he ope a ing op imiza ion p o- cess o he h ee algo i hms SOMA ATO, SHADE, and iSOMA, implemen ed on he Ro a ed Composi ion Func ion (F.26) o he CEC17. I clea ly shows how he indi iduals o he classical SOMA mo e along he edges while SHADE’s mo emen is sp ead e enly in he sea ch space. The sea ching capabili y has been imp o ed in he iSOMA e sion by applying he abo e-men ioned p ocesses p o iding iSOMA’s balanced powe as e idenced by ‘‘sp ead- ing’’ indi iduals h oughou he sea ch space and hen ‘‘ ocusing’’ owa d he bes indi idual. 4. Expe imen al se up 4.1. Tes unc ions To ho oughly e alua e he iSOMA pe o mance, h ee com- mon es sui es o he IEEE Cong ess on E olu iona y Compu a ion (IEEE CEC) we e used, including a o al o 73 unc ions as lis ed in Tables 1,2, and 3and p esen ed below: •The i s benchma k se is he IEEE CEC 2013 Special Session on Real Pa ame e Single Objec i e Op imiza ion, consis ing o 28 unc ions (CEC13, see de ail a [30]); •The second is he IEEE CEC 2015 Compe i ion on Lea ning- based Real Pa ame e Single Objec i e Op imiza ion (CEC15, see de ail a [31]); •And he las one is he IEEE CEC 2017 Special Session and Compe i ion on Single Objec i e Real Pa ame e Nume ical Op imiza ion (CEC17, see de ail a [32]). These single objec i e benchma k p oblems we e used o e alua ion because hey a e he basis o esea ch on mo e com- plex op imiza ion p oblems such as mul i-objec i e, dynamic, niching composi ion, compu a ionally expensi e, and so on. They a e ca ego ized in o a ious ypes o unc ions including uni- modal, basic mul imodal, simple mul imodal, hyb id, and com- posi ion, (non-)sepa able, shi ed, and o a ed unc ions ha a e challenging enough o e alua e an algo i hm. De ini ions and de ails can be ound in [30–32]. 4.2. Compa ison algo i hms To demons a e imp o emen o e p e ious e sions o he iSOMA, he esul s we e compa ed o he o iginal and la es e sions o SOMA, as lis ed below. On he SOMA amily: •Sel -o ganizing mig a ing algo i hm AllToOne and AllToAll (SOMA ATO; SOMA ATA) [8,9]; •Pa e o-based sel -o ganizing mig a ing algo i hm (SOMA Pa e o) [22]; •Sel -o ganizing mig a ing algo i hm eam o eam adap i e (SOMA T3 A) [20]. To in es iga e he iSOMA le el o pe o mance and e ec- i eness compa ed o some well-known exis ing algo i hms, we ca y ou expe imen s on he a ious ypes o algo i hms such as DE, PSO, and ABC shown below, including algo i hms ha ha e pa icipa ed in he co esponding yea s’ compe i ions. Compa e iSOMA wi h o he SOMAs o igu e ou he impac o he imp o e- men s we ha e p oposed and compa e wi h o he algo i hms ou side he SOMAs o asce ain he posi ion o iSOMA on he op imiza ion algo i hm map. IEEE CEC 2013 (CEC13): •Success-his o y based pa ame e adap a ion o di e en ial e olu ion (SHADE) [33]; •Supe - i mul ic i e ia adap i e di e en ial e olu ion (SMADE) [34]; •A CMA-ES supe - i scheme o he e-sampled inhe i ance sea ch (CMAES-RIS) [35]; •A pa icle swa m op imiza ion and a i icial bee colony hy- b id algo i hm (SPSOABC) [36]; •A gene ic algo i hm o sol ing he CEC’2013 compe i ion p oblems on eal-pa ame e op imiza ion (TPC-GA) [37]. IEEE CEC 2015 (CEC15): •A di e en ial e olu ion algo i hm wi h success-based pa- ame e adap a ion o CEC2015 lea ning-based op imiza- ion (DEsPA) [38]; •Tuning ma u i y model o ecogeog aphy-based op imiza ion on CEC 2015 single-objec i e op imiza ion es p oblems (TEBO) [39]; •A Sel -adap i e Dynamic Pa icle Swa m Op imize (SaDPSO) [40]; •An imp o ed co a iance ma ix leaning and sea ching p e - e ence algo i hm o sol ing CEC 2015 benchma k p oblems (ICMLSP) [41]; •Dynamic sea ch i ewo ks algo i hm wi h co a iance mu a- ion o sol ing he CEC 2015 lea ning based compe i ion p oblems (dynFWACM) [42]. 5 Q.B. Diep, T.C. T uong, S. Das e al. Applied So Compu ing 116 (2022) 108270 Fig. 8. The ope a ion o h ee algo i hms SOMA-ATO, SHADE, and iSOMA on he unc ion 26 h o CEC17 es ed on 2D. Table 1 The lis o he IEEE CEC 2013 special session on eal pa ame e single objec i e op imiza ion unc ions. No. Func ions F∗No. Func ions F∗ 1 Sphe e unc ion −1400 15 Ro a ed Schwe el’s unc ion 100 2 Ro a ed high condi ioned ellip ic unc ion −1300 16 Ro a ed Ka suu a unc ion 200 3 Ro a ed Ben Ciga unc ion −1200 17 Lunacek Bi-Ras igin unc ion 300 4 Ro a ed discus unc ion −1100 18 Ro a ed Lunacek Bi-Ras igin unc ion 400 5 Di e en powe s unc ion −1000 19 Expanded G iewank’s plus Rosenb ock’s unc ion 500 6 Ro a ed Rosenb ock’s unc ion −900 20 Expanded Sca e ’s F6 unc ion 600 7 Ro a ed Scha e s F7 unc ion −800 21 Composi ion Func ion 1 (n =5,Ro a ed) 700 8 Ro a ed Ackley’s unc ion −700 22 Composi ion Func ion 2 (n =3,Un o a ed) 800 9 Ro a ed Weie s ass unc ion −600 23 Composi ion Func ion 3 (n =3,Ro a ed) 900 10 Ro a ed G iewank’s unc ion −500 24 Composi ion Func ion 4 (n =3,Ro a ed) 1000 11 Ras igin’s unc ion −400 25 Composi ion Func ion 5 (n =3,Ro a ed) 1100 12 Ro a ed Ras igin’s unc ion −300 26 Composi ion Func ion 6 (n =5,Ro a ed) 1200 13 Non-con inuous o a ed Ras igin’s unc ion −200 27 Composi ion Func ion 7 (n =5,Ro a ed) 1300 14 Schwe el’s unc ion −100 28 Composi ion Func ion 8 (n =5,Ro a ed) 1400 IEEE CEC 2017 (CEC17): •A di e en ial e olu ion s a egy (DES) [43]; •A e sion o IPOP-CMA-ES algo i hm wi h midpoin o CEC 2017 single objec i e bound cons ained p oblems (RB- IPOP-CMA-ES) [44]; 6 Q.B. Diep, T.C. T uong, S. Das e al. Applied So Compu ing 116 (2022) 108270 Table 2 The lis o he IEEE CEC 2015 compe i ion on lea ning-based eal pa ame e single objec i e op imiza ion unc ions. No. Func ions F∗No. Func ions F∗ 1 Ro a ed high condi ioned ellip ic unc ion 100 9 Composi ion Func ion 1 (N =3) 900 2 Ro a ed Ciga unc ion 200 10 Composi ion Func ion 2 (N =3) 1000 3 Shi ed and o a ed Ackley’s unc ion 300 11 Composi ion Func ion 3 (N =5) 1100 4 Shi ed and o a ed Ras igin’s unc ion 400 12 Composi ion Func ion 4 (N =5) 1200 5 Shi ed and o a ed Schwe el’s unc ion 500 13 Composi ion Func ion 5 (N =5) 1300 6 Hyb id Func ion 1 (N =3) 600 14 Composi ion Func ion 6 (N =7) 1400 7 Hyb id Func ion 2 (N =4) 700 15 Composi ion Func ion 7 (N =10) 1500 8 Hyb id Func ion 3 (N =5) 800 – – – Table 3 The lis o he IEEE CEC 2017 special session and compe i ion on single objec i e eal pa ame e nume ical op imiza ion unc ions. No. Func ions F∗No. Func ions F∗ 1 Shi ed and o a ed Ben Ciga unc ion 100 16 Hyb id Func ion 6 (N =4) 1600 2 Shi ed and o a ed sum o di e en powe unc ion 200 17 Hyb id Func ion 6 (N =5) 1700 3 Shi ed and o a ed Zakha o unc ion 300 18 Hyb id Func ion 6 (N =5) 1800 4 Shi ed and o a ed Rosenb ock’s unc ion 400 19 Hyb id Func ion 6 (N =5) 1900 5 Shi ed and o a ed Ras igin’s unc ion 500 20 Hyb id Func ion 6 (N =6) 2000 6 Shi ed and o a ed expanded Sca e ’s F6 unc ion 600 21 Composi ion Func ion 1 (N =3) 2100 7 Shi ed and o a ed Lunacek Bi-Ras igin unc ion 700 22 Composi ion Func ion 2 (N =3) 2200 8 Shi ed and o a ed non-con inuous Ras igin’s unc ion 800 23 Composi ion Func ion 3 (N =4) 2300 9 Shi ed and o a ed Le y unc ion 900 24 Composi ion Func ion 4 (N =4) 2400 10 Shi ed and o a ed Schwe el’s unc ion 1000 25 Composi ion Func ion 5 (N =5) 2500 11 Hyb id Func ion 1 (N =3) 1100 26 Composi ion Func ion 6 (N =5) 2600 12 Hyb id Func ion 2 (N =3) 1200 27 Composi ion Func ion 7 (N =6) 2700 13 Hyb id Func ion 3 (N =3) 1300 28 Composi ion Func ion 8 (N =6) 2800 14 Hyb id Func ion 4 (N =4) 1400 29 Composi ion Func ion 9 (N =3) 2900 15 Hyb id Func ion 5 (N =4) 1500 30 Composi ion Func ion 10 (N =3) 3000 •P oac i e pa icles in swa m op imiza ion: A se ings- ee algo i hm o eal-pa ame e single objec i e op imiza ion p oblems (PPSO) [45]; •Dynamic Yin–Yang pai op imiza ion and i s pe o mance on single objec i e eal pa ame e p oblems o cec 2017 (DYYPO) [46]; •Teaching lea ning based op imiza ion wi h ocused lea ning and i s pe o mance on CEC2017 unc ions (TLBO-FL) [47]. 4.3. Pa ame e se ings Tes s on 10Dand 30Da e ca ied ou , wi h a sea ch ange o [−100,100]D o hose es ing p oblems. The MaxFEs was used a 10000 ∗D(MaxFEs o 10D=100000; o 30D=300000). E o alue smalle han 10−8will be aken as ze o. Each algo i hm was independen ly un 51 imes o each unc ion, as he expe - imen al se ings eques ed in [30–32]. To de e mine i he gaps be ween he indings a e meaning ul, he Wilcoxon ank-sum es (WRT) was used a he 5% signi icance [48,49]. The con ol pa ame e o iSOMA: PopSize =100, Njump =10, n=5, m=10, k=15, S ep =0.3, and PRT as in Eq. (5). The con ol pa ame e alues o he es algo i hms we e used jus as hey we e in he o iginal a icles in he ci a ions, wi h no modi ica ions. The iSOMA is a ailable a Ma hWo ks si e he e. 5. Compa ison esul s The e o alues o 51 con inuous uns a e used as a basis o compa ing he pe o mance o algo i hms wi h dimensions D=10 and D=30. I is ob ained by he di e ence be ween he bes alue ound by he algo i hm and he global op imal alue wi hin he gi en sea ch anges o hose unc ions (F(x)−F(x∗)). No e ha he e o alue is conside ed ze o when smalle han 10−8as men ioned in he con es ules o [30–32]. The esul s o compa isons be ween he iSOMA algo i hm wi h o he s we e eco ded in ables whe e each ow ep esen s he alues o he mean and s anda d de ia ion o 51 uns o each Fig. 9. The summa ized compa ison esul s be ween he iSOMA and o he SOMAs es ed on 73 benchma k unc ions. es ing unc ion. The signs (+), (−), and (≈) show he compa ison ou come a 5% o he WRT whe e i is signi ican ly be e (iSOMA loses), signi ican ly wo se (iSOMA wins), and no signi ican ly be e o wo se (d aw) compa ed o iSOMA [48,49]. Wi hou s a- is ical checks, he bes ou comes o each ow in he pa icipa ing algo i hms we e bold. The las h ee ows in each able show he o al o (+), (−), and (≈). 5.1. Ou pe o m o he SOMAs Tables 4,5, and 6p esen he compa ison esul s be ween some la es e sions o he SOMA amily, in u n, pe o med on he CEC13, CEC15, and CEC17 es sui es wi hin only 30D. In pa icula , compa ed o SOMA ATO and ATA e sions, iSOMA has signi ican ly be e esul s on 3 es sui es wi h a o al o 59 and 57 o e 73 cases wins, while iSOMA only loses 9 and 10, d aws 5 and 6, espec i ely, as summa ized in Fig. 9. These esul s clea ly show ha he imp o emen s o he iSOMA b ing supe io pe o mance compa ed o he classical e sion as well as he SOMA Pa e o and T3 A. 7 Q.B. Diep, T.C. T uong, S. Das e al. Applied So Compu ing 116 (2022) 108270 Table 4 Compa ison o iSOMA wi h SOMA amily on he CEC13 benchma k unc ions (30 dimensions, 51 uns). F iSOMA SOMA ATO SOMA ATA SOMA Pa e o SOMA T3A Mean (S d De ) Mean (S d De ) Mean (S d De ) Mean (S d De ) Mean (S d De ) F10.00e+00 (0.00e+00) 0.00e+00 (0.00e+00)≈0.00e+00 (0.00e+00)≈0.00e+00 (0.00e+00)≈0.00e+00 (0.00e+00)≈ F29.80e+04 (4.13e+04) 1.70e+07 (4.03e+06)−1.40e+07 (2.62e+06)−8.79e+04 (3.68e+04)≈3.48e+05 (2.01e+05)− F33.13e+06 (4.17e+06) 9.75e+07 (1.20e+08)−1.89e+08 (1.60e+08)−3.60e+07 (5.49e+07)−2.06e+07 (3.02e+07)− F43.23e+02 (1.65e+02) 2.49e+04 (5.35e+03)−2.08e+04 (4.85e+03)−7.43e+02 (1.12e+03)≈4.79e+02 (3.26e+02)− F50.00e+00 (0.00e+00) 0.00e+00 (0.00e+00)≈0.00e+00 (0.00e+00)≈0.00e+00 (0.00e+00)≈0.00e+00 (0.00e+00)≈ F61.95e+01 (1.70e+01) 3.24e+01 (2.34e+01)−3.28e+01 (2.01e+01)−1.71e+01 (1.95e+01)+3.13e+01 (2.46e+01)− F71.37e+01 (4.39e+00) 8.21e+01 (1.24e+01)−8.66e+01 (1.55e+01)−3.74e+01 (7.86e+00)−3.48e+01 (9.38e+00)− F82.09e+01 (5.36e−02) 2.09e+01 (5.24e−02)≈2.10e+01 (4.79e−02)≈2.09e+01 (4.89e−02)≈2.09e+01 (4.62e−02)≈ F92.10e+01 (3.43e+00) 3.11e+01 (1.30e+00)−2.82e+01 (2.60e+00)−2.35e+01 (4.33e+00)−2.85e+01 (2.33e+00)− F10 1.91e−01 (7.57e−02) 3.88e−01 (2.44e−01)−3.03e−01 (1.21e−01)−3.00e−01 (1.48e−01)−1.87e−01 (9.30e−02)≈ F11 7.33e+00 (2.10e+00) 7.02e−01 (8.74e−01)+2.93e−01 (5.73e−01)+1.55e+01 (4.86e+00)−2.58e+00 (1.40e+00)+ F12 1.84e+01 (5.98e+00) 1.65e+02 (1.87e+01)−1.22e+02 (2.02e+01)−3.57e+01 (8.45e+00)−3.91e+01 (1.16e+01)− F13 4.42e+01 (1.61e+01) 1.80e+02 (1.41e+01)−1.60e+02 (2.17e+01)−8.06e+01 (2.32e+01)−8.03e+01 (2.67e+01)− F14 1.00e+03 (3.27e+02) 1.19e+01 (6.64e+00)+4.00e+00 (3.26e+00)+1.26e+03 (4.01e+02)−1.56e+01 (8.22e+00)+ F15 2.84e+03 (6.80e+02) 5.51e+03 (3.16e+02)−4.62e+03 (3.52e+02)−3.34e+03 (6.66e+02)−3.87e+03 (6.38e+02)− F16 2.44e+00 (2.58e−01) 2.13e+00 (2.29e−01)+1.73e+00 (3.15e−01)+1.87e+00 (3.60e−01)+2.08e+00 (5.02e−01)+ F17 4.31e+01 (4.42e+00) 3.14e+01 (6.21e−01)+3.07e+01 (2.44e−01)+5.22e+01 (5.37e+00)−3.36e+01 (1.19e+00)+ F18 5.01e+01 (8.52e+00) 2.12e+02 (1.43e+01)−1.85e+02 (1.77e+01)−5.25e+01 (8.29e+00)≈6.37e+01 (1.27e+01)− F19 2.42e+00 (4.96e−01) 1.88e+00 (3.01e−01)+1.48e+00 (2.66e−01)+2.78e+00 (7.47e−01)−1.97e+00 (3.23e−01)+ F20 9.18e+00 (6.49e−01) 1.33e+01 (5.48e−01)−1.34e+01 (5.34e−01)−9.86e+00 (6.76e−01)−1.05e+01 (7.97e−01)− F21 3.03e+02 (6.21e+01) 3.21e+02 (8.91e+01)−2.73e+02 (5.81e+01)≈3.28e+02 (7.87e+01)−3.16e+02 (8.90e+01)− F22 6.51e+02 (2.49e+02) 1.42e+02 (5.39e+01)+5.33e+01 (3.75e+01)+1.30e+03 (4.06e+02)−1.29e+02 (4.65e+01)+ F23 2.84e+03 (6.93e+02) 6.27e+03 (3.41e+02)−5.39e+03 (3.87e+02)−3.48e+03 (6.40e+02)−4.61e+03 (8.02e+02)− F24 2.21e+02 (5.18e+00) 2.76e+02 (9.04e+00)−2.75e+02 (7.61e+00)−2.27e+02 (4.44e+00)−2.50e+02 (1.06e+01)− F25 2.74e+02 (7.54e+00) 3.05e+02 (3.74e+00)−2.97e+02 (4.48e+00)−2.78e+02 (8.97e+00)−2.95e+02 (6.61e+00)− F26 2.00e+02 (3.09e−03) 2.01e+02 (3.01e−01)−2.01e+02 (3.97e−01)−2.00e+02 (2.11e−03)≈2.00e+02 (7.75e−03)− F27 5.54e+02 (5.88e+01) 1.04e+03 (2.08e+02)−9.13e+02 (2.70e+02)−6.38e+02 (8.02e+01)−1.01e+03 (9.22e+01)− F28 3.00e+02 (0.00e+00) 3.00e+02 (0.00e+00)−3.00e+02 (0.00e+00)−3.00e+02 (0.00e+00)≈3.00e+02 (0.00e+00)− +6 6 2 6 – 19 18 18 18 ≈3 4 8 4 Table 5 Compa ison o iSOMA wi h SOMA amily on he CEC15 benchma k unc ions (30 dimensions, 51 uns). F iSOMA SOMA ATO SOMA ATA SOMA Pa e o SOMA T3A Mean (S d De ) Mean (S d De ) Mean (S d De ) Mean (S d De ) Mean (S d De ) F15.61e+03 (5.25e+03) 2.00e+06 (7.21e+05)−1.79e+06 (6.16e+05)−1.11e+04 (8.30e+03)−5.20e+04 (4.99e+04)− F21.14e−01 (4.52e−01) 3.19e+03 (2.99e+03)−9.12e+02 (1.21e+03)−3.14e−01 (1.10e+00)−1.93e−04 (9.21e−04)+ F32.10e+01 (4.44e−02) 2.03e+01 (3.12e−02)+2.03e+01 (3.82e−02)+2.08e+01 (9.95e−02)+2.04e+01 (1.06e−01)+ F41.47e+01 (3.76e+00) 6.80e+01 (7.81e+00)−4.89e+01 (7.40e+00)−2.95e+01 (6.53e+00)−5.11e+01 (1.52e+01)− F51.91e+03 (5.53e+02) 2.78e+03 (2.58e+02)−2.17e+03 (2.44e+02)−2.59e+03 (5.71e+02)−2.16e+03 (4.96e+02)− F62.43e+03 (1.61e+03) 1.19e+06 (6.92e+05)−1.07e+06 (5.29e+05)−5.47e+03 (5.55e+03)−1.30e+04 (9.02e+03)− F72.60e+00 (7.40e−01) 9.85e+00 (1.46e+00)−8.46e+00 (1.80e+00)−4.00e+00 (9.00e−01)−3.79e+00 (1.05e+00)− F81.88e+03 (2.41e+03) 2.70e+05 (1.28e+05)−2.49e+05 (1.32e+05)−5.71e+03 (5.02e+03)−6.47e+03 (5.73e+03)− F91.02e+02 (1.23e−01) 1.03e+02 (2.13e−01)−1.04e+02 (3.29e−01)−1.03e+02 (1.61e−01)−1.03e+02 (1.55e−01)− F10 2.45e+03 (1.71e+03) 3.89e+05 (2.05e+05)−4.35e+05 (2.12e+05)−4.55e+03 (4.50e+03)−4.95e+03 (3.57e+03)− F11 3.10e+02 (3.25e+01) 3.21e+02 (9.10e+00)−3.35e+02 (5.25e+01)−3.14e+02 (5.70e+01)−3.03e+02 (1.99e+00)+ F12 1.04e+02 (4.28e−01) 1.07e+02 (5.68e−01)−1.07e+02 (6.28e−01)−1.04e+02 (4.13e−01)−1.05e+02 (5.77e−01)− F13 9.68e+01 (5.16e+00) 1.04e+02 (2.67e+00)−1.01e+02 (3.53e+00)−1.03e+02 (5.72e+00)−1.07e+02 (4.87e+00)− F14 3.26e+04 (5.55e+02) 3.19e+04 (6.16e+02)+3.23e+04 (5.69e+02)+3.27e+04 (4.54e+02)≈3.21e+04 (7.25e+02)+ F15 1.00e+02 (1.22e−13) 1.00e+02 (1.35e−13)−1.00e+02 (1.09e−13)−1.00e+02 (2.59e−13)−1.00e+02 (5.50e−13)− +2 2 1 4 – 13 13 13 11 ≈0 0 1 0 5.2. Compe e agains o he algo i hms Fo CEC13: Tables 7 and 8show he compa ison esul s on 10Dand 30D wi h well-known DE e sions o SHADE and SMADE, and o he well-a ended algo i hms such as PSO, GA and ABC lis ed in he p e ious sec ion and summa ized in Fig. 10. Fo 10Dp oblems, iSOMA p o ed weake han he wo e sions o DE, when losing 16 and 13 ou o 28 unc ions, winning only 4 and 9 unc ions. Howe e , he si ua ion changed o 30Dp oblems when iSOMA won 14 and los 9 compa ed o SMADE. These esul s show p omising po en ial. Compa ed o TPC-GA and CMAES-RIS, i is clea ha iSOMA is on pa wi h hem on 10D, and ou pe o ms on 30Dp oblems. This shows ha TPC-GA and CMAES-RIS a e mo e e ec i e on unimodal unc ions compa ed o iSOMA, as well as as con e - gence bu po en ially be apped in local op ima o he complex unc ions. In con as , iSOMA has p o en i s syne gy on basic mul imodal and composi ion unc ions. Fo CEC15: Tables 9 and 10, in u n, show he simula ion esul s be- ween iSOMA compa ed o DEsPA, TEBO, SaDPSO, ICMLSP and dynFWACM, on bo h 10Dand 30Dand summa ized in Fig. 11. Con on ed wi h ano he DE ep esen a i e, iSOMA was a bi weake o lose 8 ou o 15 cases on bo h 10Dand 30D, winning only 4 and 6 cases. Fo TEBO, iSOMA has compa able op imiza ion esul s on 30D p oblems and is somewha weake on 10D. The esul s we e mod- e a ely be e meanwhile an agonizing o SaDPSO, ICMLSP, and 8 Q.B. Diep, T.C. T uong, S. Das e al. Applied So Compu ing 116 (2022) 108270 Table 6 Compa ison o iSOMA wi h SOMA amily on he CEC17 benchma k unc ions (30 dimensions, 51 uns). F iSOMA SOMA ATO SOMA ATA SOMA Pa e o SOMA T3A Mean (S d De ) Mean (S d De ) Mean (S d De ) Mean (S d De ) Mean (S d De ) F16.63e−10 (3.55e−09) 1.48e+03 (2.41e+03)−5.52e+02 (1.14e+03)−2.49e−08 (1.48e−07)≈0.00e+00 (0.00e+00)≈ F23.92e−02 (2.80e−01) 3.61e+08 (2.47e+09)−9.10e+04 (5.96e+05)−1.67e+03 (9.51e+03)−2.97e+09 (1.46e+10)− F39.71e−04 (2.82e−03) 1.54e+04 (4.40e+03)−9.89e+03 (3.34e+03)−3.81e−05 (1.45e−04)+1.90e−02 (6.43e−02)− F44.52e+01 (3.20e+01) 8.46e+01 (2.78e+01)−8.54e+01 (2.19e+01)−3.77e+00 (9.58e+00)+5.12e+01 (3.12e+01)≈ F51.41e+01 (4.02e+00) 6.81e+01 (6.19e+00)−4.99e+01 (9.35e+00)−3.00e+01 (7.20e+00)−5.20e+01 (1.70e+01)− F63.68e−06 (1.39e−05) 0.00e+00 (0.00e+00)+0.00e+00 (0.00e+00)+1.41e−03 (1.83e−03)−4.22e−04 (5.76e−04)− F74.35e+01 (3.73e+00) 1.06e+02 (7.50e+00)−8.27e+01 (8.37e+00)−5.13e+01 (7.60e+00)−7.77e+01 (1.48e+01)− F81.56e+01 (4.00e+00) 7.07e+01 (6.25e+00)−5.46e+01 (8.48e+00)−2.85e+01 (7.89e+00)−5.65e+01 (1.48e+01)− F93.63e−01 (5.77e−01) 7.17e−01 (1.25e+00)−3.05e+00 (4.67e+00)−6.37e+00 (5.03e+00)−4.14e+00 (4.32e+00)− F10 2.13e+03 (4.94e+02) 3.08e+03 (2.21e+02)−2.35e+03 (2.98e+02)−2.84e+03 (5.85e+02)−2.51e+03 (4.67e+02)− F11 1.26e+01 (1.53e+01) 6.00e+01 (2.77e+01)−1.75e+01 (1.32e+01)−2.80e+01 (1.94e+01)−2.35e+01 (2.15e+01)− F12 7.00e+03 (4.23e+03) 3.96e+05 (2.70e+05)−5.09e+05 (3.26e+05)−1.13e+04 (5.68e+03)−1.04e+04 (5.79e+03)− F13 2.61e+01 (1.44e+01) 1.31e+04 (1.39e+04)−8.30e+03 (7.59e+03)−7.26e+01 (5.11e+01)−1.63e+02 (2.24e+02)− F14 4.35e+01 (1.64e+01) 4.28e+04 (3.36e+04)−8.75e+04 (1.14e+05)−1.21e+02 (3.14e+02)−6.86e+01 (7.42e+01)≈ F15 1.79e+02 (6.81e+02) 7.45e+03 (7.70e+03)−2.12e+03 (2.42e+03)−1.49e+02 (3.22e+02)+2.52e+01 (1.77e+01)+ F16 3.54e+02 (2.09e+02) 7.83e+02 (1.27e+02)−5.89e+02 (1.72e+02)−6.93e+02 (2.43e+02)−5.61e+02 (1.66e+02)− F17 3.80e+01 (2.62e+01) 2.34e+02 (7.51e+01)−1.45e+02 (8.89e+01)−8.78e+01 (8.76e+01)−9.63e+01 (7.21e+01)− F18 9.13e+03 (6.75e+03) 2.09e+05 (1.09e+05)−2.04e+05 (1.17e+05)−1.15e+04 (6.72e+03)−1.24e+04 (1.34e+04)≈ F19 1.46e+01 (6.05e+00) 7.99e+03 (8.78e+03)−2.92e+03 (3.60e+03)−4.29e+01 (4.28e+01)−1.91e+01 (9.23e+00)− F20 1.28e+02 (4.87e+01) 2.91e+02 (9.03e+01)−1.85e+02 (8.71e+01)−1.75e+02 (8.22e+01)−1.57e+02 (8.33e+01)− F21 2.17e+02 (4.66e+00) 2.79e+02 (8.92e+00)−2.50e+02 (2.86e+01)−2.28e+02 (8.31e+00)−2.45e+02 (4.41e+01)− F22 1.00e+02 (3.44e−01) 5.43e+02 (1.02e+03)−6.45e+02 (1.08e+03)−1.45e+02 (3.21e+02)−3.85e+02 (8.73e+02)− F23 3.65e+02 (8.16e+00) 4.26e+02 (9.35e+00)−4.04e+02 (1.05e+01)−3.82e+02 (8.72e+00)−4.01e+02 (1.70e+01)− F24 4.37e+02 (6.12e+00) 5.49e+02 (1.40e+01)−5.12e+02 (4.29e+01)−4.52e+02 (7.26e+00)−4.74e+02 (1.79e+01)− F25 3.87e+02 (3.12e−01) 3.87e+02 (1.09e+00)≈3.87e+02 (9.42e−01)≈3.88e+02 (3.32e+00)−3.88e+02 (1.11e+00)− F26 1.15e+03 (8.28e+01) 1.18e+03 (6.92e+02)≈1.00e+03 (5.68e+02)≈1.41e+03 (2.01e+02)−6.61e+02 (5.68e+02)+ F27 5.17e+02 (5.00e+00) 5.20e+02 (6.22e+00)−5.12e+02 (6.48e+00)+5.34e+02 (6.75e+00)−5.12e+02 (6.49e+00)+ F28 3.17e+02 (4.13e+01) 4.04e+02 (1.14e+01)−4.02e+02 (4.98e+00)−3.04e+02 (2.05e+01)+3.23e+02 (4.25e+01)− F29 4.61e+02 (3.84e+01) 6.67e+02 (7.73e+01)−5.29e+02 (7.50e+01)−5.20e+02 (9.75e+01)−5.63e+02 (9.71e+01)− F30 2.82e+03 (6.75e+02) 7.12e+03 (2.83e+03)−4.60e+03 (9.44e+02)−3.22e+03 (2.87e+02)−4.34e+03 (2.00e+03)− +1 2 4 3 – 27 26 25 23 ≈2 2 1 4 Table 7 Compa ison o iSOMA wi h well-known algo i hms on he CEC13 benchma k unc ions (10 dimensions, 51 uns). F iSOMA SHADE SMADE CMAES-RIS SPSOABC TPC-GA Mean (S d De ) Mean (S d De ) Mean (S d De ) Mean (S d De ) Mean (S d De ) Mean (S d De ) F10.00e+00 (0.00e+00) 0.00e+00 (0.00e+00)≈0.00e+00 (0.00e+00)≈0.00e+00 (0.00e+00)≈0.00e+00 (0.00e+00)≈0.00e+00 (0.00e+00)≈ F21.67e+03 (1.92e+03) 0.00e+00 (0.00e+00)+0.00e+00 (0.00e+00)+0.00e+00 (0.00e+00)+1.47e+05 (1.65e+05)−0.00e+00 (0.00e+00)+ F32.77e+05 (1.15e+06) 1.27e−01 (8.84e−01)+2.48e−01 (1.24e+00)+7.04e−01 (4.61e+00)+1.27e+05 (6.22e+05)+0.00e+00 (0.00e+00)+ F41.83e+01 (4.14e+01) 0.00e+00 (0.00e+00)+0.00e+00 (0.00e+00)+0.00e+00 (0.00e+00)+1.37e+03 (1.46e+03)−0.00e+00 (0.00e+00)+ F50.00e+00 (0.00e+00) 0.00e+00 (0.00e+00)≈0.00e+00 (0.00e+00)≈0.00e+00 (0.00e+00)≈0.00e+00 (0.00e+00)≈0.00e+00 (0.00e+00)≈ F64.04e+00 (4.88e+00) 7.89e+00 (3.93e+00)≈5.41e+00 (4.81e+00)−1.10e+00 (2.88e+00)≈0.00e+00 (0.00e+00)+0.00e+00 (0.00e+00)+ F72.30e+00 (2.66e+00) 3.26e−03 (4.54e−03)+2.27e+00 (4.50e+00)+5.33e+01 (4.68e+01)−0.00e+00 (0.00e+00)+4.24e−02 (2.10e−01)+ F82.03e+01 (7.12e−02) 2.04e+01 (8.95e−02)≈2.03e+01 (1.04e−01)≈2.03e+01 (1.37e−01)≈0.00e+00 (0.00e+00)+2.04e+01 (8.44e−02)− F92.73e+00 (8.60e−01) 3.39e+00 (7.35e−01)−2.29e+00 (7.26e−01)+3.59e+00 (1.04e+00)−0.00e+00 (0.00e+00)+3.39e+00 (2.88e+00)≈ F10 3.33e−01 (2.05e−01) 1.20e−02 (8.99e−03)+1.42e−02 (9.67e−03)+1.24e−02 (1.35e−02)+0.00e+00 (0.00e+00)+3.87e−02 (2.83e−02)+ F11 1.16e+00 (1.10e+00) 0.00e+00 (0.00e+00)+9.75e−02 (2.99e−01)+3.57e+00 (1.48e+00)−0.00e+00 (0.00e+00)+2.73e−01 (4.91e−01)+ F12 4.97e+00 (2.24e+00) 3.14e+00 (9.73e−01)+7.80e+00 (4.14e+00)−1.29e+01 (5.42e+00)−0.00e+00 (0.00e+00)+6.03e+00 (2.18e+00)− F13 7.73e+00 (5.20e+00) 3.77e+00 (1.85e+00)+1.21e+01 (6.47e+00)−2.56e+01 (1.08e+01)−0.00e+00 (0.00e+00)+9.87e+00 (6.24e+00)≈ F14 8.58e+01 (8.20e+01) 4.90e−03 (1.70e−02)+3.64e+00 (4.44e+00)+1.02e+02 (7.39e+01)≈0.00e+00 (0.00e+00)+2.45e+01 (2.47e+01)+ F15 5.13e+02 (2.99e+02) 4.21e+02 (1.14e+02)≈7.36e+02 (2.63e+02)−6.17e+02 (1.74e+02)−5.96e+02 (1.37e+02)−7.34e+02 (2.44e+02)− F16 1.16e+00 (2.08e−01) 7.08e−01 (2.12e−01)+4.04e−01 (3.17e−01)+1.64e−01 (7.56e−02)+2.00e+02 (1.25e−01)−1.25e+00 (3.29e−01)− F17 1.31e+01 (1.69e+00) 1.01e+01 (0.00e+00)+1.03e+01 (1.56e−01)+1.04e+01 (3.73e+00)+3.10e+02 (1.96e+00)−1.12e+01 (7.76e−01)+ F18 2.02e+01 (5.81e+00) 1.69e+01 (1.54e+00)+2.46e+01 (4.73e+00)−2.98e+01 (6.16e+00)−4.17e+02 (1.95e+00)−1.80e+01 (3.13e+00)≈ F19 7.05e−01 (2.21e−01) 3.44e−01 (4.90e−02)+3.95e−01 (1.26e−01)+8.14e−01 (2.74e−01)≈5.00e+02 (5.21e−02)−5.01e−01 (1.21e−01)+ F20 2.02e+00 (5.98e−01) 2.16e+00 (3.52e−01)≈2.65e+00 (4.52e−01)−4.16e+00 (3.99e−01)−6.02e+02 (4.82e−01)−3.17e+00 (4.81e−01)− F21 3.98e+02 (1.25e+01) 4.00e+02 (0.00e+00)−3.83e+02 (5.56e+01)+1.61e+02 (6.03e+01)+1.10e+03 (2.80e+01)−2.90e+02 (5.00e+01)+ F22 9.20e+01 (8.29e+01) 4.84e+00 (6.20e+00)+4.93e+01 (5.38e+01)+2.44e+02 (1.09e+02)−8.13e+02 (5.48e+00)−9.07e+01 (6.14e+01)≈ F23 3.84e+02 (2.45e+02) 4.61e+02 (1.78e+02)≈5.78e+02 (3.20e+02)−8.35e+02 (1.90e+02)−1.50e+03 (1.81e+02)−8.40e+02 (2.83e+02)− F24 1.45e+02 (4.32e+01) 1.93e+02 (2.46e+01)−2.02e+02 (1.78e+01)−1.19e+02 (5.69e+00)≈1.20e+03 (2.33e+01)−2.13e+02 (6.62e+00)− F25 2.01e+02 (1.02e+01) 2.00e+02 (7.02e−01)+2.02e+02 (1.93e+00)≈1.93e+02 (3.42e+01)+1.30e+03 (2.16e+01)−2.17e+02 (6.59e+00)− F26 1.05e+02 (2.26e+00) 1.33e+02 (4.36e+01)−1.26e+02 (3.73e+01)−1.61e+02 (4.06e+01)−1.33e+03 (3.99e+01)−1.96e+02 (1.81e+01)− F27 3.03e+02 (3.65e+00) 3.00e+02 (1.46e−08)+3.37e+02 (5.29e+01)≈3.13e+02 (2.30e+01)−1.65e+03 (7.13e+01)−4.24e+02 (6.83e+01)− F28 2.92e+02 (3.92e+01) 3.00e+02 (0.00e+00)≈3.17e+02 (6.94e+01)≈2.06e+02 (1.07e+02)+1.69e+03 (7.01e+01)−2.92e+02 (3.92e+01)≈ +16 13 9 10 11 – 4 9 12 16 10 ≈86727 9 Q.B. Diep, T.C. T uong, S. Das e al. Applied So Compu ing 116 (2022) 108270 om he obs acle. The ajec o y o he d one is a se o eal- ime con inuous poin s gene a ed by he iSOMA o each d one membe , independen o any cen al con olle . Howe e , mul- iple dynamic obs acles and e en mo ing a ge s ha e no been in es iga ed in his s udy. Wha i he numbe o obs acles and d ones is ela i ely la ge? Will i lead o he d ones being apped and unable o mo e o collide wi h each o he ? The solu ions o such issues will be add essed in ou subsequen s udies. Wi h ou s anding pe o mance, eal-wo ld applica ions ha use he iSOMA algo i hm will p omise o deli e supe io powe , ca ching up wi h he e e -inc easing demands o echnical de el- opmen . CRediT au ho ship con ibu ion s a emen Quoc Bao Diep: Concep ualiza ion, Me hodology, So wa e, Valida ion, Fo mal analysis, In es iga ion, Resou ces, Visualiza- ion, W i ing – o iginal d a , W i ing – e iew & edi ing. Thanh Cong T uong: Concep ualiza ion, Me hodology, So wa e, Vali- da ion, Fo mal analysis, In es iga ion, Resou ces, Visualiza ion, W i ing – o iginal d a , W i ing – e iew & edi ing. Swaga am Das: Concep ualiza ion, Me hodology, So wa e, Valida ion, Fo - mal analysis, In es iga ion, Resou ces, Visualiza ion, W i ing – o iginal d a , W i ing – e iew & edi ing. I an Zelinka: Concep- ualiza ion, Me hodology, So wa e, Valida ion, Fo mal analysis, In es iga ion, Resou ces, Visualiza ion, W i ing – o iginal d a , W i ing – e iew & edi ing. Decla a ion o compe ing in e es The au ho s decla e ha hey ha e no known compe ing inan- cial in e es s o pe sonal ela ionships ha could ha e appea ed o in luence he wo k epo ed in his pape . Acknowledgmen s The ollowing g an s a e acknowledged o he inancial sup- po p o ided o his esea ch: G an o SGS, Czech Republic No. SP2021/72, VSB-Technical Uni e si y o Os a a. Re e ences [1] J.D. Se , E. Osaba, D. Molina, X.-S. Yang, S. Salcedo-Sanz, D. Camacho, S. 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