optimization ship hydrodynamics swarm for problems Parameter synchronous selection in and asynchronous deterministicparticle Applied Soft Computing

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ContentslistsavailableatScienceDirect

Applied

Soft

Computing

jo u r n al hom e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / a s o c

Parameter

selection

in

synchronous

and

asynchronous

deterministic

particle

swarm

optimization

for

ship

hydrodynamics

problems

Andrea

Serani

a,b

_,

_Cecilia

_Leotardi

a

_,

_Umberto

_Iemma

b

_,

_Emilio

_F.

_Campana

a

_,

Giovanni

Fasano

c

_,

_Matteo

_Diez

a,∗

a_CNR-INSEAN,_National_Research_{Council—Marine}_Technology_Research_Institute,_Rome,_Italy b_Department_of_Mechanical_and_Industrial_Engineering,_Roma_Tre_University,_Rome,_Italy c_Department_of_Management,_Ca’_Foscari_University_of_Venice,_Venice,_Italy

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received13November2015 Receivedinrevisedform15July2016 Accepted12August2016

Availableonline20August2016 Keywords:

Simulation-baseddesign Derivative-freeoptimization Globaloptimization Particleswarmoptimization Shiphydrodynamicsoptimization

a

b

s

t

r

a

c

t

Deterministicoptimizationalgorithmsareveryattractivewhentheobjectivefunctionis computation-allyexpensiveandthereforethestatisticalanalysisoftheoptimizationoutcomesbecomestooexpensive. Amongdeterministicmethods,deterministicparticleswarmoptimization(DPSO)hasseveralattractive characteristicssuchasthesimplicityoftheheuristics,theeaseofimplementation,anditsoftenfairly remarkableeffectiveness.TheperformancesofDPSOdependonfourmainsettingparameters:the num-berofswarmparticles,theirinitialization,thesetofcoefficientsdefiningtheswarmbehavior,and(for box-constrainedoptimization)themethodtohandletheboxconstraints.Here,aparametricstudyof DPSOispresented,withapplicationtosimulation-baseddesigninshiphydrodynamics.Theobjectiveis theidentificationofthemostpromisingsetupforbothsynchronousandasynchronousimplementations ofDPSO.Theanalysisisperformedundertheassumptionoflimitedcomputationalresourcesandlarge computationalburdenoftheobjectivefunctionevaluation.Theanalysisisconductedusing100 analyt-icaltestfunctions(withdimensionalityfromtwotofifty)andthreeperformancecriteria,varyingthe swarmsize,initialization,coefficients,andthemethodfortheboxconstraints,resultinginmorethan 40,000optimizations.Themostpromisingsetupisappliedtothehull-formoptimizationofahighspeed catamaran,forresistancereductionincalmwaterandatfixedspeed,usingapotential-flowsolver.

1. Introduction

Particleswarm optimization(PSO)wasoriginallyintroduced in [1], based on the social-behaviour metaphor of a ﬂock of birds or a swarm of bees searching for food. PSO belongs to theclassofheuristicalgorithmsforsingle-objectiveevolutionary derivative-free global optimization. Derivative-free global opti-mizationapproachesareoftenpreferredtolocalapproacheswhen objectivesarenonconvexand/ornoisy,andwhenmultiplelocal optimacannotbeexcluded,asoftenencounteredin simulation-based design (SBD) optimization. Thecomputational burden of globaloptimizationtechniquesisusuallymuch largercompared tolocalmethods,sothattheaccuracyofthesolutionsoughtoften dependsontheavailablecomputationalresources.

Zhangetal.[2]presentsacomprehensivesurveyonthePSO variantsandtheirapplicationinseveralengineeringﬁelds,suchas

∗ Correspondingauthor.

E-mailaddress:matteo.diez@cnr.it(M.Diez).

mechanicalorchemical. RecentapplicationsofPSO toshipSBD include medium-tohigh-fidelityhull-form andwaterjet design optimization of fast catamarans, by morphing techniques [3,4] andgeometrymodificationsbasedonKarhunen–Loèveexpansion (KLE)[5–7],andlow-tomedium-fidelityoptimizationof uncon-ventionalmulti-hullconfigurations[8].Whenglobaltechniquesare usedindesignoptimization,withCPU-timeexpensivesolvers,the optimizationprocessiscomputationallyexpensiveandits effec-tivenessandefficiencyremainanalgorithmicand technological challenge.Although complex SBD applicationsare oftensolved bymetamodels[9,10],theirdevelopmentandassessmentrequire benchmarksolutions,withsimulationsdirectlyconnectedtothe optimizationalgorithm.Thesesolutionsareachievedonlyif afford-ableandeffectiveoptimizationproceduresareavailable.

The original PSO makes use of random coefﬁcients, aiming atsustaining thevarietyofthe swarmdynamics.Thisproperty impliesthatstatisticallysigniﬁcantresultscanbeobtainedonly throughextensivenumericalcampaigns.Suchanapproachcanbe tooexpensiveinSBDoptimizationforindustrialapplications,when CPU-time expensivecomputer simulations are used directlyas http://dx.doi.org/10.1016/j.asoc.2016.08.028

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314 A.Seranietal./AppliedSoftComputing49(2016)313–334 analysistools.Furthermore,ifthedesignprobleminhandis

sched-uledwithinanaccurateprojectplanning,timeresourcesmightbe atightrequirementfortheoptimizationprocess.Forthese rea-sonsefficientdeterministicapproaches(suchasdeterministicPSO, DPSO)havebeendeveloped,andtheireffectivenessandefficiency inindustrialapplicationsinship hydrodynamicsproblemshave beenshown,includingcomparisonswithlocalmethods[11]and randomPSO[5].Moreover,theavailabilityofparallelarchitectures andhighperformancecomputing(HPC)systemshasofferedthe opportunitytoextendtheoriginalsynchronousimplementationof PSO(SPSO)toCPU-timeefficientasynchronousmethods(APSO), assessedontestfunctionsin[12],andappliedinseveral engineer-ingproblemssuchasmultidisciplinaryoptimizationofcommercial aircraft[13],biomechanics[14],andswarmrobotics[15].Using dis-tributedcomputing,synchronousimplementationsofPSOimply thatatiteration k+1thepositionandvelocityofanyparticleis updatedafterevaluatingthefunctionatalltheparticlespositions atiterationk.InanasynchronousimplementationofPSOthe posi-tionandvelocityofaparticleispossiblybasedonthefitnessvalue atasubsetofallparticlespositions.

The effectiveness and efficiency of PSO for box constrained optimization are significantly influenced by four main setting parameters:(a)thenumberofswarmparticlesinteractingduring theoptimization,(b)theinitializationoftheparticlesintermsof initiallocationandvelocity,(c)thesetofcoefficientsdefiningthe personalorsocialbehaviouroftheswarmdynamics,and(d)the methodtohandletheboxconstraints.Theseparametersandtheir effectsonPSOhavebeenstudiedbyanumberofauthors[16–18]. Morerecentlytheeffectsoftheparticleinitializationhavebeen studiedin[19,20],theeffectsofthecoefficientshavebeenshown in[21],whereasboundhandlingtechniqueshavebeenpresented in[22].AcomprehensivestudyonthePSOparameterselectionhas beenpresentedin[23]andapreliminaryassessmentofthe per-formancesofDPSO,varying(a),(b)and(c),ispresentedin[24]. Asurveyofapproachesforgeneralconstrainedoptimization prob-lemsinindustrialdesignandmultidisciplinarydesignoptimization maybefoundin[25],includingalsogeneralnonlinearconstraints. However,thediscussionontheapplicationofDPSOinSBD prob-lemsisstilllimited,lackingasystematicandcomparativeanalysis. Theobjectiveofthepresentworkistheidentificationofthemost effectiveandefficientparametersforbothsynchronousand asyn-chronousdeterministicparticleswarmoptimization(SDPSOand ADPSO),fortheiruseinSBDprocedures.Thefocusisonindustrial problems,directlyusingCPU-timeexpensiveanalyses.Thesemake thestatisticalanalysisoftheresultstooexpensiveandtherefore demandfordeterministicalgorithms.Due totheattractive fea-turesofDPSO(suchasthesimplicityoftheheuristics,theease ofimplementation, anditsoftenfairlyremarkableeffectiveness inindustrialproblems),thecurrentstudyislimitedtoDPSOand itsimplementations.AsystematiccomparisonofDPSOwithother deterministicandstochasticmethodsisbeyondthescopesofthe presentwork.

Theapproach includes a preliminaryparametric analysison 100analytical test functions[26–29]characterized bydifferent degreesofnon-linearitiesandnumberoflocalminima,with full-factorialcombinationof:(a)numberofparticles(usingapower oftwo times thenumber ofdesign variables); (b) initialization oftheparticlepositionandvelocity(usingHammersleysequence sampling(HSS)[30]);(c)setofcoefﬁcients,chosenfromliterature [12,16,17,31,32];(d)inelasticandsemi-elasticwall-typeapproach forboxconstraints[22].Inorder tohandletheboxconstraints, wall-typeapproachesarepreferredinsteadofpenaltyapproaches orLagrangianfunctions,whichmightintroduceadditionalbiasin theanalysis.Thenumberofdesignvariablesrangesfromtwotoﬁfty andthesimulationbudget(maximumnumberofobjectivefunction evaluations)isupto4096timesthenumberofdesignvariables.

Thepreliminaryparametricanalysisisconductedontwosubsets ofproblems,respectivelywithlessandmorethantendesign vari-ables.AIntelXeonE5-1620v23.70GHzisusedforthepreliminary tests.Threeabsolutemetricsaredefinedandappliedforthe evalua-tionofthealgorithmperformances,basedonthedistancebetween PSO-found solutions and analytical optima. Accordingwiththe numericaltests,themosteffectiveparameterchoiceamong(a), (b),(c),and(d)isidentified,basedontheassociatedrelative vari-abilityoftheresults.Then,themostpromisingsetupsforSDPSO andADPSOaredeterminedandappliedtoanindustrialproblem, namelyafastcatamaranhull-formoptimization,forcalmwaterand fixedspeed.TheobjectivefunctionistheratioRT/Wbetweenthe

totalresistance(RT)andtheshipweight(W).Thehull-form

modi-ficationisperformedusingaKLE-basedmorphingapproach[5–7], usingrespectivelyfour-andsix-dimensionaldesignspaces. Com-putersimulationsareperformedusingthepotentialflow(PF)solver CNR-INSEANWARP[33],onaclusterofIntelXeonE54622.80GHz. Eachfunctionevaluationtakesabout10minpernode. Addition-ally,theoptimizationresultsarecomparedwiththoseobtainedin earlierresearch,basedonahigh-fidelityURANSsolver[5].

2. PSOformulations

Considerthefollowingobjectivefunction:

f(x):RNdv−→R ₍₁₎

andtheglobaloptimizationproblem min

x∈Lf(x), L⊂R

Ndv ₍₂₎

whereLisaclosedandboundedsubsetofRNdv_and_N_dv_is_the

num-berofdesignvariables.Theglobalminimizationoftheobjective functionf(x)requirestoﬁndavectora_∈_Lsuchthat:

∀

b∈L:f(a)≤f(b) (3)

Then,ais aglobalminimum forthefunctionf(x)over L.Since thesolutionofEq.(2)isingeneralanNP-hardproblem,theexact identificationofaglobalminimummightbeverydifficult. There-fore,solutionswithsufficientgoodfitness,providedbyheuristic procedures,areoftenconsideredacceptableforseveralpractical purposes.Amongdifferentheuristicprocedures,PSOisoftenthe methodofchoiceforitscapabilitytooutreachasuitable approxi-matesolutionwithinafewiterations.InPSO,thepositionsofthe particlesrepresentthecandidate solutionsand willbedenoted byx∈L,withassociatedfitnessf(x).Moreover,inthispaperthe compactsetLrepresentstheboxconstraints.

2.1. Originalformulation

TheoriginalformulationofthePSOalgorithm,aspresentedin [16],reads

vk+1_i ₌wvk i +c1r k 1,i(xi,pb−xki)+c2r k 2,i(xgb−xki) xk+1_i =xk i +vk+1i (4)

Theaboveequationsupdatevelocityandpositionoftheith parti-cleatthekthiteration,wherewistheinertiaweight;c1andc2are

respectivelythesocialandcognitivelearningrate;rk

1,iandr

k

2,iare

uniformlydistributedrandomnumbersin[0,1];xi,pbisthepersonal

bestpositioneverfoundbytheithparticleintheprevious itera-tionsandxgbistheglobalbestpositioneverfoundintheprevious

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Fig.1.BlockdiagramsforparallelSDPSO(a)andADPSO(b).Thegreenboxesrepresenttheﬁrstsetofparticlesevaluatedbythealgorithm[14].(Forinterpretationofthe referencestocolourinthisﬁgurelegend,thereaderisreferredtothewebversionofthearticle.)

Anoverallconstrictionfactorisusedin[34–38],inplaceofthe inertiaweightw.Accordingly,thesysteminEq.(4)isrecastinthe followingequivalentform

vk_i+1=

vk i +c1r k 1,i(xi,pb−x k i)+c2r k 2,i(xgb−x k i)

xk_i+1=xk i +v k+1 i (5)

Inordertoprovidenecessary(butpossiblynotsufﬁcient) condi-tionswhichavoiddivergenceofparticlestrajectories,thefollowing condition:

=

2

2−ϕ−

ϕ2₋_4ϕ

, whereϕ=c1+c2, ϕ>4 (6)

is indicated in [35], where setting the value of ϕ to 4.1, with =0.729,c1=c2=2.050issuggested[36].NotethatPSOschemes

includingboththeparameterswandhavebeenalsoproposed intheliterature.Furthermore,recentsurveysonPSOcoefﬁcients, inthelightofpossibledivergenceofparticlestrajectories,canbe foundin[39,40].

2.2. Deterministicformulation

TwoessentialrequisitesinSBDproblemsaresurelyrepresented bytheefﬁciencyoftheoptimizationprocedureadopted,alongwith thepossibilitytoreplicatethesameprocedureondifferentsettings oftheproblem.In ordertocomplywithboththelatterfacts,a deterministicversion ofthePSO algorithm (namely DPSO)was formulatedin[11] bysettingrk

1,i=r

k

2,i=1inEq.(5).Ofcourse,

skippingrandomparametersdoesnotallowtoassessastatistical analysisoftheresultsonseveralpracticalproblems.Nevertheless, sinceSBDproblemsareusuallycomputationallyveryexpensive, andrepresentjustonestepwithinacomplexandformalproject plan,theuseofDPSOrevealedtobemandatorywithrespectto adoptingPSO.Ontheoverall,theﬁnalDPSOschemebecomes

vk_i+1=

vk i +c1(xi,pb−xki)+c2(xgb−xki)

xk_i+1=xk i +v k+1 i (7)

InthecontextofSBDforshipdesignoptimization,asmentioned inSection1,theformulationofEq.(7)wascomparedtotheoriginal in[5].DPSOisthereforeusedforallthesubsequentanalyses.

Usingtheaboveformulation,itispossibletoprovethatthe nec-essary(butpossiblynotsufﬁcient)conditionswhichensurethatthe trajectoryofeachparticledoesnotdiverge[41],are

0<

<1 0<ω<2 (+1) (8) whereω=(c1+c2).Introducing ˇ= ω 2(+1) (9)

andassuming>0asusuallyintheliterature,theconditionsofEq. (8)reduceto

0<<1

0<ˇ<1 (10)

2.3. Synchronousandasynchronousimplementations

AsreportedinSection1,twodifferentimplementationsofDPSO are comparatively applied in this paper. In particular, the syn-chronousimplementationofDPSO(SDPSO)updatesthepersonal bests{xi,pb}andtheglobalbestxgb,alongwithparticlesvelocity

andposition,attheendofeachiteration.SDPSOispresentedasa pseudo-codeinAlgorithm1,andasablockdiagraminFig.1a.The synchronousupdateensuresanyparticleofaperfectandcomplete informationofitsneighborhood.However,synchronousupdateis acostlychoice[12],asaparticleneedstowaitforthewholeswarm tobeupdated,beforeitcanmovetoanewpositionandcontinueits search.Hence,theﬁrstparticleevaluatedmightbeidleforalong time,waitingforthewholeswarmtobeupdated.

Algorithm1. SDPSOpseudo-code

1: InitializeaswarmofNpparticles

2: while(k<Maxnumberofiterations)do

3: fori=1,Npdo

4: Evaluatef(xk

i)

5: endfor

6: Update{xi,pb},xgb

7: Updateparticlepositionsandvelocities{xk+1 i },{vk+1i }

8: endwhile

9: Outputthebestsolutionfound

Furthermore,in parallelarchitectures,if theamountof time required to evaluatethe objective function at each iteration is notuniform(e.g.,duetoiterativeprocess/convergenceof analy-sistools),thewall-clocktimeandCPU-timereservationofSDPSO maysigniﬁcantlyincrease.IncontrasttoSDPSO,theasynchronous implementationofPSO(namelyADPSO[12])updatespersonaland globalbests,alongwithparticlesvelocityandposition,assoonas

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316 A.Seranietal./AppliedSoftComputing49(2016)313–334

Fig.2. ExamplesofinitializationsinL=[−0.5,0.5]×[−0.5,0.5]with32particles.(Forinterpretationofthereferencestocolourintext,thereaderisreferredtotheweb versionofthearticle.)

theinformationrequiredfortheirupdateisavailable,anda parti-cleisreadyforanewanalysis.Thereforeinthisversionaparticle’s position/velocityisupdatedusingpartialor“imperfect” informa-tionofitsneighborhood.I.e.,someinformationcomefromparticles withposition/velocityupdatedinthecurrentiteration,while par-tialinformationisprovidedbyparticleswhoseposition/velocity wasupdatedinthepreviousiterations.Thislastaspectcontributes todiversityintheswarm,whichisadesirabletrait.ADPSOis pre-sentedasapseudo-codeinAlgorithm2,andasablockdiagramin Fig.1b.

ThedifferencesbetweenADPSOandSDPSOaresummarizedin Table1.

Algorithm2. ADPSOpseudo-code 1: InitializeaswarmofNpparticles

2: while(k<Maxnumberofiterations)do

3: fori=1,Npdo

4: evaluatef(xk

i)

5: Update{xi,pb},xgb

6: Updateparticlepositionsandvelocities{xk+1 i },{vk+1i }

7: endfor

8: endwhile

9: Outputthebestsolutionfound

3. PSOsettingsandevaluationmetrics

ThechoiceofPSOparametersusedinthecurrentanalysisis deﬁnedinthefollowingsection.Theirfull-factorialcombinationis considered,resultinginatotalof420PSOsetups.

3.1. Numberofparticles

Thenumberofparticlesused(Np)isdeﬁnedas

Np=2mNdv, withm∈N[1,7] (11)

thereforerangingfromNp=2NdvtoNp=128Ndv.

3.2. Particlesinitialization

In order toavoid possible undesired bias due to thechoice of the swarm initialization, parameters from the literature are investigatedinthecurrentstudy.Inparticular,theinitializationof

Table1

SynchronousandasynchronousDPSOcharacteristics.

SDPSO ADPSO

Update Attheendofeachiteration Assoonastheobjective functionisevaluated Information Thewholeswarmuses

informationfromthesame iteration

Aparticlecoulduse informationfromboth currentandprevious iterations

particles’locationandvelocityfollowsadeterministicand homo-geneous distribution,according withtheHammersley sequence sampling(HSS)[30].Speciﬁcally,letq={q1,...,qNdv−1}beavector

ofprimenumberswithqi /= qj,

∀

i /= j.Anypositiveintegericanbe

expressedusingthesequence{qj}by

i=

r

k=0

akqkj (12)

whererisasuitableintegerandakisanintegerin[0,qj−1].Finally,

theithparticlelocationisdeﬁnedas xi=

i Np ,q1(i),...,q_Ndv−1(i)

fori=0,1,2,...,Np−1 (13) whereqj(i)=

r k=0ak/qk+1j .

Eq.(13)isappliedtothreedifferentsub-domains,deﬁnedas: 1.entiredomainL(reddotsinFig.2a)

2.domainbounds(blutrianglesinFig.2b)

3.domainandbounds(reddotsandbluetrianglesinFig.2c). Ontheotherhand,theinitialvelocityisdeﬁnedbyeitherthe fol-lowing:

• nullvelocity:

vi=0,

∀

i∈[1,Np] (14)

• non-nullvelocity,basedoninitialparticleposition: v_i₌

2 Ndv

x_i₋l+u 2

(15) wherelandurepresentthelowerandupperboundforx, respec-tively[5].

Combininginitialpositionand velocitysettingsresultsinsix differentinitializations,summarizedinTable2.

3.3. Coefﬁcientset

WithasimilarapproachusedinSection3.2,inthispaperthe authorsareﬁrstconsideringacomparisonamongdifferentsetups

Table2

Swarminitialization.

HSS,sub-domain v=0 v /=0

Domain A.0 A.1

Bounds B.0 B.1

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Table3

Coefﬁcientset.

IDset Name c1 c2 ˇ

1 ShiandEberhart(1998) 0.729 2.050 2.050 0.864 2 CarlisleandDozier(2001) 0.729 2.300 1.800 0.864 3 Trelea(2003) 0.600 1.700 1.700 0.638 4 Clerc(2006) 0.721 1.655 1.655 0.693 5 PeriandTinti(2012) 0.754 2.837 1.597 0.953

ofDPSO,inordertofindapromisingconfiguration,tobeappliedfor thesolutionofanengineeringdesignproblem.Onthisguideline, ononehandinthispapertheauthorsarenotproposingtheirown setofpreferablePSOcoefficients.Ontheotherhand,afair com-parisonwithinapanelofdifferentDPSOcoefficientsetsrequiresto testoriginalsettingsfromtheliterature,withoutintroducingany hybridcombination/integrationamongthem.

Tothelatterpurpose,fivecoefficientsetsaretakenfrom liter-ature,asproposedbyseveralauthors.Thefirstsetistheoriginal byShiandEberhart[16],thesecondwassuggestedbyCarlisleand Dozier[12],thethirdwasproposedbyTrelea[17],thefourthisa furthersuggestionbyClerc[31],whereasthefifthwassuggestedby PeriandTinti[32].Theassociatednumericalvaluesareincludedin Table3,andtheyallsatisfyEq.(10).For possibleprosandcons ofeach coefficientsetchoice,thereadercanrefer totheabove references.

3.4. Boxconstraints

TheoriginalPSOprovidesanupdateofpositionandvelocityof particlesforunconstrainedproblems.Thus,incasepossible con-straintsarepresent,theoriginalformulationmightbeinadequate, evenincasesimpleboxconstraintsareconsidered.Thisimplies that,duringtheevolutionoftheswarm,simplyusingEq.(7)the particlescantraveloutsidethedomainbounds.Thiscanbeacritical issueinSBDproblems,whenthedomainboundscannotbeviolated duetophysical/geometrical/gridsconstraints.Abarrier(wall)ora penaltyapproachcanbeusedontheboundsoftheresearchspace, inordertoconﬁnetheparticles[25,42–44].Inparticular,theuse ofexactorsequentialpenaltyfunctionsmaybeinadequateinthe currentwork,foratleastacoupleofreasons:

• only box constraintsare used (i.e.simple) or inequality con-straints,whichhavebeenhandledintheliteratureofPSOand otherheuristicsinaspeciﬁcway(seeAlgorithms3and4inthis section);

• the use of general penalty approaches described in [25] is expectedtobeappealingwhenhighlynonlinearconstraintsare

involved.Moreover,acorrectimplementationofthelatter meth-odsdeﬁnitelyrequiresaﬁnetuningofthepenaltyparameters involved,includingtheiriterativeupdate.

Thus,accordingwiththeabovemotivations,andinordertoavoid thepossiblebiasintroducedbythepenaltyparameterupdate(see also [45] for details), herein theapproach presented in [22] is applied,sinceitisspeciﬁcallydesignedforthecurrentconstrained problems,intheframeworkofPSOliterature.

Speciﬁcally,theparticlesareconﬁnedwithinLusingan inelas-ticwall-typeapproach(IW).Ifaparticleisfoundtoviolateoneof theboundsinthetransitionfromkthto(k+1)thPSOiteration,itis placedontheboundsettingtozerotheassociatedvelocity compo-nent(seeFig.3a).Thisapproachhelpsthealgorithmtoexplorethe domainbounds.TheIWapproachisimplementedinAlgorithm3. Algorithm3. Inelasticwall-typeapproach(IW)

1: fori=1,numberofparticlesdo

2: forj=1,numberofvariablesdo

3: ifxk i,j>ujthen 4: xk i,j=uj,vki,j=0 5: else[ifxk i,j<lj] 6: xk i,j=lj,vki,j=0 7: endif 8: endfor 9: endfor

TheuseofIWhassomelimitations:intheunlikelyeventthatall theparticlestendtoleavethedomainfromthesamehyper-corner, theIWsetsallvelocitiestozeroandthePSOprogressmaystop pre-maturely.Forthisreason,asemi-elasticwall-typeapproach(SEW) isalsousedinthiswork.Specifically,incasetheparticleposition violatesaboundconstraint,thentheparticlepositionismodified inordertomakethatconstraintactive(i.e.theparticleismoved ontheboundaryofthatconstraint),whiletheassociated veloc-itycomponentisdefinedasinthefollowingAlgorithm4(seealso Fig.3b).

Algorithm4. Semi-elasticwall-typeapproach(SEW)

1: fori=1,numberofparticlesdo

2: forj=1,numberofvariablesdo

3: ifxk i,j>ujthen 4: vk i,j=−v k i,j/[␹(c1+c2)] 5: else[ifxk i,j<lj] 6: vk i,j=−vki,j/[␹(c1+c2)] 7: endif 8: endfor 9: endfor

Observethatthedampingfactor[(c1+c2)]−1inAlgorithm4is

usedtoconﬁnetheparticleinthefeasibledomain.

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318 A.Seranietal./AppliedSoftComputing49(2016)313–334 3.5. NumberoffunctionevaluationsandPSOiterations

ThemaximumnumberoffunctionevaluationsNfeval(evaluation

budget)inthepresentworkisdeﬁnedas

Nfeval=2nNdv, wheren∈N[7,12] (16)

andthereforerangesfrom128Ndvto4096Ndv.Asaconsequence,

accordingwiththesettinginEq.(11),thenumberofPSOiterations Niterperformedisgivenas

Niter= Nfeval Np = 2n_N dv 2m_N dv = 2n−m (17) 3.6. Evaluationmetrics

Threeabsoluteperformancecriteriaareusedtoassessthe algo-rithms,whicharedeﬁnedasfollows[24]:

x=

1 Ndv Ndv

j=1

_x j,min−xj,min Rj

2 , f = fmin−f_min f max−fmin , t=

2 x+2f 2 (18)

wherexrepresentsanormalizedEuclideandistancebetweenthe

minimumpositionfoundbythealgorithm(xmin)andthe

analyt-icalminimumposition(x

min), andRj=|uj−lj|istherangeofthe

jthvariable.fistheassociatednormalizeddistanceintheimage

space,wheref_ministheminimumfoundbythealgorithm,f

minis

theanalyticalminimum,andf

maxistheanalyticalmaximumofthe

functionf(x)inthedomainL.tisacombinationofxandfand

isusedforanoverallassessment.

Additionally,therelative variability2

r,k [29]for eachmetric

x, f,t (Eq. (18)) is usedto assessthe impactof each

set-tingparameterskonthealgorithms’performance.LetPbeaset

of|P|problems,deﬁningthealgorithm’ssettingparameter vec-torass=[s1,s2,...,sS]T ∈RS,therelativeperformancevariability

associatedtoitskthcomponentis 2 r,k= 2 k

|S| k=12k (19) where _k2= 1 |˝|

ω∈˝ [ ˆk(ω)] 2 −

1 |˝|

ω∈˝ ˆ k(ω)

2 (20)

with˝containingthepositionsωassumedbytheparametersk,

ˆ k(ω)= 1 |B|

s∈B ¯ (s), B={s:sk=ω} (21) and ¯ (s)=_|P|1

p∈P [(s)]p (22) 4. Optimizationproblems 4.1. Analyticaltestfunctions

One hundred analytical test functions are used in the pre-liminary numerical experience, including a wide variety of problems,suchascontinuousanddiscontinuous,differentiableand

non-differentiable,separableandnon-separable,scalableand non-scalable,unimodalandmultimodal,withdimensionalityranging fromtwo toﬁfty [26–29,46].The numericalexperience is con-ducted setting apart problems withrespectively less and more than 10 design variables, as summarized in Appendix A (see TablesA.10andA.11).

4.2. Hull-formSBDoptimizationofahigh-speedcatamaran Thehigh-speedDelftcatamaran[3]isusedasSBDtest prob-lem.Thisisaninternationalbenchmarkgeometryusedforboth experimentalandnumericalstudies,includingcapabilitiesofCFD predictionsforcomplexﬂowswithassociatedvalidationandglobal shapeoptimizationforcalmwaterandwaveconditions.Forthe currentproblem,theobjectivefunctionf(x)isdeﬁnedas

f(x)= RT

W (23)

whereRTis thetotal resistanceatFroudenumber(Fr)equalto

0.5incalmwater,andWistheweightforcemodulus.Geometry modificationshave tofit inabox, definedbymaximumoverall length,beam,and draught.Twofeasibledesign spacesare con-sidered.Thefirstincludesoveralldimensionbounds,whereasthe secondincludesoveralldimensionboundsand,inaddition, con-stantlengthbetweenperpendiculars(LBP)[5].Modificationsofthe parenthullareperformedusinghigh-dimensionalfree-form defor-mation(FFD)and95%-confidencedimensionalityreductionbased onKLEasshownin[6].Fourvariablesareusedforthefirstdesign spaceandsixforthesecond,referredtointhefollowingas4Dand 6Dspace,respectively.Newdesignsgareproducedas

g(x)=

⎛

⎝

1− Ndv

j=1 xj

⎞

⎠

g0+ Ndv

j=1 xjgj (24)

where−1≤xj≤1,

∀

j∈[1,Ndv]arethedesignvariables;Ndv=4for

theﬁrstdesignspace,whereasNdv=6forthesecond;g0isthe

orig-inalgeometryandgjarethegeometriesassociatedtothedesign

spaceprincipal directions (oreigenmodes), asprovided by KLE fordimensionalityreduction.Fordetails,thereaderisreferredto [5,6].

SimulationsareconductedusingtheWARP(WAveResistance Program)code,developedatCNR-INSEAN.Waveresistance com-putationsarebasedonlinearpotentialflowtheoryanddetailsof equations,numericalimplementation,andvalidationofthe numer-icalsolveraregivenin[33].Thefrictionalresistanceisestimated usingaflat-plateapproximation,basedonlocalReynoldsnumber [47].Themodelisfreetosinkandtrim(2degreesoffreedom). Simulationsareperformedfortherightdemi-hull,sincethe prob-lemissymmetricalwithrespecttothexz-plane.Thefreesurface isdiscretizedasfollows:20×1panelsontheinner-upstream sub-domain,20×40ontheouter-upstream,20×1ontheinner-hull, 20×40ontheouter-hull,80×1ontheinner-downstream,80×2 onthetransom-downstream,80×40ontheouter-downstream; thebody is discretizedby125×50 panels (see Fig.4).Domain boundsaredefinedby1LBPupstream,4LBPdownstreamand2LBP sideways.

Sincetheanalyticalsolutionoftheoptimizationproblemisnot available,thealgorithmperformanceisevaluatedconsideringthe objectivereductionanditsconvergence.Thedesignsolutionsare ﬁnallycomparedwithearlieroptimizationresultsproducedusing ahigh-ﬁdelityCFDsolver[5].

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X Y Z hull body X Y Z outer hull outer upstream outer downstream inner downstream innerhull inner upstream

Fig.4.Panel-gridfortheCNR-INSEANWARPpotential-ﬂowcode.

5. Numericalresults

5.1. TestfunctionsandDPSOparametersguidelineidentiﬁcation Resultsonthe100analyticaltestfunctionsarepresentedinthe followingsectionsandusedtodeﬁnetheguidelinesforSDPSOand ADPSO,adoptedlaterfortheSBDproblem.

5.1.1. SDPSO

Figs.5and7showtheperformancesofSDPSOversusthe bud-getoffunctionevaluations(Nfeval/Ndv),intermsofx,f,t,for

Ndv<10and≥10respectively.Averagevaluesarepresented,

condi-tionaltonumberofparticles,particleinitialization,coefﬁcientset, andwall-typeapproach,respectively.Figs.6and8showtherelative variance2

rofx,f,tforNdv<10and≥10respectively,retained

byeachoftheaforementionedparameters.Theparticles initial-izationis foundthemostsigniﬁcantparametertoaffectSDPSO performance,especiallyforNdv≥10,whereasthecoefﬁcientsset

(selectedherein)andthewall-typeapproachareshowntobethe leastrelevant.Tables4and5showtheﬁvebestperformingsetups foreachx,f,andt,forNdv<10and≥10respectively,varying

thebudgetoffunctionevaluations.Averagevaluesandstandard deviationsforallSDPSOsetupsarealsoprovided.

5.1.2. ADPSO

Generally,ADPSOresultsarefoundsimilartoSDPSO. Speciﬁ-cally,Figs.9and11showtheperformancesofADPSOversusthe budgetoffunctionevaluations,intermsofx,f,t,forNdv<10

and≥10respectively. Averagevaluesarepresented,conditional tonumberofparticles,particleinitialization,coefﬁcientsset,and wall-typeapproachrespectively.Figures10and12showthe rela-tivevariance2

r ofx,f,t forNdv<10and≥10respectively,

retainedby each of the aforementionedparameters. The parti-cleinitializationisagainthemostsigniﬁcantparametertoaffect ADPSOperformance,especiallyforlowbudgetsandNdv≥10.The

coefﬁcientsetand thewall-typeapproachareshown tohave a limited effect ontheperformance, compared toother parame-ters.Tables6and7summarizetheﬁvebestperformingsetupsfor eachx,f,andt,for Ndv<10and ≥10,respectively,varying

thebudgetoffunctionevaluations.Overallaveragesandstandard deviationsforallADPSOsetupsarealsoincluded.

5.1.3. Suggestedguidelines

ThemostpromisingsetupsareselectedfromTables4–7,inorder todeﬁneareasonableandrobustguidelinefortheuseofSDPSOand ADPSO.ThesearesummarizedinTable8.

Figs.13and14showtheperformanceofthesuggestedsetups, forSDPSOandADPSOandNdv<10and≥10,respectively.Average

performance, standard deviation, and best performing setup amongallcombinationsisalsoshownforeachbudget.The guide-linesetups(“Guide”)are foundalwaysvery closeorcoincident tothe“Best”.Inaddition,itmaybenotedhowADPSOisalways equivalentorslightlybetterthanSDPSO.

5.2. High-speedcatamaranSBDoptimization

A preliminarysensitivityanalysisfor each design variableis shown in Fig. 15 showing f(%) compared to the parent hull. Changesinfrevealareductionoftheobjectivefunctioncloseto 9%forthe4Ddesignspaceandcloseto10%forthe6D.Forthe cur-rentstudy,alldesignvariablesaredeemedrequiredforthehull formoptimization,eachoneprovidingasigniﬁcantvariationofthe designobjective.Accordingly,thesensitivityanalysisisnotapplied fordimensionalityreduction.

TheoptimizationisperformedwithbothSDPSOandADPSO,as pertheguidelinesuggestedinTable8.SDPSOandADPSOiterations areshowninFig.16,revealingaquitefastconvergence.Itmaybe notedthatthefinalconfigurationforthefirstdesignspace(4D)is fairlyclose(exceptforthesecondvariable)tothatobtainedusing metamodelsanda URANSsolverin[5],usedhereforreference (Fig.17a).Fortheseconddesignspace(6D)thedifferencesaremore significant(Fig.17b).Asageneralresult,onecanalsoobservebythe optimizationresults(seeTable9)thatSDPSOandADPSOwithboth IWandSEWleadtoareductionoftheobjectivefunctioncloseto 20%,forthe4Ddesignspace,andgreaterthan20%forthe6Ddesign space.Furthermore,theoptimumconfigurationleadstoa consid-erablereductionofwave’selevationcomparedtotheoriginalshape (Figs.18and19).Therearenotsignificantdifferencesbetweenthe resultsobtainedbySDPSOandADPSO,exceptforSDPSOwithIWfor the6Ddesignspace.InthiscasetheIWapproachinducesthe opti-mizationtostopafter6iterations.AsshowninFig.17,differences inoptimaldesignvariablesaremainlyduetoIWorSEW.

6. Discussion

TheanalysesofSDPSOandADPSOperformancesonthe ana-lyticaltestfunctionshaveshowntheimportanceoftheparticle initializationespeciallyforproblemswithahighnumberofdesign variables. In particular, the initialization on the only domain boundary is alwaystheworst and should beavoided, whereas the use of a non-null initial velocity is always advisable (see Figs.5b,7b,9band11b).

Thenumber ofparticles becomes moderatelyrelevantwhen thebudgetoffunctionevaluationsincreases,speciallyfor prob-lems with a smaller number of variables. More in detail,

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Fig.5.SDPSOaverageperformanceforNdv<10.

Fig.6. Relativevariance2

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Fig.7. SDPSOaverageperformanceforNdv≥10.

Fig.8.Relativevariance2

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322 A. Serani et al. / Applied Soft Computing 49 (2016) 313–334 Table4

BestperformingsetupsforSDPSO,Ndv<10.

Nfeval/Ndv Average(STD) BestSDPSO

x f t NNdvp Init Coef Wall x

Np

Ndv Init Coef Wall f

Np

Ndv Init Coef Wall t

128 0.193_(0.078) 0.155_(0.078) 0.205_(0.081)

32 A.1 3 SEW 0.099 4 C.1 4 IW 0.033 4 C.1 4 SEW 0.086

4 C.1 4 SEW 0.102 4 C.0 4 SEW 0.034 4 C.1 4 IW 0.086

4 C.1 4 IW 0.106 2 A.1 4 SEW 0.035 2 A.1 4 SEW 0.091

4 C.1 3 IW 0.106 4 C.1 4 SEW 0.036 4 C.0 4 SEW 0.098

32 A.1 3 IW 0.107 2 A.1 4 IW 0.046 2 A.1 4 IW 0.100

256 0.178_(0.076) 0.134_(0.079) 0.185_(0.081)

32 A.1 3 SEW 0.079 4 C.1 4 SEW 0.031 4 C.1 4 IW 0.082

64 A.1 3 SEW 0.082 4 C.1 4 IW 0.032 4 C.1 4 SEW 0.082

32 A.1 3 IW 0.092 4 C.0 4 SEW 0.032 16 A.1 3 SEW 0.086

64 A.1 3 IW 0.092 2 A.1 4 SEW 0.035 32 A.1 3 SEW 0.089

64 A.0 4 SEW 0.097 16 A.1 3 SEW 0.042 2 A.1 4 SEW 0.091

512 0.167 (0.077) 0.119 (0.080) 0.170 (0.082)

64 A.1 4 IW 0.070 4 C.1 4 SEW 0.031 32 A.1 3 SEW 0.068

32 A.1 3 SEW 0.072 4 C.1 4 IW 0.032 64 A.1 3 IW 0.071

64 A.1 2 IW 0.072 4 C.0 4 SEW 0.032 64 A.1 3 SEW 0.073

64 A.1 3 IW 0.074 2 A.1 4 SEW 0.035 64 A.1 4 IW 0.078

64 A.1 3 SEW 0.076 32 A.1 3 SEW 0.036 4 C.1 4 IW 0.081

1024 0.161 (0.078) 0.109 (0.083) 0.161 (0.085)

64 A.1 4 IW 0.063 64 A.1 3 SEW 0.021 64 A.1 3 SEW 0.052

64 A.1 3 SEW 0.063 32 A.1 3 SEW 0.021 64 A.1 4 IW 0.054

64 A.1 2 IW 0.065 64 A.0 4 IW 0.023 32 A.1 3 SEW 0.056

64 A.1 2 SEW 0.066 64 A.1 3 IW 0.023 64 A.1 3 IW 0.057

32 A.1 3 SEW 0.070 32 A.0 4 SEW 0.023 64 A.1 4 SEW 0.062

2048 0.157_(0.080) 0.103_(0.085) 0.154_(0.088)

64 A.1 2 IW 0.053 64 A.1 3 SEW 0.020 64 A.1 2 IW 0.045

64 A.1 4 IW 0.059 64 A.1 4 SEW 0.020 64 A.1 4 IW 0.049

64 A.1 3 SEW 0.061 64 A.1 4 IW 0.020 64 A.1 3 SEW 0.050

64 A.1 2 SEW 0.063 64 A.0 4 SEW 0.020 64 A.1 2 SEW 0.052

64 A.1 3 IW 0.067 32 A.0 4 SEW 0.021 64 A.1 3 IW 0.055

4096 0.156_(0.081) 0.100_(0.086) 0.152_(0.089)

64 A.1 2 IW 0.049 128 A.1 3 IW 0.020 64 A.1 2 IW 0.042

64 A.1 4 IW 0.058 128 C.1 3 SEW 0.020 64 A.1 4 IW 0.048

64 A.1 2 SEW 0.059 64 A.1 2 IW 0.020 64 A.1 2 SEW 0.049

64 A.1 3 SEW 0.060 128 C.1 4 SEW 0.020 64 A.1 3 SEW 0.050

128 C.1 4 IW 0.064 64 C.0 4 SEW 0.020 128 C.1 4 IW 0.053 Av. 0.169 (0.077) 0.120 (0.079) 0.171 (0.082)

32 A.1 3 SEW 0.076 4 C.1 4 SEW 0.032 4 C.1 4 SEW 0.081

64 A.1 3 SEW 0.078 4 C.1 4 IW 0.032 4 C.1 4 IW 0.081

64 A.1 2 IW 0.078 4 C.0 4 SEW 0.033 32 A.1 3 SEW 0.082

64 A.1 4 IW 0.079 2 A.1 4 SEW 0.035 64 A.1 3 SEW 0.082

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A. Serani et al. / Applied Soft Computing 49 (2016) 313–334 323

BestperformingsetupsforSDPSO,Ndv≥10.

Nfeval/Ndv Average(STD) BestSDPSO

Np

128 0.390 (0.101) 0.274 (0.150) 0.368 (0.128)

4 A.1 4 SEW 0.261 4 A.1 4 SEW 0.118 4 A.1 4 SEW 0.221

2 A.1 4 SEW 0.266 8 A.1 3 SEW 0.121 2 A.1 2 IW 0.224

8 A.1 3 SEW 0.267 4 A.1 3 SEW 0.122 2 A.1 4 SEW 0.225

4 A.1 3 SEW 0.267 2 A.1 4 SEW 0.122 8 A.1 3 SEW 0.227

8 C.1 3 SEW 0.268 4 C.1 4 SEW 0.122 4 A.1 3 SEW 0.227

256 0.379_(0.102) 0.256_(0.148) 0.354_(0.128)

4 A.1 4 SEW 0.257 8 A.1 3 SEW 0.107 4 A.1 4 SEW 0.214

8 A.1 3 SEW 0.260 4 A.1 4 SEW 0.111 8 A.1 3 SEW 0.216

16 A.1 3 SEW 0.260 4 C.1 4 SEW 0.113 8 C.1 4 SEW 0.219

8 C.1 4 SEW 0.261 8 C.1 4 SEW 0.114 8 C.1 3 SEW 0.220

8 C.1 3 SEW 0.261 8 C.1 3 SEW 0.115 2 A.1 2 IW 0.220

512 0.367_(0.103) 0.240_(0.147) 0.340_(0.128)

32 A.1 3 SEW 0.250 16 A.1 3 SEW 0.100 16 A.1 3 SEW 0.208

16 A.1 3 SEW 0.253 8 A.1 3 SEW 0.104 32 A.1 3 SEW 0.212

16 A.1 4 SEW 0.253 8 C.1 4 SEW 0.106 8 C.1 4 SEW 0.212

8 C.1 4 SEW 0.255 16 C.1 3 SEW 0.107 8 A.1 3 SEW 0.213

64 A.1 3 SEW 0.255 32 A.1 3 SEW 0.109 4 A.1 4 SEW 0.213

1024 0.361 (0.104) 0.230 (0.148) 0.332 (0.129)

64 A.1 3 SEW 0.242 32 A.1 3 SEW 0.095 32 A.1 3 SEW 0.200

32 A.1 3 SEW 0.244 16 A.1 3 SEW 0.097 64 A.1 3 SEW 0.204

16 A.1 4 SEW 0.246 32 C.1 3 SEW 0.099 16 A.1 4 SEW 0.205

32 A.1 4 SEW 0.247 16 C.1 4 SEW 0.101 16 A.1 3 SEW 0.205

16 A.1 3 SEW 0.250 64 A.1 3 SEW 0.103 32 A.1 4 SEW 0.207

2048 0.357 (0.105) 0.223 (0.149) 0.326 (0.130)

64 A.1 3 SEW 0.235 32 A.1 3 SEW 0.092 64 A.1 3 SEW 0.194

32 A.1 4 SEW 0.240 64 A.1 3 SEW 0.092 32 A.1 3 SEW 0.198

32 A.1 3 SEW 0.242 32 C.1 3 SEW 0.096 32 A.1 4 SEW 0.199

64 A.1 4 SEW 0.245 16 A.1 3 SEW 0.097 16 A.1 4 SEW 0.203

16 A.1 4 SEW 0.245 64 C.1 3 SEW 0.097 64 A.1 4 SEW 0.204

4096 0.354_(0.105) 0.220_(0.150) 0.323_(0.131)

64 A.1 3 SEW 0.233 64 A.1 3 SEW 0.090 64 A.1 3 SEW 0.191

128 A.1 3 SEW 0.239 32 A.1 3 SEW 0.092 128 A.1 3 SEW 0.196

128 A.1 4 SEW 0.239 128 C.1 3 SEW 0.094 64 A.1 4 SEW 0.197

64 A.1 4 SEW 0.239 64 C.1 4 SEW 0.094 128 A.1 4 SEW 0.197

32 A.1 4 SEW 0.239 64 C.1 3 SEW 0.094 64 A.1 4 IW 0.197

Av. 0.368_(0.102) 0.240_(0.147) 0.340_(0.128)

32 A.1 3 SEW 0.254 8 A.1 3 SEW 0.107 16 A.1 3 SEW 0.214

16 A.1 3 SEW 0.256 16 A.1 3 SEW 0.108 4 A.1 4 SEW 0.215

4 A.1 4 SEW 0.257 4 A.1 4 SEW 0.111 32 A.1 3 SEW 0.216

16 A.1 4 SEW 0.257 8 C.1 4 SEW 0.112 8 A.1 3 SEW 0.216

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Fig.9. ADPSOaverageperformanceforNdv<10.

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Fig.11.ADPSOaverageperformanceforNdv≥10.

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326 A. Serani et al. / Applied Soft Computing 49 (2016) 313–334 Table6

BestperformingsetupsforADPSO,Ndv<10.

Nfeval/Ndv Average(STD) BestADPSO

Np

128 0.191_(0.078) 0.155_(0.076) 0.205_(0.080)

4 C.1 4 IW 0.095 4 C.1 3 IW 0.023 4 C.1 4 IW 0.075

4 C.1 4 SEW 0.097 4 C.1 4 IW 0.024 4 C.1 4 SEW 0.084

8 A.1 3 IW 0.100 4 C.0 3 IW 0.034 4 C.1 3 IW 0.087

8 A.1 3 SEW 0.107 4 C.0 4 SEW 0.039 4 C.0 3 IW 0.094

2 A.1 4 SEW 0.110 4 C.0 4 IW 0.039 4 C.0 4 SEW 0.096

256 0.176_(0.075) 0.134_(0.076) 0.185_(0.079)

4 C.1 4 IW 0.085 4 C.1 4 IW 0.021 4 C.1 4 IW 0.068

32 A.1 3 SEW 0.092 4 C.1 3 IW 0.023 4 C.1 4 SEW 0.080

4 C.1 4 SEW 0.093 4 C.0 4 IW 0.031 4 C.1 3 IW 0.085 8 A.1 3 IW 0.096 4 C.0 3 IW 0.033 4 C.0 4 SEW 0.087 32 A.1 4 IW 0.097 4 C.0 4 SEW 0.034 4 C.0 3 IW 0.090 512 0.163 (0.073) 0.118 (0.077) 0.168 (0.078) 32 A.1 4 IW 0.073 4 C.1 4 IW 0.020 4 C.1 4 IW 0.065

64 A.1 3 SEW 0.080 4 C.1 3 IW 0.023 16 A.0 4 IW 0.074

4 C.1 4 IW 0.082 16 A.0 4 IW 0.024 4 C.1 4 SEW 0.078

32 C.1 4 IW 0.084 4 C.0 4 IW 0.031 32 A.1 3 SEW 0.079

32 A.1 3 SEW 0.084 4 C.0 3 IW 0.033 64 A.1 3 SEW 0.082

1024 0.156 (0.075) 0.107 (0.079) 0.157 (0.081) 32 A.1 4 IW 0.064 4 C.1 4 IW 0.020 32 A.1 4 IW 0.057

64 A.1 3 SEW 0.071 16 A.0 4 IW 0.021 64 A.1 3 SEW 0.059

64 A.1 4 SEW 0.072 4 C.1 3 IW 0.023 4 C.1 4 IW 0.065

32 A.0 4 SEW 0.077 64 A.1 3 SEW 0.024 64 A.0 3 SEW 0.067

64 A.0 4 IW 0.077 64 A.0 3 SEW 0.024 64 A.0 4 SEW 0.067

2048 0.151_(0.076) 0.100_(0.082) 0.150_(0.083)

64 A.1 4 SEW 0.064 4 C.1 4 IW 0.020 64 A.1 4 SEW 0.053

32 A.1 4 IW 0.064 32 C.1 4 SEW 0.021 32 A.1 4 IW 0.056

64 A.1 3 SEW 0.070 16 A.0 4 IW 0.021 64 A.1 3 SEW 0.057

64 A.1 1 IW 0.073 64 A.0 4 IW 0.022 64 A.0 4 IW 0.059

64 A.0 4 SEW 0.074 32 A.1 2 IW 0.022 64 A.0 4 SEW 0.060

4096 0.150_(0.077) 0.097_(0.083) 0.147_(0.085)

64 A.1 4 SEW 0.063 128 A.1 4 SEW 0.020 64 A.1 4 SEW 0.052

32 A.1 4 IW 0.064 4 C.1 4 IW 0.020 32 A.1 4 IW 0.056

64 A.1 3 SEW 0.069 32 A.1 2 IW 0.021 64 A.1 3 SEW 0.056

64 A.1 1 IW 0.070 32 C.1 4 SEW 0.021 64 A.1 1 IW 0.057

128 A.0 2 SEW 0.072 16 A.0 4 IW 0.021 64 A.0 4 IW 0.059

Av. 0.164 (0.073) 0.119 (0.076) 0.169 (0.078) 32 A.1 4 IW 0.079 4 C.1 4 IW 0.021 4 C.1 4 IW 0.067 4 C.1 4 IW 0.084 4 C.1 3 IW 0.023 4 C.1 4 SEW 0.080

64 A.1 3 SEW 0.089 4 C.0 4 IW 0.033 32 A.1 4 IW 0.085

32 A.1 3 SEW 0.090 4 C.0 3 IW 0.033 4 C.1 3 IW 0.085

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A. Serani et al. / Applied Soft Computing 49 (2016) 313–334 327

BestperformingsetupsforADPSO,Ndv≥10.

Nfeval/Ndv Average(STD) BestADPSO

Np

128 0.386 (0.100) 0.268 (0.146) 0.363 (0.125)

4 A.1 4 SEW 0.253 4 A.1 3 SEW 0.112 4 A.1 4 SEW 0.213

2 A.1 4 SEW 0.255 2 C.1 4 SEW 0.113 2 A.1 4 SEW 0.216

4 C.1 4 SEW 0.256 4 A.1 4 SEW 0.114 4 C.1 4 SEW 0.217

4 C.1 2 SEW 0.257 2 A.1 1 IW 0.115 2 A.1 2 IW 0.218

2 A.1 2 SEW 0.260 4 C.1 4 SEW 0.117 4 A.1 3 SEW 0.220

256 0.374_(0.101) 0.249_(0.143) 0.349_(0.125)

4 A.1 4 SEW 0.246 4 A.1 4 SEW 0.100 4 A.1 4 SEW 0.202

4 C.1 4 SEW 0.250 8 A.1 3 SEW 0.108 4 C.1 4 SEW 0.209

4 C.1 2 SEW 0.252 4 C.1 4 SEW 0.109 8 C.1 4 SEW 0.212

8 C.1 4 SEW 0.252 4 A.1 3 SEW 0.110 4 C.1 2 IW 0.213

2 A.1 4 SEW 0.254 8 C.1 3 SEW 0.110 2 A.1 4 SEW 0.215

512 0.361_(0.101) 0.232_(0.141) 0.333_(0.124)

8 C.1 4 SEW 0.245 8 C.1 4 SEW 0.098 4 A.1 4 SEW 0.200

4 A.1 4 SEW 0.245 4 A.1 4 SEW 0.099 8 C.1 4 SEW 0.200

16 A.1 4 SEW 0.246 16 A.1 3 SEW 0.103 16 A.1 4 SEW 0.206

4 C.1 4 SEW 0.249 8 A.1 3 SEW 0.104 8 A.1 4 SEW 0.206

16 C.1 4 SEW 0.250 8 A.1 4 SEW 0.104 4 C.1 4 SEW 0.207

1024 0.354 (0.101) 0.221 (0.142) 0.324 (0.125)

32 A.1 4 SEW 0.237 8 C.1 4 SEW 0.095 16 A.1 4 SEW 0.197

16 A.1 4 SEW 0.239 16 A.1 4 SEW 0.097 8 C.1 4 SEW 0.198

16 C.1 4 SEW 0.242 16 C.1 4 SEW 0.097 16 C.1 4 SEW 0.199

32 A.1 4 IW 0.243 4 A.1 4 SEW 0.099 32 A.1 4 SEW 0.200

8 C.1 4 SEW 0.243 16 A.1 3 SEW 0.099 4 A.1 4 SEW 0.200

2048 0.349 (0.102) 0.214 (0.143) 0.317 (0.126)

32 A.1 4 SEW 0.228 32 A.1 4 SEW 0.091 32 A.1 4 SEW 0.187

64 A.1 3 SEW 0.233 32 C.1 4 SEW 0.093 64 A.1 3 SEW 0.194

16 A.1 4 SEW 0.238 64 A.1 3 SEW 0.095 16 A.1 4 SEW 0.195

32 A.1 4 IW 0.239 16 C.1 4 SEW 0.095 32 A.1 4 IW 0.196

16 C.1 4 SEW 0.239 16 A.1 4 SEW 0.095 16 C.1 4 SEW 0.196

4096 0.346_(0.103) 0.210_(0.144) 0.314_(0.127)

32 A.1 4 SEW 0.227 64 C.1 4 SEW 0.088 32 A.1 4 SEW 0.186

64 A.1 3 SEW 0.231 32 A.1 4 SEW 0.089 64 A.1 3 SEW 0.190

128 A.1 3 SEW 0.236 64 A.1 3 SEW 0.092 64 A.1 4 IW 0.193

64 A.1 4 IW 0.236 32 C.1 4 SEW 0.092 128 A.1 3 SEW 0.193

64 A.1 4 SEW 0.237 64 A.1 4 SEW 0.092 64 A.1 4 SEW 0.193

Av. 0.362_(0.100) 0.232_(0.141) 0.333_(0.124)

4 A.1 4 SEW 0.246 4 A.1 4 SEW 0.101 4 A.1 4 SEW 0.203

8 C.1 4 SEW 0.248 8 C.1 4 SEW 0.104 8 C.1 4 SEW 0.206

16 A.1 4 SEW 0.249 8 A.1 3 SEW 0.108 4 C.1 4 SEW 0.209

32 A.1 4 SEW 0.250 4 C.1 4 SEW 0.109 16 A.1 4 SEW 0.211

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Table8

SuggestedguidelineforSDPSOandADPSO.

Ndv Np/Ndv Initialization Coefﬁcientsset Wall-type

SDPSO <10_≥10 4₄ C.1_A.1 4₄ IW/SEW_SEW

ADPSO <10_≥10 4₄ C.1_A.1 4₄ IW/SEW_SEW

Fig.13. PerformanceofsuggestedguidelinesusingSDPSO.

Figs. 5a, 7a,9a, and 11a showthat a higher number of parti-clesperformsbetterwithahighbudgetoffunctionevaluations. Thiscanbeexplainedrecallingtherelationbetweenthenumber ofparticles and thenumber ofiterations, fora ﬁxedbudget of

functionevaluations.Ontheonehand,alownumberofparticles requires alarger numberof iterations,providing a fast conver-gence butpossibly alsoa faststagnation. On theotherhand, a highnumber ofparticlesrequiresa lowernumberofiterations,

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Fig.15.Sensitivityanalysis.

Fig.16.ConvergenceofSDPSOandADPSO.

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330 A.Seranietal./AppliedSoftComputing49(2016)313–334 Table9

SBDresults.

Wall-type RT[N] W[N] f[–] f[%]

Designspace Original 50.15 852.5 5.88e−2 –

4D

SDPSO IW 39.92 850.9 4.69e−2 −20.24 SEW 40.36 850.6 4.67e−2 −20.57 ADPSO IW_SEW 39.85_40.68 851.1_851.7 4.68e−2_4.71e−2 −20.41_−19.90

6D

SDPSO IW_SEW 39.41_34.29 849.1_835.4 4.64e−2_4.10e−2 −21.10_−30.27 ADPSO IW_SEW 34.31_34.08 835.5_830.8 4.11e−2_4.09e−2 −30.10_−30.30

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Fig.19.6DSBDresults.

providingaslowerconvergenceandpossiblydelayingstagnation. Thisisthereasonwhythenumberofparticlesshouldbe deter-mined considering theavailable budget, that is usually low in theSBDcontext,especiallywhenCPU-timeexpensivesolversare used.

The coefficient set and the wall-type approach seem to be the less relevant parameters. Nevertheless, it can be said that Clerc’scoefficients(=0.721,c1=c2=1.655[31])andtheSEW-type approacharethemosteffectiveandefficientchoicesforbothSDPSO andADPSO(seeFigs.5c,7c,11c,5d,7d,9d,and11d).Moreover,the

useofSEWissuggestedinordertopreventanearlystopofthe swarmparticledynamics,asshownforthe6Ddesignspaceofthe navalengineeringproblem.

The asynchronous mechanism of ADPSO shows equivalent or slightly better performance (in terms of number of objec-tive function evaluations and objective reduction) compared to SDPSO, probably due to the diversity in the information update. This aspect of ADPSO provides a further interesting opportunityfortheexploitationofparallelarchitecturesin HPC systems.

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332 A.Seranietal./AppliedSoftComputing49(2016)313–334 7. Conclusions

A guideline for an effective and efficient useof SDPSO and ADPSO,in a computationalframeworkcharacterizedby limited resources,has been suggested. A parametric analysis hasbeen performedvaryingthe numberof particles,theinitialization of theswarm, the setof coefficients, and the wall-type approach fortheboxconstraints.Theassessmentisbasedon100 analyt-ical test functions (with dimensionality from two to fifty) and three differentabsoluteperformance criteria. Allpossible com-binationsofDPSOparametersledto420optimizationsforeach function.ThemostpromisingDPSOsetupshavebeendiscussed, identified, and successfully applied to a ship SBD optimization problem, namely thehull-form optimization of the high-speed Delftcatamaran,advancingincalmwateratfixedspeed,usinga potential-flowsolver.Theoptimizationaimedatthereductionof theratiobetweenthetotalresistanceandtheshipweight,using four-andsix-dimensionaldesignspaces.

Theparticlesinitializationhasbeenfoundthemostsigniﬁcant parameterfortheDPSOperformance,especiallyforanumberof designvariablesNdv≥10andlowbudgetsoffunctionevaluations.

Conversely,thecoefﬁcientssetandthewall-typeapproachhave beenfoundhavinga little inﬂuenceon theDPSO performance, comparedtotheotherparameters.

Forproblemswithlessthen10variables,thesuggestedSDPSO andADPSOsetupsfollow:(a)numberofparticlesNp equalto4

timesthenumberofdesignvariables;(b)particleinitializationwith HSSdistributionondomainandboundswithnon-nullvelocity;(c)

setof coefﬁcientsproposedin[31],i.e.,=0.721,c1=c2=1.655;

(d)inelasticandsemi-elasticwall-typeapproach,forSDPSOand ADPSOrespectively.

Forproblemswithmorethen10designvariables,thesuggested setupsforSDPSOandADPSOfollow:(a)number ofparticlesNp

equalto4timesthenumberofdesignvariables;(b)particles ini-tialization includingHSS distribution on domainwith non-null velocity;(c)setofcoefﬁcientsproposedin[31];(d)semi-elastic wall-typeapproach.

Theperformanceoftheguidelinesuggestedforthetest func-tions are very close or coincident to the best setup for each evaluation budget, among all 420conﬁgurations available (see “Guide”and“Best”inFigs.13and14).Finally,theguidelinesetups haveprovedtoperformwellonboththe4Dand6DshipSBD opti-mizationproblems.

Acknowledgments

Thepresent researchissupportedby theUS Ofﬁceof Naval Research,NICOPGrantN62909-15-1-2016,underthe administra-tionofDr.Woei-MinLinandDr.Ki-HanKim,andbytheItalian Flagship Project RITMARE, coordinated by the Italian National ResearchCouncilandfundedbytheItalianMinistryofEducation. AppendixA. Listoftestfunctions

TablesA.10andA.11summarizetheanalytical testfunctions withNdv<10andNdv≥10,respectively,usedinthecurrentwork.

TableA.10

AnalyticaltestfunctionswithNdv<10.

fp Name Dimension Bounds Optimum

Ndv [l,u]j,...,Ndv _f

min

f1,2 5nloc.minima(Levy) 2,5 [−5,5]Ndv _0.000

f7 Ackley 2 [−5,4]Ndv 0.000 f8,9 Alpine 2,5 [−9,7]Ndv _0.000 f10 Beale 2 [−4.5,4.5]Ndv _0.000 f11 Booth 2 [−10,10]Ndv _0.000 f12 Bukinn.6 2 [−15,−5]×[−3,3] 0.000 f13 Colville 4 [−10,10]Ndv _0.000 f14,15 CosineMixture 2,4 [−1,0.5]Ndv −0.100·_N_dv f16,17 Dixon-Price 2,5 [−1,1]Ndv 0.000 f18 Easom 2 [−100,100]Ndv −1.000 f19,20 Exponential 2,4 [−9,7]Ndv −1.000 f21 Freudenstein-Roth 2 [−5,5]Ndv _0.000 f22 Goldstein-Price 2 [−2,2]Ndv _3.000 f23,24 Griewank 2,5 [−9,7]Ndv _0.000 f25 Hartmann.3 3 [0,1]Ndv −3.860 f26 Hartmann.6 6 [0,1]Ndv −3.320 f27 Matyas 2 [−9,7]Ndv _0.000 f28,29 MultiModal 2,5 [−1,0.5]Ndv _0.000 f30 Powell 8 [−4,5]Ndv _0.000 f31 Quartic 2 [−10,10]Ndv −0.352 f32 Rosenbrock 2 [−30,30]Ndv _0.000 f33 Schaffern.2 2 [−100,90]Ndv _0.000 f34 Schaffern.6 2 [−100,90]Ndv 0.000 f35 Schubertpenalty1 2 [−10,10]Ndv −186.731 f36 Schubertpenalty2 2 [−10,10]Ndv −186.731 f37 Shekeln.5 4 [0,10]Ndv −10.153 f38 Shekeln.7 4 [0,10]Ndv −10.403 f39 Shekeln.10 4 [0,10]Ndv −10.536

f40 Six-HumpCamelBack 2 [−2.5,2.5]×[−1.5,1.5] −1.032

f41 Sphere 2 [−5,4]Ndv _0.000

f42,43 Styblinski-Tang 2,4 [−5,5]Ndv −39.166·_N

dv

f44 TestTubeHolder 2 [−10,10]Ndv −10.872

f45 Three-HumpCamelBack 2 [−5,4]Ndv _0.000

f46 Treccani 2 [−5,4]Ndv _0.000

f47 Tripod 2 [−100,100]Ndv _0.000

f48 Vincent 5 [0.25,10]Ndv −Ndv

f49 Xin-SheYangn.2 5 [−2,]Ndv _0.000

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TableA.11

AnalyticaltestfunctionswithNdv≥10.

fp Name Dimension Bounds Optimum

Ndv [l,u]j,...,Ndv fmin

f7,8,9 Ackley 10,30,50 [−5,4]Ndv _0.000 f10,11 Alpine 10,20 [−9,7]Ndv _0.000 f12,13,14 Dixon-Price 10,25,50 [−1,1]Ndv 0.000 f15,16 Griewank 10,20 [−9,7]Ndv _0.000 f17,18,19 Mishran.11 10,25,50 [−10,9]Ndv _0.000 f20,21 MultiModal 10,20 [−1,0.5]Ndv _0.000 f22,23,24 Pathological 10,25,50 [−100,90]Ndv _0.000 f25 Paviani 10 [2.0001,9.9999]Ndv −45.778 f26,27 Powell 16,24 [−4,5]Ndv _0.000 f28,29,30 Rastrigin 10,30,50 [−5.12,5.12]Ndv 0.000 f31,32,33 Rosenbrock 10,25,50 [−30,30]Ndv _0.000 f34,35,36 Schwefel 10,25,50 [−500,500]Ndv −0.100·_N dv f37,38,39 Trigonometricn.2 10,25,50 [−500,500]Ndv _1.000 f40,41 Vincent 15,30 [0.25,10]Ndv −N_dv f42,43,44 Xin-SheYangn.2 10,25,50 [−2,]Ndv _0.000 f45,46,47 Xin-SheYangn.4 10,25,50 [−10,10]Ndv −1.000 f48,49,50 Zacharov 10,25,50 [−5,10]Ndv _0.000 References

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