A Dynamic System Model of Biogeography-Based Optimization

Cleveland State University

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Electrical Engineering & Computer Science

Department

2011

A Dynamic System Model of Biogeography-Based

Optimization

Daniel J. Simon

Cleveland State University

, [email protected]

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Changes may have been made to this work since it was submitted for publication. A definitive version

was subsequently published in Applied Soft Computing, 11, 8, (01-01-2011); 10.1016/

j.asoc.2011.03.028.

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Simon, Daniel J., "A Dynamic System Model of Biogeography-Based Optimization" (2011).Electrical Engineering & Computer Science Faculty Publications. 140.

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Original Citation

Dan Simon. (2011). A dynamic system model of biogeography-based optimization. Applied Soft Computing, 11(8), 5652-5661, doi:

10.1016/j.asoc.2011.03.028.

A dynamic

system

model

of

biogeography-based optimization

"

Dan

Si

mon

Clf>Veland Siale University. Df>partmenr ofElectrical and Computer Engineering. CIne/and. OH. United Slaies

1.

Introduction

Biogeography is [he study of the migration, speciation. and

extinct

i

on

of species

[1.2[.

Biogeography has

often

been considered

as a process that enforces equilibrium in the number of species in

habitats. However.

equilibrium

in a

system

can

also

be considered

as a minimum-energy configuration,

so

we

see

that biogeography

can

be

viewed

as

a

n

optimization

process. This idea

is

further dis­

cussed in

13

[

.

Biogeography-based optimization (BBO) is an evolutionary

algorithm

(

EA

)

morjv.ned

by

the

optimality perspective of

nat­

ural biogeography, and was initially developed in

[4

[.

Just

as

species

migrate back and forth between islands, BBO operates

by

s

h

a

rin

g

info

r

mation between individuals in a

populati

on

of

candidate solutions.

BBO

has shown good

performance

both on

benchmark problems

[

J-5

[

and on real-world

problems, includ

­

ing aircraft engine

se

n

sor se

lection [4

J

,

power

system

optimization

[6,

7

J

.

groundwater

detec

t

ion

[8

j,

mech

anical gear

train design

[9J

,

sate

l

lite

image classification

[

10

J

. and neuro-fuzzy

system

training

for

biomedical

applications

1

11

J.

Section

1

.

1

summarizes some

of the impor

t

ant notation

u

se

d

in

this paper.

Sec

t

ion

1

.2 gives an

overview

of

BBO.

Section

1.3

gives

an

overview

and outline of

the

remainder of this paper.

t)' This work was supponed by NSF erant 0816114 in the CMMI Division of the Enginet'ring Directorate.

E-mail address: [email protected]

1.1. Notation

This

section summarizes some

of the notation

used

in BBO and

in

this paper.

Some

of these terms may be more

gen

eral

or more

speCific in other

contexts. The definitions

indi

cated

here are not

universal. but

are commonly

used. and more importantly for our

purposes, are

specifically

used in this paper.

BBO:

Biogeo

graphy

-

based

optimization, an EA

whic

h

is mo

t

i­

vated

by the

math

ematical

p

r

inciples

of

natural biogeography.

Dynamic system:

A process whose

state

at time

step

(t -+-

1

)

depends

only on the

state

at

time

t.

The transition of the

state

from

one time

step

to the next is deterministic.

Emigration

:

The sharing of

a

solution

feature in BBO from

o

ne

individual

to another. The emigrating

so

luti

on

feature remains in

the

emigrating

individual.

Thi

s

is

similar

to emigration

of a

s

pecies

in

biogeography. in which

representatives of a species

le

ave

an

isl

and

but

the

s

peci

es

does not become extinct from t

h

e emigrating

island.

GASP:

A

genetic a

l

gorithm

wit

h

single-poi

nt

crossove

r.

GAGUR: Agenetic

algorithm

with

global

uniform

recombination.

This means that a

solution

feature of an offspring

can

rece

i

ve

each

solution feature

from

a different parent. The likelihood

that

any

given

solut

i

on feature

in

the offspring

comes

from any

give

n

pare

n

t

is

proportional

to

that parent

's

fi

t

ness.

Individual:

Acand

i

date solution in an EA.

Immigration: The replacement

of

an old

solution

feature

in

an

individual

with a

new solution

feature from

another

individual.

T

he

emigration.Theimmigratingsolutionfeaturereplacesafeaturein theimmigratingindividual.

Markov process: A process whose state at time step (t+1) dependsonlyonthestateattimet.Thetransitionofthestatefrom onetimesteptothenextisprobabilistic.

Populationdistribution:The distributionof individuals inthe searchspace.Forexample,ifthesearchspaceconsistsoffourpossi­ blesolutions,{x1,x2,x3,x4},andthepopulationsizeisthree,thena

populationdistributionmightconsistofonecopyofx1,zerocopies

ofx2,twocopiesofx3,andonecopyofx4.

Solutionfeature: Anindependentvariableof anoptimization problem.Forexample,ifthesolutiondomainisabitstring,thena solutionfeatureisabit.

GenerationalEA:AnEAinwhichrecombinationisperformedto createanentirenewpopulationbeforeanyoftheoldpopulation membersarereplaced.

Steady-stateEA:AnEAinwhichrecombinationisperformedto createasinglenewindividualwhichreplacesoneoftheoldindivid­ ualsinthepopulationbeforethenextrecombinationisperformed.

1.2. Biogeography-basedoptimization

This section gives an overview of BBO. BBO operates by migrating information betweenindividuals, thus resulting in a modificationofexistingindividuals.Individualsdonotdieatthe endofageneration.Inaddition,ahigh-fitnessBBOindividualis unlikelytoacceptinformationfromalow-fitnessindividual.This is motivated by biogeography and does not have an analog in GAs.Innaturalbiogeography,averyhabitable islandisunlikely toacceptimmigrantsfromalesshabitableisland [12].Thisisdue totworeasons.First,theveryhabitableislandisalreadysaturated withspeciesanddoesnothavemanyadditionalresourcestosup­ portimmigrants.Second,theinhabitableislanddoesnothavevery manyspeciestobeginwith,andsoitdoesnothavemanypotential emigrants.

BBOismotivatedbybiogeographybutisnotintendedtobea simulationofbiogeography. Theanalogybetweenbiogeography andBBObreaksdownatseveralpoints.Forexample,inbiogeog­ raphythenumberofspeciesvariesfromislandtoisland,whilein BBOthenumberofsolutionfeaturesisconstantforallindividuals andisequaltotheproblemdimension.BBOcancurrentlyonlydeal withoptimizationproblemsofconstantdimension;itsextension tovariable-sizedproblemsisatopicforfutureresearch.Although theanalogyisnotperfect,thekeypointinBBOisthatthemigra­ tionofsolutionfeaturesbetweenindividualsismotivatedbythe mathematicaltheoryofspeciesmigrationinbiogeography.

LikeotherEAs,BBOoperatesprobabilistically.Theprobability thatanindividual sharesa featurewiththerestof thepopula­ tionisproportionaltoitsfitness.Theprobabilitythatanindividual receivesafeaturefromtherestofthepopulationdecreaseswith itsfitness.Whenacopyoffeaturesfromindividualzjreplacesa

featureinindividualyk,wesaythatshasemigratedfromzjand

immigratedtoyk;thatis,yk(s)←zj(s).

Althoughnonlinearmigrationcurvesmaygivebetteroptimiza­ tionresults [5],inthispaperweuselinearcurvesasshownin Fig.1. Fig.1depictstwoBBOindividuals.S1depictsapoorsolutionandS2

depictsagoodsolution.TheimmigrationprobabilityforS1willthus

behigherthanthatofS2,andtheemigrationprobabilityforS1will

belowerthanthatofS2.Notethatifagoodsolutionisobtainedin

thepopulation,thentheremaybeahighprobabilitythatthepop­ ulationwillconvergetowardsthatsolution,resultinginpremature convergence.AswithanyotherEA,anappropriatemutationrate needstobeusedinBBOtobalanceexplorationandexploitation.

Thereareseveralways toimplementBBO. TheoriginalBBO algorithm,whichweuseinthispaper,iscalledpartial immigration-basedBBO [13].Inthisapproach,foreachfeatureineachindividual

1

immigration

λ

emigration

μ

probability

S

1

S

2

fitness

Fig.1.IllustrationoftwoBBOindividualsusinglinearmigrationcurves.S1repre­

sentsapoorsolutionandS2representsagoodsolution.

weprobabilisticallydecidewhetherornottoimmigrate.Ifimmi­ grationisdecidedupon,thenafitness-basedprobabilisticmethod (e.g.,roulettewheelselection)isusedtoselecttheemigratingindi­ vidual.ThisgivesthealgorithmshowninFig.2asadescription ofoneBBOgeneration.Weperformmigrationandmutationfor eachindividualinthecurrentgenerationbeforeanyindividualsare replaced,resultinginagenerationalEA [14].Themigrationdecision requiresthattheindividualsbesortedinorderoffitness,whichisa computationalconsideration.However,inalmostallreal-worldEA applications,fitnessfunctionevaluationcomprisesthevastmajor­ ityofcomputationaleffort.

1.3. Overviewandoutline

Inpreviousworkwederiveda MarkovchainmodelforBBO [15,16].AMarkovchainisarandomprocesswhichhasTpossible statevalues [17,chap.11].Theprobabilitythatthesystemtransi­ tionsfromstateitostatejisgivenbytheprobabilityPij.TheT×T

matrixP= [Pij]iscalledthetransitionmatrix.EachstateintheBBO

Markovmodelrepresentsapopulationdistribution,thatis,adis­ tributionofindividualsinthesearchspace.ProbabilityPij isthe

probabilitythatthepopulationtransitionsfromtheithpopulation distributiontothejthpopulationdistributioninonegeneration.

AlthoughweuseBBOMarkovtheoryinthispapertoderivea dynamicsystemmodel,thestatesinthedynamicsystemmodel arenotthesameasthestatesintheMarkovmodel.BBOMarkov states represent populationdistributions. BBO dynamic system statesrepresenttheproportionofeachindividualinthepopula­ tion;thatis,theithstateistheproportionoftheithindividualin thepopulation.ThestatedimensionoftheBBOdynamicsystem willthusbeonlyasmallfractionofthestatedimensionoftheBBO Markovmodel.Thismakesthedynamicsystemmodelapplicable tolargerproblemsthantheMarkovmodel.

Section 2reviewstheBBOMarkovmodelwhichformsthebasis forthisworkandwhichwasderivedin [15,16].Section 3andfol­ lowing comprisethenewcontributionsofthis paper.Section3 derivestheBBOdynamicsystemmodelandsomeofitsproper­ ties.WealsoextendthedynamicsystemmodeltoaGAwithglobal uniformrecombination(GAGUR).Section 4comparesthedynamic systemmodelsofBBO,GAGUR,andGAwithsingle-pointcrossover (GASP).Weprovideconcludingremarksandsuggestionsforfuture workinSection 5.

Fig.2. OnegenerationoftheBBOalgorithm.zistheentirepopulationofindividuals,zk isthekthindividual,andzk(s)isthesthfeatureofzk.

2. AMarkovmodelforBBO

In [15,16]wederivedaMarkovmodelforBBO.Inthissection wereviewthatmodelasafoundationforourlaterdevelopment ofadynamicsystemmodelinSection 3.TheBBOMarkovmodel inthis section isbased onseveralassumptions.First, we usea generationalBBOalgorithmratherthanasteady-stateBBOalgo­ rithm,ascanbeseenfromtheuseofthetemporarypopulation

yin Fig.2.Second,anindividualcanemigrateafeaturetoitself.

bit of xi is denoted as xi(s). We use Ji(s) to denote the

set of search space indices j such that xj(s) =xi(s). That

is,

Ji(s)= {j:xj(s)=xi(s)}, i∈[1, n]. (3)

Note that |Ji(s)| =n/2 for all i and s. From (2) we see

that Thismeans that in the statement “usethe values to proba­ bilisticallyselecttheemigratingindividualzj” inFig.2,jmight

bechosen tobeequal tok.This issimilar tothepossibility of

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

x1 for k=1, . . . ,

v

1 x2 for k=

v

1+1, . . . ,

v

1+

v

2 x3 for k=

v

1+

v

2+1, . . . ,

v

1+

v

2+

v

3 . .

selectingthesameparenttwiceina GASP, which resultsin an yk= ._. ._. (4)

offspring which is a cloneof its parent. However,in BBOit is nottheindividual,butratherthesolutionfeature,thatiscloned in this situation.Third, themigration rates are independent of

n−1

xn for k=

v

i+1, . . . , N

thepopulationdistribution;thatis,absolutefitnessvaluesrather i=1 than rank-based fitness values [18] are used to obtain migra- _{whichcanalsobewrittenas} tionrates.Fourth,ouroptimizationproblemhasabinarysearch

space.Fifth,weignorethepossibilityofmutation;mutationwill _y

k=xm(k) for k=1, . . . , N

beconsideredinSection3whenwederivethedynamicsystem model.

Suppose that the search space consists of all bit strings r

xi which have q bits each. The cardinality of the search m(k)=minr suchthat

v

i≥k. (5)

space is therefore n= 2q_{. The population size is denoted}

i=1

by N. The n-element population-count vector

v

denotes

the number of each xi individual in the population. There- We use the additional subscript t to denote generation num­

fore, ber. For example, yk(s)t is the value of the sth bit of the

kth individual at generation t. With these definitions, it is

n

shown in[15,16] thatthe probabilitythat yk(s)t+1=xi(s)canbe

v

_i=N. (1)

writtenas

i=1

The kth individual in the population is denoted by yk. The yk

v

jj

values are ordered so that identical individuals are grouped. _Pr( j∈Ji(s)

yk(s)t+1=xi(s))=(1−m(k))1[xm(k)(s)=xi(s)]+m(k)

Furthermore, the order of the yk values is same as that n

of the xi values. The BBO population can thus be depicted

v

_jj

as

j=1

Population= {y1,...,yN} (6)

= {x1, x1

, . . . , x1, x2, x2, . . . , x

2,...xn,xn

,...,xn}. (2) where1[·]isthepredicatefunction;thatis,1[A]=1ifAistrue,and v1 copies v2 copies vn copies 0_countotherwise._vectorThere_isequalare_toq

v

bits_attheineach_{tthgeneration,}yk.Therefore,_thenifthe_thepopulation-_probability The emigration probability of xi is denoted as i, and the thatimmigrationresultsinyk,t+1=xiisdenotedbyPki(

v

)andcanbe

Pki(

v

)=Pr(yk,t+1=xi) Wecancombine (4),(5)and(11)toget

⎤

⎡

m(k−1) m(k)

v

_jj _Pr(y k1=xi)=Pr(yk2 =xi) if (k1, k2)∈

v

i+1,

v

i . i=1 i=1

⎢

⎢

⎢

⎢

⎣

⎥

⎥

⎥

⎥

⎦

. = q s=1

I

_j∈Ji(s) (1₋m(k))1[xm(k)(s)=xi(s)]+m(k) (12) n

v

_jj

Therefore (10)canbewrittenas

j=1 (7)

⎧

⎪

⎪

⎪

⎪

⎨

⎫

⎪

⎪

⎪

⎪

⎬

⎪

⎪

⎪

⎪

⎭

.

⎤

⎡

vjj

⎢

⎢

⎢

⎢

⎣

(1−k)1[xk(s)=xi(s)]+k

⎥

⎥

⎥

⎥

⎦

Pki(

v

)canbecomputedforeachk∈[1,N]andeachi∈[1,n]inorder q

s=1

I

n 1 j∈Ji(s) n Pr(Mh=xi)= (13)

toformtheN×nmatrixP(

v

).Thisgivestheprobabilitythateach _N vk

⎪

⎪

⎪

⎪

⎩

ofNBBOmigrationtrialsresultsineachofnsearchspaceindivid- k=1

uals.Weusewitodenotethetotalnumberoftimesthatindividual

vjj

j=1 xiisobtainedafterallNmigrationtrialshavebeencompletedfor

agivengeneration,andwedefinew=[w1 ··· wn]T.Theprob- Nowwedefinetheproportionalityvectorpas

abilitythatweobtainthepopulation-countvectorwatthe(t+1)

v

(14) p= .

stgeneration,giventhat

v

isthepopulation-countvectoratthe _N

tthgeneration,canbeobtainedfromthegeneralizedmultinomial

Thatis,piistheproportionofxiindividualsinthepopulationfor

theorem [15,16,19]as

i∈[1,n],andtheelementsofpaddupto1.Eq. (13)canthenbe writtenas N

I

n

I

Pr(w|v)= PJ_kiki(v)

⎧

⎪

⎪

⎪

⎪

⎨

⎫

⎪

⎪

⎪

⎪

⎬

⎪

⎪

⎪

⎪

⎭

⎤

⎡

J∈Y(w)

Y(w)= J∈RN×n_{:Jki∈ {0,1}, Jki=1forallk, Jki=wi foralli} k=1 i=1 (8) _vjj n N

⎢

⎢

⎢

⎢

⎣

(1−k)1[xk(s)=xi(s)]+k

⎥

⎥

⎥

⎥

⎦

n

I

q s=1 j∈Ji(s) n . Pr(Mh=xi)= pk i=1 k=1

TheMarkovtransitionmatrixisobtainedbycomputing (8)foreach

⎪

⎪

⎪

⎪

⎩

k=1 _vjj j=1

possible

v

vectorandeachpossiblewvector.Thetransitionmatrix

I

q

isthusaT×Tmatrix,whereTisthetotalnumberofpossiblepop- n pk

(pT)q T(1−k)1[xk(s)=xi(s)]+k vjj

ulationdistributions.Thatis,Tisthenumberofallpossiblen×1

integervectorssuchthat Ti=Nand0≤Ti≤N.In [20]itisshown k=1

p

= .

s=1 j∈Ji(s)

thatTcanbecalculatedusingtheformulaforcombinations: (15)

Thequantitiesontherightsideoftheaboveequationaredefined atthetthgeneration.Theleftsideoftheaboveequationgivesthe n+N−1

T= . (9)

N

I

probabilityofobtainingxiatthe(t+1)stgeneration.Wecanuse

[21,Theorem2]toseethattheleftsideoftheaboveequationis OthermethodsforcalculatingTarediscussedin [15].

equaltotheproportionofxiindividualsinthepopulationatthe

(t+1)stgeneration:

q

3. AdynamicsystemmodelforBBO

n

pk(t) _pT_(t)(1−

k)1[xk(s)=xi(s)]

pi(t+1)=

InthissectionweusetheMarkovmodelofSection 2toderive

(pT_(t))q

adynamicsystemmodelforBBOandGAGUR.Eq. (7)givesPki(

v

), k=1 s=1

whichistheprobabilitythatthekthmigrationresultsinyk=xiat

the(t+1)stgeneration,assumingthat

v

isthepopulation-count

+ k

v

j(t)j

vectoratthetthgeneration.Inordertoderiveadynamicsystem

⎫

⎬

⎭

⎤

⎦

(16)

modelforBBO,wemakeaslightchangeinthealgorithmof Fig.2. WestillcyclethroughtheimmigrationloopNtimes,whereNis thepopulationsize.However,insteadofdeterministicallycycling througheachpopulationmemberykforimmigration,werandomly

selectapopulationmemberforimmigrationeachoftheNtimes throughtheloop.Therefore,each timethroughtheimmigration loop,eachykhasa1/Nchanceofbeingselectedforimmigration.

NotethatasN→ ∞,thisisequivalenttothealgorithmof Fig.2.With thischange,theBBOalgorithmismodifiedtobecomethealgorithm shownin Fig.3.

Withthisalgorithm,theprobabilitythatthehthmigrationtrial

Mhresultsinxiis

N

1

Pr(Mh=xi)= _N Pr(yk=xi). (10)

k=1

Nownotethat

Pr(yk1=xi)=Pr(yk2=xi) if yk1 =yk2. (11)

j∈Ji(s)

where we have now explicitlyshown that p is a functionof t. Writingtheaboveequationfori∈[1,n]givesanonlineardynamic systemfortheevolutionoftheproportionalityvector:

p(t+1)=f(p(t)). (17)

Toincludethepossibilityofmutation,wedenotethen×nmuta­ tionmatrixasU,whereUijistheprobabilitythatxjmutatestoxi.

Iftheelementsofthesearchspace{x}areinnaturalbinaryorder andeachbithasaprobabilityuofmutation,then

Uij=udij(1−u)q−dij (18)

wheredijistheHammingdistancebetweenxiandxj[22,p.153].

Thisgivesthedynamicsystemequation

p(t+1)=Uf(p(t)). (19)

IfmutationisnotusedintheBBOalgorithm,thenUistheidentity matrixand (19)reducesto (17).

Fig.3. OnegenerationoftheBBOalgorithmwithrandomselectionoftheimmigratingindividual.

3.1. Specialcase:= 0 writtenas

Itisinstructivetoconsiderthedynamicsystemwhenk=0for Pr(yk,t+1=xi|s)=Pr[yk,t(r:r=/s)=xi(r:r=/s)]Pr(yk,t+1(s)=xi(s)).

allk.Inthiscase,thereisnopossibilityofimmigrationand (15)

(23) reducesto

Thefirsttermontherightsideof (23)istheproportionofthepopu­

q

pi(t+1)= pk(t) 1[xk(s)=xi(s)] . (20) lationwhichhasallbitsrsuchthatr =/ s,equaltothecorresponding s=1 bitsinxi.WedenotetheindicesoftheseindividualsasLi(s): k=1

Sinceeachxiisdistinct,weseethat Li(s)= {j:xj(r:r=/s)=xi(r:r=/s)}, i∈[1, n]. (24) q

I

I

n

Notethat|Li(s)| =2forall(i,s).Nowwecanwrite (23)as

1[xk(s)=xi(s)]=1[k=i] (21)

⎛

⎞

⎛

⎞

s=1

⎜

⎜

⎜

⎜

⎝

v

_j j∈Li(s) n

⎜

⎜

⎜

⎜

⎝

⎟

⎟

⎟

⎟

⎠

j∈Ji(s) n

v

_jj

⎟

⎟

⎟

⎟

⎠

whichgives n Pr(yk,t+1=xi|s)= . (25) pi(t+1)= pk(t)1[k=i]=pi(t). (22)

v

j

v

jj j=1 j=1 k=1

Thatis,withnoimmigrationandnomutation,theproportionality Wecanuse (1)and(14)towritetheaboveequationas vectordoesnotchangefromonegenerationtothenext,which

agreeswithintuition. _pjj

j∈Ji(s)

Pr(yk,t+1=xi|s)= pj _n . (26)

3.2. Specialcase:=1andrandomfeatureselection

j∈L_i(s) _p jj

Ifk=1forallk,thentheBBOalgorithm of Fig.3becomesa j=1

specialtypeofageneticalgorithmwithglobaluniformrecombina­

tion(GAGUR) [23].GAGURcanbeimplementedinmanydifferent Fig.4showsthateachbits∈[1,q]hasa1/qprobabilityofbeing ways,butifitisimplementedwiththeentirepopulationaspoten- selectedasthemigratingfeature.Therefore,

tialcontributorstothenextgeneration [24],andwithfitness-based

⎡

⎤

⎡

⎤

selectionforeachsolutionfeatureineachoffspring,thenitisequiv- q

1

alenttoBBOwithk=1forallk.Inthiscase,immigrationtakes Pr(yk,t+1=xi)=

⎣

pj

⎦

⎣

pjj

⎦

. (27)

qpT

placeforallindividualsinthepopulation,andthenewindividual _{s=1 j∈L}

i(s) j∈Ji(s) thatresultsfromeachimmigrationcanbethoughtofasanoffspring

ofthepreviousgeneration.Supposealsothatinadditiontok=1 Thisisaquadraticfunctionofthepitermsandcanthusbewritten

forallk,eachimmigrationtrialmigratesonerandomlyselectedbit. _as ThentheBBOalgorithmof Fig.3becomestheGAGURalgorithmof

n n

Fig.4.

Pr(yk,t+1=xi)= Yi,abpapb. (28)

Theprobabilitythat yk atthe(t+1)stgenerationisequalto

Fig.4. OnegenerationofGAGURwithrandomselectionoftheimmigratingindividualandrandomselectionofthemigratingsolutionfeature.

Eq. (27)showsthatthep2 _{coefficientontherightsideof(28),where} m m∈[1,n],canbewrittenas q q Yi,mm= m1[m∈Ji(s)]1[m∈Li(s)]= m1[m∈(Ji(s)∩Li(s))]. s=1 s=1 (29) FromthedefinitionofLi(s)in (24)weknowthati∈Li(s).Wealso

knowthatthereisonlyoneotherelementinLi(s).Theotherele­

mentinLi(s),say˛,hasabitstringsuchthatx˛(r) =xi(r)forall

r=/ s.Butsince˛ =/ iweknowthatx˛(s)=/ xi(s),which means

that˛ /∈Ji(s).Therefore

Ji(s)∩Li(s)= {i} forall s. (30)

Eq. (29)canthereforebewrittenas

q

Yi,mm= m1[m=i]=qm1[m=i]. (31) s=1

Wecanuse (27)toshowthatthepmpkcoefficient(k =/ m)onthe

rightsideof (28)canbewrittenas

q Yi,mk+Yi,km=m 1[m∈Ji(s)]1[k∈Li(s)] s=1 q +k 1[k∈Ji(s)]1[m∈Li(s)] for m=/k. (32) s=1

TheGAGURdynamicsystemmodelcanthusbewrittenasthefol­ lowingsetofncoupledquadraticequations:

pi(t+1)=pT(t)Yip(t), i∈[1, n] (33)

whereYi,mkistheelementinthemthrowandkthcolumnofYi.If

mutationisincludedintheGAGURalgorithm,then

p(t+1)=Udiag(pT(t)Yip(t)) (34)

wherediag(pT_(t)Y

ip(t))isthen×ndiagonalmatrixconsistingof

pT_(t)Y

1p(t),. . .,pT(t)Ynp(t).

4. Dynamicsystemmodelresults

Section 4.1verifiesthedynamicsystemmodelderivedinthe previoussection.Section 4.2comparesthedynamicsystemmodels ofGAwithsingle-pointcrossover(GASP),GAGUR,andBBO.

4.1. Verificationofdynamicsystemmodels

ThedynamicsystemmodelforBBOisgivenin (16)–(19)withk

proportionaltofitness,andk= 1−k.Thedynamicsystemmodel

forGAGURisgivenin(16)–(19)withk=1,whichisequivalent

to (34).ThedynamicsystemmodelforGASPwithroulette-wheel selectionwasoriginallydevelopedin [25].Itissummarizedin [22, chap.6]as

pT_(t)diag()UT_{C(i)Udiag()p(t)}

pi(t+1)= (35)

(pT_(t))2

wherediag()isthen×ndiagonalmatrixconsistingoftheele­ mentsof(fitness),andUisthemutationmatrixgivenin (18).

C(i)isann×nmatrixsuchthattheelementinitsmthrowandkth columnistheprobabilitythatxmandxkcrossovertoproducexi.

Toverify thedynamicsystemmodels,we considera simple three-bitproblem(n=8)withaper-bitmutationrateu=0.2.Thefit­ nessvalues,whichareequivalenttounnormalizedBBOemigration rates,aregivenasfollows:

(000)=8, (001)=1,

(010)=1, (011)=1, ₍₃₆₎

(100)=1, (101)=1, (110)=1, (111)=9.

Thisisarelativelydifficultoptimizationproblembecausex1=000

hasahighfitness,andeverytimeweadda1bittoitthefitness decreasesdramatically,buttheindividualwithall1’shasthehigh­ estfitness.Webeginwithaninitialpopulationwithproportionality vector

p(0)=[ 0.8 0.1 0.1 0 0 0 0 0 ]T. (37) Figs.5–7showsomedynamicsystemmodelresultsandsimulation resultsforEAswithapopulationsizeof1000.Theplotsprovide confirmation for thedynamic systemmodels presentedearlier. Thesimulationresultsoscillatearoundtheirmeanvalue,whichis expectedbecauseofthemutationoperator.Thesimulationresults willvaryfromonesimulationtothenext,andwillneverexactly matchthetheory,duetothestochasticnatureofthesimulations. Thatiswhythedynamicsystemmodelscanbemoreusefulthan simulation;themodelsareexactwhilesimulationresultsareonly approximate.

0 5 ₁₀ 4 −2 10 simulation simulation mean theory 0 20 40 60 80 100 −6 10 −2 −1 −3 BBO GA/sp GA/gur

percent of optimum percent of optimum

3 2 1 −4 10 0 _{10 10} probability of mutation generation 10

Fig.5.BBOdynamicsystemresultsgivingconfirmationthatsimulationmatches theory.Thetracesshowtheproportionoftheoptimalindividualsforatypical simulation,themeanofthatproportionoverallgenerations,andtheproportion accordingtothedynamicsystemmodel.

3

2.5

Fig.8. Dynamicsystemmodelresultsfora3-bitproblem(searchspacecardinality

n=8)showingthesteady-stateproportionofoptimalindividuals.

4.2. Comparisonofdynamicsystemmodels

NextwecomparedynamicsystemmodelresultsbetweenBBO, GAGUR,andGASP.Weconsideraproblemwhosefitnessvaluesare givenas 8 for xi=(0· · ·0) simulation simulation mean theory i= 9 for xi=(1· · ·1) (38) 2

1 forallother xi.

Thisisthesameas (36)exceptthatitisgeneralizedforanarbitrary

1.5

numberofbits.Theproportionalityvectoroftheinitialpopulation isgivenas 1 percent of optimum p(0)= 1 [ 1 . . . 1 0 ]T. (39) n−1 0.5

Thatis,theinitialpopulationisevenlydistributedamongthesub­ optimalindividuals,andtherearenooptimalindividuals. Figs.8–10 showsteady-statedynamicsystemmodelresultsforthreediffer­

0

0 20 40 60 80 100

generation

Fig.6. GAGURdynamicsystemresultsgivingconfirmationthatsimulationmatches theory.Thetracesshowtheproportionoftheoptimalindividualsforatypical simulation,themeanofthatproportionoverallgenerations,andtheproportion accordingtothedynamicsystemmodel.

6 5 simulation simulation mean theory percent of optimum 4 3 2 1

entsearchspacecardinalities,plottedasfunctionsofmutationrate. Fig.8showsthatBBOismuchbetteratachievingahighpercentage ofoptimalindividualsthanGASPandGAGURforsmallproblems. Figs.9and10showthatastheproblemdimensiongetslarger,BBO performancegetsworserelativetotheGAsforlargemutationrates. However,BBOremainsmanyordersofmagnitudebetterthanthe GAsforsmallmutationrates,whicharemoretypicalforreal-world problems. percent of optimum −5 10 −10 10 BBO GA/sp GA/gur 0 0 20 40 60 80 100 generation _{−3 −2 −1} 10 10 10

Fig.7. GASPdynamicsystemresultsgivingconfirmationthatsimulationmatches

theory.Thetracesshowtheproportionoftheoptimalindividualsforatypical probability of mutation simulation,themeanofthatproportionoverallgenerations,andtheproportion

accordingtothedynamicsystemmodel. Fig.9. Dynamicsystemmodelresultsfora5-bitproblem(searchspacecardinality

−5 10 0 10 −2 10 BBO GA/sp GA/gur BBO GA/sp GA/gur 1 2 10 10 percent of optimum percent of optimum −10 10 −15 10 10−4 −3 −2 −1 10 10 10 ₁₀−6 probability of mutation problem dimension

Fig.10.Dynamicsystemmodelresultsfora7-bitproblem(searchspacecardinality

n=128)showingthesteady-stateproportionofoptimalindividuals. Fig.13. Dynamicsystemmodelresultsformutationrate=10%perbitshowingthe steady-stateproportionofoptimalindividuals.

−5 10 percent of optimum −10 10 −15 10 −20 10 BBO GA/sp GA/gur 1 2 10 10 problem dimension

Fig.11.Dynamicsystemmodelresultsformutationrate=0.1%perbitshowingthe

mutation rate is low, as is typical of real-world problems. Figs. 12 and 13 showthat as themutation rateincreases,BBO remainsbetterthantheGAsfor smallproblemdimensions,but becomesworsethanGASPastheproblemdimensionincreases.

Asseenin Fig.11,withrealisticmutationratesBBOismuch betterthantheGAsforallproblemdimensions.Furthermore,the relative advantage of BBO increases as theproblem dimension increases.Thisisconsistentwiththeconclusionspresentedin [23] whichwerebasedonadifferenttypeofanalysisandwhichwere confirmedwithavarietyofstandardbenchmarksimulations.

NextwecomparethedynamicsystemmodelresultsofBBO, GASP, and GAGUR,onstandard benchmarkfunctions.TheNee­ dleFunctionisgivenin (38).TheOnemaxFunctionhasafitness thatisproportionaltothenumberofone-bitsineachbitstring. The Deceptive Function is the same as the Onemax Function, exceptthat thebitstring withall zeroshasthehighestfitness. The continuousfunctionsthatwe usearelistedin Table1and steady-stateproportionofoptimalindividuals.

Figs. 11–13 depict the same information as that shown in Figs. 8–10,but presented in a differentway. Figs. 11–13show dynamicsystemmodelresultsforthreedifferentmutationrates, plottedasfunctionsofproblemdimension. Fig.11showsthatBBO is much betterthan theGAs for all problemdimensions ifthe

aredocumentedin [26–28].Weimplementedthecontinuousfunc­ tionsastwo-dimensionalfunctionswhoseindependentvariables arecodedwiththreeorfourbitsperindependentvariable.This givesanoptimizationproblemwitheithersixoreightbitstotal, whichresultsinasearchspacecardinalityofeither64or256.We initializedthepopulationwithauniformdistributionoverallof thenon-optimalsolutions.Theinitialpopulationdidnothaveany optima.Werecordedthepercentofoptimalsolutionsinthepop­ ulationafter10generations,whichgivesanideaofhowfasteach algorithmconverges. Table1showstheresults.Notethattheseare

BBO GA/sp GA/gur 1 2 10 10 −5 10 percent of optimum −10 10 problem dimension

Fig.12.Dynamicsystemmodelresultsformutationrate=1%perbitshowingthe

notsimulationresults,butexactdynamicsystemmodelresults. Forboththe64-bitand256-bitproblems,BBOperformedthe bestin15outof19benchmarks.Moreimportantly,forverydifficult problems(theNeedleandDeceptiveFunctions),BBOperformed betterthanGASPandGAGURbyordersofmagnitude.

Theseresultsarenotintendedtogiveacomprehensivecom­ parisonofBBOandGAs;extensivecomparisonsbetweenBBOand otherEAsusingstandardbenchmarkfunctionsareshownin [3]. Thetheoryand resultshereare insteadintended toshowhow dynamicsystemmodelscanbeusedtocompareEAsinsituations whereprobabilitiesareextremelysmallandwhereMonteCarlo simulationsarethereforenotuseful.Dynamicsystemmodelscan alsobeusedtostudytheeffectofvariousparametersettingsand learningapproaches.Dynamicsystemmodelscanalsoaidinthe developmentofadaptivealgorithmsorparameterupdateschemes thatworkwellonmanydifferenttypesofproblems.Ourdynamic systemmodelscanalsobeusedtohelpunderstandthebehav­ steady-stateproportionofoptimalindividuals. iorofBBO;forexample,howandwhyitworkswell,ordoesnot

Table 1

Dynamic system results on benchmark functions. The number in each cell indicates the percentage of optimal individuals in the population after 10 generations. The best result for each benchmark/cardinality combination is shown in boldface font.

Function Cardinality = 64 Cardinality = 256

BBO GAGUR GASP BBO GAGUR GASP

Needle 6.97×10-3 _2.32×10−10 _2.17×10−6 _6.72×10-3 _9.95×10−6 _6.48×10−6 Onemax 46.04 47.67 42.64 22.44 23.84 19.43 Deceptive 3.89×10-3 _2.25×10−4 _2.33×10−4 _1.03×10-3 _5.36×10−5 _5.13×10−5 Rosenbrock 10.37 3.56 9.13 6.60 4.18 6.49 Ackley 58.92 25.40 71.38 43.40 54.17 70.00 Fletcher 74.30 88.41 28.88 32.19 23.15 23.73 Griewank 67.82 44.18 49.00 36.12 15.07 25.61 Penalty #1 29.69 24.98 27.77 26.22 25.02 25.68 Penalty #2 32.83 24.96 28.98 13.03 12.51 14.08 Quartic 39.09 26.13 34.12 18.64 12.49 17.01 Rastrigin 43.93 50.05 41.79 46.85 14.64 42.43 Schwefel 1.2 69.37 25.29 49.21 35.91 14.02 25.07 Schwefel 2.22 69.14 25.80 51.27 38.10 13.36 26.84 Schwefel 2.21 75.65 66.84 62.89 49.03 39.82 38.60 Schwefel 2.26 15.93 0.47 1.30 38.07 15.69 8.70 Sphere 69.63 47.69 49.21 35.95 12.50 24.94 Ellipsoid 65.82 56.07 48.84 34.15 16.08 24.80 Michalewicz 34.01 33.30 30.38 14.47 12.32 15.67 Step 50.04 25.04 37.56 25.51 12.52 19.45

work well, on certain type of problems. This paper does not explore these many possible research directions, but we envision that the groundwork laid here will encourage these ideas and make their pursuit possible.

5. Conclusion

Mature EAs, such as GAs, have a well-structured theory. This paper attempts to extend BBO theory in that direction to enable its use and study as an alternative EA. We have summarized a pre-viously derived BBO Markov model and used it to obtain a new dynamic system model for BBO. We have further shown how it can be used to derive a new dynamic system model for GAGUR. We compared the dynamic system models between BBO, GAGUR, and GASP on some simple optimization problems. We have seen that BBO far outperforms GAs when mutation rates are low (0.1% per bit), as are typically used for real-world problems. In addition, the relative advantage of BBO increases as the problem dimension increases. This indicates that BBO can be especially useful for large, real-world problems when small mutation rates are used.

For some real-world problems where small population sizes are required due to huge computational complexity for fitness evalua-tion, it may be desirable to use large mutation rates on the order of 10% per bit[29]. In this case BBO still outperforms the GAs for small problem dimensions, but GASP is the best performer for large prob-lems. Note that these conclusions are similar to those of[23], which were obtained using a different approach. Reference[23]also pro-vides a more extensive discussion of the conceptual similarities and differences between GAs and BBO.

There are many interesting possibilities for future work. One is to extend BBO using features of natural biogeography theory[1,2]. This could include modeling nonlinear migration curves, modeling species populations and their effects on migration, modeling preda-tor/prey relationships, including species mobilities in the migration model, including directional migration momentum as a temporal effect from one generation to the next, modeling habitat area and isolation, and many others.

In this paper we used Markov theory for partial immigration-based BBO to derive a dynamic system model. In addition, we analyzed a generational BBO algorithm; that is, modification of each individual in the current generation occurs before any individuals are replaced in the population[14]. Future work could include the extension of our Markov and dynamic system models for other

variations of BBO. These variations include partial emigration-based BBO, single immigration-emigration-based BBO, single emigration-emigration-based BBO[13], oppositional BBO[30], and a steady-state BBO in which individuals in the population are modified with replacement. It would also be of interest to extend the Markov theory and dynamic system model to BBO with nonlinear migration curves[5]. Also, our results have been restricted to problems with binary representa-tions, and it would be interesting and useful to develop Markov and dynamic system models of BBO with other types of representations. Finally, our results are exact only in the limit as the population size approaches infinity. Many EA applications have large populations, but many others do not. Further work could focus on obtaining a dynamic system model for small population sizes, perhaps by combining states into “meta-states,” or perhaps by using the BBO Markov model in either[13]or[15].

These extensions would allow analytical comparisons between different types of BBO algorithms rather than a reliance on sim-ulation results. Although simsim-ulation results are important and necessary, if used apart from theory they can be misleading. For example,Fig. 11 shows that BBO is orders of magnitude better than GAs for large problems with a low mutation rate. However, since the probabilities inFig. 11are so small, it would be difficult to derive such a conclusion from simulation results because of the huge amount of computation that would be required.

Another suggestion for future work is to combine BBO with other EAs. Hybrid EAs are an important topic of research and can exploit the strengths of multiple methods in a single optimization algo-rithm[31]. BBO has already been combined with opposition-based learning[30]. Future work could focus on combining BBO with the many other EAs that are available. These hybrid algorithms should be studied not only with benchmarks and real-world optimization problems, but also with analytical approaches such as the Markov and dynamic system theory used in this paper. Such analytical tools include the stochastic character of EAs, but do not depend on the random results of simulation studies to draw general conclusions.

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