Sch i en eihe CIplus, Band 2/2015
He ausgebe : T. Ba z-Beiels ein, W. Konen, H. S enzel, B. Naujoks
O e iew: E olu iona y
Algo i hms
Thomas Ba z-Beiels ein, J¨u gen B anke, J¨o n Mehnen,
Ola Me smann
O e iew: E olu iona y Algo i hms1
Thomas Ba z-Beiels ein
Cologne Uni e si y o Applied Sciences
J¨
u gen B anke
Wa wick Uni e si y
J¨
o n Mehnen
C an ield Uni e si y
Ola Me smann
Cologne Uni e si y o Applied Sciences
Keywo ds
E olu iona y algo i hms, Bio-inspi ed sea ch heu is ics, E olu ion s a egies,
Gene ic algo i hms, Gene ic p og amming
Abs ac
E olu iona y algo i hm is an umb ella e m used o desc ibe popula ion based
s ochas ic di ec sea ch algo i hms ha in some sense mimic na u al e olu ion.
P ominen ep esen a i es a e gene ic algo i hms, e olu ion s a egies, e olu-
iona y p og amming, and gene ic p og amming. Based on he e olu iona y
cycle, simila i ies and di e ences be ween heses algo i hms a e desc ibed.
We b ie ly discuss how e olu iona y algo i hms can be adap ed o wo k well in
case o mul iple objec i es, dynamic o noisy op imiza ion p oblems. We look
a he uning o algo i hms and p esen some ecen de elopmen s om heo y.
Finally, ypical applica ions o e olu iona y algo i hms o eal-wo ld p oblems
a e shown, wi h special emphasis on da a mining applica ions.
E olu iona y Algo i hms in a Nu shell
In en ion and de elopmen o he i s e olu iona y algo i hms is nowadays
a ibu ed o a ew pionee s who independen ly sugges ed ou ela ed ap-
p oaches (Ba z-Beiels ein e al., 2010b).
•Fogel e al. (1965) in oduced e olu iona y p og amming (EP) aiming a
e ol ing ini e au oma a, la e a sol ing nume ical op imiza ion p oblems.
1This is he p e-pee e iewed e sion o he ollowing a icle: Ba z-Beiels ein, T. and B anke, J. and
Mehnen, J. and Me smann, O.: E olu iona y Algo i hms. WIREs Da a Mining Knowl Disco 2014, 4:178-
195. doi:10.1002/widm.1124
1
•Holland (1973) p esen ed gene ic algo i hms (GA), using bina y s ings,
which we e inspi ed by he gene ic code ound in na u al li e, o sol e
combina o ial p oblems.
•E olu ion s a egies (ES) as p oposed by Rechenbe g (1971) and Schwe-
el (1975) we e mo i a ed by enginee ing p oblems and hus mos ly used
a eal- alued ep esen a ion.
•Gene ic p og amming (GP), sugges ed by Koza (1992b) eme ged in he
ea ly 1990s. GP explici ly pe o ms he op imiza ion o p og ams.
Since abou he same ime, hese ou echniques a e collec i ely e e ed o
as e olu iona y algo i hms (EAs), building he co e o he e olu iona y compu-
a ion (EC) ield.
E olu iona y algo i hms a e unde s ood as popula ion based s ochas ic di ec
sea ch algo i hms ha in some sense mimic he na u al e olu ion. Poin s in
he sea ch space a e conside ed as indi iduals (solu ion candida es), which
o m a popula ion. Thei i ness alue is a numbe , indica ing hei quali y o
he p oblem a hand. Besides ini ializa ion and e mina ion as necessa y con-
s i uen s o e e y algo i hm, EAs can consis o h ee impo an ac o s: A se
o o sea ch ope a o s (usually implemen ed as ’ ecombina ion’ and ’mu a ion’),
an imposed con ol low, and a ep esen a ion ha maps adequa e a iables o
implemen able solu ion candida es ( he so-called ’geno ype-pheno ype map-
ping’). A widely accep ed de ini ion eads as ollows:
E olu iona y algo i hm: collec i e e m o all a ian s o (p oba-
bilis ic) op imiza ion and app oxima ion algo i hms ha a e inspi ed
by Da winian e olu ion. Op imal s a es a e app oxima ed by suc-
cessi e imp o emen s based on he a ia ion-selec ion-pa adigm.
The eby, he a ia ion ope a o s p oduce gene ic di e si y and he
selec ion di ec s he e olu iona y sea ch (Beye e al., 2002).
Al hough di e en EAs may pu di e en emphasis on he sea ch ope a o s
mu a ion and ecombina ion, hei gene al e ec s a e no in ques ion. Mu a ion
means neighbo hood based mo emen in he sea ch space ha includes he
explo a ion o he ’ou e space’ cu en ly no co e ed by a popula ion, whe eas
ecombina ion ea anges exis ing in o ma ion and so ocuses on he ’inne
space’. Selec ion is mean o in oduce a bias owa ds be e i ness alues.
I can be applied a wo s ages: When pa en s a e selec ed om he popula ion
o gene a e o sp ing (ma ing selec ion), and a e new solu ions ha e been
c ea ed and need o be inse ed in o he popula ion, compe ing o su i al
(en i onmen al selec ion o su i al selec ion). GAs p ima ily ocus on ma ing
selec ion, ESs u ilize only en i onmen al selec ion.
A conc e e EA may con ain speci ic mu a ion, ecombina ion, o selec ion op-
e a o s, o call hem only wi h a ce ain p obabili y, bu he con ol low is usu-
ally le unchanged. Each o he consecu i e cycles is e med a gene a ion.
2
Conce ning he ep esen a ion, i should be no ed ha mos empi ic s udies
a e based on canonical o ms such as bina y s ings o eal- alued ec o s,
whe eas many eal-wo ld applica ions equi e specialized, p oblem dependen
ep esen a ions.
B¨
ack (1996) compa es GAs, ES, and EP. Fo an in-dep h co e age on he
de ining componen s o an EA and hei connec ion o na u al e olu ion, see
Eiben & Schoenaue (2002) and Eiben & Smi h (2003). Beye e al. (2002)
p o ide a e y use ul glossa y, which co e s he basic de ini ions. De Jong
(2006) p esen s an in eg a ed iew.
The emainde o his a icle is s uc u ed as ollows. A e in oducing p omi-
nen ep esen a i es o EAs, namely e olu iona y p og amming, gene ic algo-
i hms, e olu ion s a egies and gene ic p og amming, he ollowing special op-
ics a e discussed: Mul i-objec i e op imiza ion, dynamic and s ochas ic op i-
miza ion, uning, heo y, and applica ions. The a icle concludes wi h a sho
lis o EA ela ed so wa e.
The Family o E olu iona y Algo i hms
S a ing wi h i s oldes membe , namely EP, he amily o EAs is desc ibed in he ol-
lowing pa ag aphs. Al hough EP, GA, ES, and GP ha e been in en ed independen ly
and a e desc ibed sepa a ely, i is unques ioned ha hese algo i hms a e speci ic in-
s ances o he mo e gene al class o EAs (De Jong, 2006) and ha i is nowadays di -
icul o dis inguish heses algo i hms om each o he (Beye , 2001). Only one di e -
en ia ion is possible e en oday: EP algo i hms do no use ecombina ion. Today, he e
is a huge se o e y sophis ica ed and p oblem-speci ic EA implemen a ions, and his
a icle can only sco ch he su ace.
E olu iona y P og amming
E olu iona y p og amming uses a ixed p og am s uc u e, while i s nume ical pa am-
e e s a e allowed o e ol e. The essen ial s eps o he EP app oach can be desc ibed
as ollows (Yao e al., 1999): (i) gene a e o sp ing by mu a ing he indi iduals in he
cu en popula ion and (ii) selec he nex gene a ion om he o sp ing and pa en pop-
ula ion. These key ideas a e simila ly used in ES, GP, and GAs. Howe e , while ES
o en used de e minis ic selec ion, selec ion is o en p obabilis ic in EP. EP ope a es
on he na u al p oblem ep esen a ion, hus no geno ype-pheno ype mapping is used.
Con a y o GAs and ES, ecombina ion (c osso e ) is no used in EP. Only mu a ion
is used as he a ia ion ope a o .
Algo i hm 1 p o ides a pseudocode lis ing o an EP algo i hm. The s eps o an EA
a e implemen ed as ollows: i s , indi iduals, which o m he popula ion, a e an-
domly gene a ed (line 1). Random ini ializa ion is p obably he simples ini ializa ion
3
Algo i hm 1: E olu iona y p og amming algo i hm
1popula ion = Ini ializePopula ion(popula ionSize, p oblemDimension);
2e alua ePopula ion(popula ion);
3bes Solu ion = ge Bes Solu ion(popula ion);
4while es Fo Te mina ion == alse do
5o sp ing = ∅;
6 o pa en i∈popula ion do
7o sp ingi= mu a e(pa en i);
8o sp ing = {o sp ing} ∪ o sp ingi;
9e alua ePopula ion(o sp ing);
10 bes Solu ion = ge Bes Solu ion(o sp ing,bes Solu ion);
11 popula ion = {popula ion}∪{o sp ing};
12 popula ion = en i onmen alSelec ion(popula ion);
13 Re u n(bes Solu ion);
me hod. O he , mo e p oblem speci ic ini ializa ion me hods a e possible. Fo exam-
ple, al eady known good solu ions can be used as seeds (s a ing poin s) o he ini ial
popula ion. The e alua ion unc ion (line 2) assigns a quali y measu e o i ness alue
o each indi idual. I he o iginal p oblem o be sol ed is an op imiza ion p oblem, he
e m objec i e unc ion is used. Mu a ion is a s ochas ic a ia ion ope a o , which is
applied o one indi idual (line 7). Be o e he nex ound in he e olu iona y cycle is
s a ed, he en i onmen al (o su i o ) selec ion is pe o med (line 12) ha emo es
some indi iduals in o de o keep he popula ion size cons an . The decision which
indi iduals o include in he nex gene a ion is usually based on i ness alues. This
e olu iona y cycle con inues un il he e mina ion c i e ion is ul illed (line 4).
Fogel (1999) summa izes expe iences om o y yea s o EP, and Fogel & Chellapilla
(1998) e isi and compa e EP wi h o he EA app oaches.
Gene ic Algo i hms
Gene ic algo i hms a e a a ian o EA, which, in analogy o he biological DNA alpha-
be , o iginally ocused on (bi ) s ing ep esen a ions. Howe e , al e na i e encodings
ha e been conside ed o he ep esen a ion issue, such as eal-coded GAs (Goldbe g,
1990; He e a e al., 1998).
Bina y s ings can be decoded in many ways o in ege o eal alues. The s ing co -
esponds o he geno ype o he indi idual. The pheno ype o he indi idual is ealized
by a mapping on o he objec pa ame e s, he so-called ’geno ype-pheno ype mapping’.
Fo example, a bina y s ing ha is eigh bi s long can encode in ege s be ween 0 and
255. The geno ype ’00000011’ decodes he in ege alue 3. The i ness o he indi id-
ual depends on he op imiza ion p oblem.
4
A ypical (ma ing) selec ion me hod in GAs is i ness-p opo ional selec ion. The p ob-
abili y ha an indi idual is selec ed o ma ing depends on i s i ness. Fo example, i
a popula ion consis s o an indi idual wi h i ness alue 1 and a second indi idual wi h
i ness 3, hen he e is a p obabili y o 1/(1+3) = 1/4 o selec ing he i s indi idual
and o 3/(1 + 3) = 3/4o selec ing he second indi idual. Many o he selec ion me h-
ods ha e been de eloped o GAs, such as ou namen selec ion. A ou namen is a
simple compa ison o he i ness alues o a small andomly chosen se o indi iduals.
The indi idual wi h he bes i ness is he winne o he ou namen and will be selec ed
o he ecombina ion (c osso e ) s ep.
Mu a ion adds new in o ma ion o he popula ion and gua an ees ha he sea ch p ocess
ne e s ops. A simple mu a ion me hod is bi - lipping o bina y encoded indi iduals:
a a andomly chosen posi ion, a ’0’ is changed o a ’1’ and ice- e sa. Fo example,
he geno ype ’00000011’ can be mu a ed o ’00000010’, i he eigh h bi is lipped.
Mu a ion can be pe o med wi h a ce ain p obabili y, which can be dec eased du ing
he sea ch. Besides mu a ion, c osso e is an impo an a ia ion ope a o o GAs. I
has he ollowing wo pu poses: (i) educ ion o he sea ch o mo e p omising egions
and (ii) inhe i ance o good gene p ope ies. One-poin c osso e is a e y simple
o m o c osso e . A a andomly chosen posi ion, segmen s om wo indi iduals a e
exchanged. Conside wo indi iduals, say ’10101010’ and ’11110000’. One-poin
c osso e a he hi d posi ion esul s in wo new indi iduals. The i s is ’10110000’
(wi h bi s one o h ee om he i s pa en and bi s ou o eigh om he second
pa en ) and he second is ’11110000’, espec i ely. O he popula c osso e me hods
a e wo-poin c osso e and uni o m c osso e (Sywe da, 1989). Ins ead o a i ness-
based en i onmen al selec ion, o en a simple gene a ional eplacemen p ocedu e is
used whe e he newly gene a ed o sp ing eplaces hei pa en s independen o i ness.
Algo i hm 2 p o ides a pseudocode lis ing o an GA. No e ha in addi ion o he
ope a o s used by EP, GAs apply ma ing selec ion (line 6) and c osso e (o ecombi-
na ion). C osso e combines in o ma ion om wo o mo e indi iduals (line 8). The
newly gene a ed o sp ing eplaces he pa en al indi iduals in he eplacemen p oce-
du e (line 14).
Goldbe g (1989) and Whi ley (1994) a e classical in oduc ions o GAs.
E olu ion S a egies
The i s ES, he so-called (1+1)-ES o wo membe ed e olu ion s a egy, uses one pa -
en and one o sp ing only. Two ules ha e been applied o hese candida e solu ions:
Apply small, andom changes o all a iables simul aneously. I he o sp ing solu-
ion is be e (has a be e unc ion alue) han he pa en , ake i as he new pa en ,
o he wise e ain he pa en . Schwe el (1995) desc ibes his algo i hm as ’ he mini-
mal concep o an imi a ion o o ganic e olu ion.’ The i s (1+1)-ES used binomially
dis ibu ed mu a ions (Schwe el, 1965). These ha e been eplaced by con inuous a i-
ables and Gaussian mu a ions, which enable he (1+1)-ES o gene a e la ge mu a ions
and he eby possibly escape om local op ima. Rechenbe g (1971) p esen ed an ap-
p oxima e analysis o he (1+1)-ES. His analysis showed ha he op imal mu a ion
5
Algo i hm 2: Gene ic algo i hm
1pop = Ini ializePopula ion(popula ionSize, p oblemDimension);
2e alua ePopula ion(popula ion);
3bes Solu ion = ge Bes Solu ion(popula ion);
4while es Fo Te mina ion == alse do
5o sp ing = ∅;
6pa en s = ma ingSelec ion(popula ion);
7 o pa en 1, pa en 2∈pa en s do
8(o sp ing1,o sp ing2) = c osso e (pa en 1, pa en 2);
9o sp ing1= mu a e(o sp ing1);
10 o sp ing2= mu a e(o sp ing2);
11 o sp ing = {o sp ing} ∪ o sp ing1∪o sp ing2;
12 e alua ePopula ion(o sp ing);
13 bes Solu ion = ge Bes Solu ion(o sp ing);
14 popula ion = eplace(popula ion, o sp ing);
15 Re u n(bes Solu ion);
a e co esponds o a success p obabili y ha is independen o he p oblem dimen-
sion. The op imal success p obabili y is app oxima ely 1/5 o a linea and also o
a quad a ic objec i e unc ion. These esul s inspi ed he amous one- i h ule: In-
c ease he mu a ion a e i he success a e is la ge han 1/5, o he wise, dec ease he
mu a ion a e. Schwe el’s e olu ion s a egy (ES) is a a ian o EA, which gene ally
ope a es on he na u al p oblem ep esen a ion and hus uses no geno ype-pheno ype
mapping o objec pa ame e s. In addi ion o he usual se o decision a iables, he
indi idual also con ains a se o so-called s a egy pa ame e s ha in luence he mu-
a ion ope a o (e.g., he s ep size). The ES employs mu a ion and ecombina ion as
a ia ion ope a o s. Beye & Schwe el (2002) p esen a comp ehensi e in oduc ion o
ES. Algo i hm 3 p o ides a pseudocode lis ing o an ES. Please no ha he s a egy
pa ame e s a e also subjec o ecombina ion (line 8) and mu a ion (line 9).
The co a iance ma ix adap a ion e olu ion s a egy (CMA-ES) is a a ian o ES,
which was de eloped o di icul non-linea non-con ex op imiza ion p oblems in con-
inuous domains (Hansen & Os e meie , 1996). To mo i a e he de elopmen o he
CMA-ES, conside he ollowing black-box op imiza ion p oblem
minimize
x∈Rn (x)(1)
and he ela ed p oblem
minimize
x∈Rn
˜
(x) := (Rx).(2)
He e Ris an n×n o a ion ma ix2. Clea ly we would like an op imiza ion algo i hm o
show simila pe o mance cha ac e is ics when sol ing p oblem (1) o (2). Bu nei he
2A o a ion ma ix is an o hogonal ma ix (RT=R−1) wi h de e minan (de R= 1). The se o all
n×n o a ion ma ices o ms he special o hogonal g oup SO(n).
6
Algo i hm 3: E olu ion s a egy.
1popula ion = Ini ializePopula ion(popula ionSize, p oblemDimension);
2e alua ePopula ion(popula ion);
3bes Solu ion = ge Bes Solu ion(popula ion);
4while es Fo Te mina ion == alse do
5o sp ing = ∅;
6 o i = 0 o o sp ingSize do
7ma ingPop = ma ingSelec ion(popula ion);
8o sp ingi= ecombina ion (ma ingPop);
9o sp ingi= mu a e(o sp ingi);
10 o sp ing = {o sp ing} ∪ o sp ingi;
11 e alua ePopula ion(o sp ing);
12 popula ion = en i onmen alSelec ion(popula ion);
13 bes Solu ion = ge Bes Solu ion(popula ion);
14 Re u n(bes Solu ion);
he (1 + 1)-ES no Schwe el’s ES a e in a ian unde o a ion and he e o e will pe -
o m qui e di e en ly when sol ing he wo p oblems. To illus a e his, le us conside
a simple example. Le
n= 2, (x) = x2
1+ 10x2
2and R=cos(π/4) −sin(π/4)
sin(π/4) cos(π/4) .
He e Ris a clockwise o a ion o x∈Rna ound he o igin by 45 deg ees. Bo h and
˜
each hei global minimum in x∗=0. Thei espec i e unc ion landscape is shown
in Figu e 1.
The op imal sampling dis ibu ion o (x)should ha e a co a iance s uc u e ha
(locally) ma ches he con ou s o he unc ion landscape. I we es ic ou sel es o he
mul i a ia e No mal dis ibu ion, hen he op imal sampling dis ibu ion o a ound
he poin xis gi en by
Nx, σ 1 0
0 10.
This dis ibu ion is clea ly co e ed by bo h he (1 + 1)-ES and Schwe el’s ES. The
op imal sampling dis ibu ion o ˜
on he o he hand is no wi hin he scope o ei he
algo i hm:
Nx, σR1 0
0 10.
He e, we need o adap no jus he indi idual’s a iances o each pa ame e bu also
he co a iance s uc u e which models he in e dependence be ween he pa ame e s so
ha he con ou lines o ou unc ion and he con ou o he sea ch dis ibu ion a e
(locally) simila .
I was his insigh ha ga e ise o he de elopmen o he o iginal CMA-ES algo-
i hm (Hansen e al., 1995). Ins ead o only adap ing he indi idual a iances in each
7
(x)
x1
x2
1
4
9
9
16
16
25
25
36
36
−2 −1 0 1 2
−2 −1 0 1 2
~(x)= (Rx)
x1
x2
1
4
9
9
16
16
25
25
36
36
49
49
64
64
−2 −1 0 1 2
−2 −1 0 1 2
Figu e 1: Con ou plo o (x)and ˜
(x)illus a ing he e ec o o a ion on he unc-
ion landscape.
i e a ion, a ull co a iance ma ix upda e is pe o med based on an es ima e o he
co a iance s uc u e om he cu en popula ion.
In he canonical CMA-ES, o sp ing is gene a ed by mu a ing he (some imes weigh ed)
cen e o he µpa en indi iduals, which is usually deno ed as (µ/µ, λ)-ES. The s a -
egy pa ame e s include he ull co a iance ma ix ins ead o jus he a iance o each
dimension o he sea ch space. The upda e o he s a egy pa ame e s is calcula ed us-
ing a maximum-likelihood app oach. He e he he mean o he sea ch dis ibu ion is
upda ed such ha he likelihood o selec ed o sp ings is maximized. The co a iance
ma ix is inc emen ally adap ed such ha he likelihood o success ul sea ch s eps is
maximized.
Because o i s iche class o sampling dis ibu ions compa ed o a egula ES and i s in-
a iance o o a ions o he sea ch space, i is no su p ising ha he CMA-ES has been
e y success ul a sol ing bo h syn he ic as well as eal-wo ld black-box op imiza ion
p oblems (Ke n e al., 2004). I is well sui ed o p oblems ha a e non-con ex, non-
sepa able, ill-condi ioned, mul i-modal, o i he objec i e unc ion is noisy. I he
objec i e unc ion is sepa able, he CMA-ES may no be ideal because i will a emp
o lea n a co a iance s uc u e whe e he e is none o exploi . In such cases a classic
ES may ou pe o m he CMA-ES. Recen ly he upda e mechanism o he CMA-ES has
been ecas as a o m o na u al g adien descen (Wie s a e al., 2008). This has made
i possible o adap he co e ideas o o he ypes o con inuous sea ch dis ibu ions. The
u o ial ’E olu ion S a egies and CMA-ES’ (Auge & Hansen, 2013) migh se e as
an in oduc ion o he ecen de elopmen s in he ield o CMA-ES.
8
Pa ame e con ol (on-line) is used o change EA pa ame e alues du ing a un and
o e s he g ea es lexibili y and p omises he bes pe o mance. Howe e , i poses
g ea challenges o EA designe s. Eiben e al. (1999) se es as a good s a ing poin .
Theo y
The heo e ical analysis made some p og ess o e he las decades, bu s ill many open
p oblems a e emaining. Rudolph (1997) in es iga ed con e gence p ope ies o ES.
Beye (2001) p esen s a amewo k and he i s s eps owa d he heo e ical analysis o
ES and a ecen a icle by Auge & Hansen (2011) p esen s global con e gence esul s
o ES. The Gene ic P og amming Theo y and P ac ice (GPTP) Wo kshop se ies dis-
cuss he mos ecen de elopmen s in GP heo y and p ac ice (Yu e al., 2006). Ree es
& Rowe (2002) p esen heo e ical esul s o GAs. The u o ial slides om Rowe
(2012) migh se e as a good s a ing poin o GA heo y.
The exis ence o a popula ion and he combina ion o se e al andomized p ocedu es
(mu a ion, ecombina ion, selec ion) make EA analysis di icul . Compu a ional com-
plexi y heo y, which can be seen as he co ne s one o compu e science, is a popula
app oach o he heo e ical analysis o EAs (Wegene , 2005). In applica ion o an-
domized sea ch heu is ics i akes he o m o black-box complexi y. Jansen (2013)
discusses black-box op imiza ion om a complexi y- heo e ical pe spec i e. Howe e ,
complexi y heo y can gi e pa adoxical esul s, such as assigning a low complexi y
o e y ha d p oblems. New de ini ions such as pa ame e ized complexi y ha e been
ecen ly p oposed (Downey & Fellows, 1999). Pa ame e ized complexi y classi ies
compu a ional p oblems wi h espec o hei numbe o inpu pa ame e s. The com-
plexi y o a p oblem is a unc ion in hose pa ame e s. Because he complexi y o a
p oblem is only measu ed by he numbe o bi s in he inpu in classical complexi y
heo y, p oblems can be classi ied on a ine scale.
Much o he ad ances in he heo y o e olu iona y algo i hms has s udied simpli ied
algo i hms on a i icial ( oy) p oblems. The applica ion o eal-wo ld p oblems is much
mo e di icul . An in e es ing app oach is based on landscape analysis, Kau man &
Le in (1987) in oduced NK i ness landscapes o cap u e he in ui ion ha bo h he
o e all size o he landscape and he numbe o i s local ’hills and alleys’ in luence
he complexi y o objec i e unc ions. Compu ing hese ea u es can be used o guide
he EA sea ch p ocess.
Explo ing new me hods o designing EAs is also subjec o cu en esea ch (Wie s a
e al., 2011; Ro hlau , 2011). And, las bu no leas , he e a e many undamen al ques-
ions on hei wo king p inciples o mul i-objec i e op imiza ion p oblems, which s ill
emain unsol ed (Coello e al., 2006).
15
Applica ions
Analyzing he pape s published in he mos popula EA con e ences (i.e., GECCO,
WCCI, PPSN) e eals ha he op ou ields o EA applica ions a e in enginee ing
(pa ame e op imiza ion), medicine, scheduling, and image analysis. The ollowing
pa ag aphs desc ibe impo an conside a ions ha a e necessa y o applying EAs in
p ac ice.
Choosing he Righ Model
Many classical algo i hms equi e simpli ied p oblems (e.g., quad a ic unc ions o di -
e en iabili y) o gua an ee exac solu ions, whe eas EAs gene a e app oxima e solu-
ions on he na u al p oblem. EAs a e able o wo k on a model o he eal p oblem, bu
canno gua an ee con e gence o he global op imum (Michalewicz & Fogel, 2004).
The design o a i ness unc ion should be concise. O e ly de ailed unc ions may use
oo many pa ame e s which impac nega i ely on he pe o mance o he sea ch algo-
i hm. As a ule o humb maybe a dimension o 30 is ypically well manageable by
an EA while any numbe abo e 300 may be called high dimensional in e olu iona y
e ms. Linea p og amming such as CPLEX can easily deal wi h se e al housand pa-
ame e s while being limi ed o linea p oblems only. Ko don e al. (2005) desc ibe a
me hodology how o deal wi h p oblems in indus y. They in eg a e EAs wi h s a is i-
cal me hods, neu al ne wo ks, and suppo ec o machines and desc ibe applica ions
in he a eas o in e en ial senso s, empi ical emula o s o mechanis ic models, accele -
a ed new p oduc de elopmen , complex p ocess op imiza ion, and e ec i e indus ial
design o expe imen s.
Cons ain s
When dealing wi h eal-wo ld p oblems cons ain s play a majo ole as almos all
eal-wo ld p oblems ha e o ake some kind o limi a ions o he pa ame e space in o
accoun . E olu iona y algo i hms can deal wi h cons ain s hough special echniques
such as he applica ion o penal y unc ions, decode s, epai mechanisms, cons ain
p ese ing ope a o s, o o he echniques need o be used. Some imes i could be wo h-
while e o mula ing a cons ain as an objec i e and ice e sa. This echnique can be
pa icula ly use ul in mul i-objec i e op imiza ion whe e he algo i hms a e designed
o deal wi h se e al objec i es a he same ime. Cons ain s can also be imposed g ad-
ually so ha he algo i hm may iola e cons ain s in i s ea ly explo a i e s age while
ge ing cons ained o he ue easible solu ion space as he popula ion ma u es. Al-
hough heo e ically e olu iona y algo i hms can ind solu ions anywhe e in he solu-
ion space, in eali y cons ain s can di ec he algo i hms in o a subop imal a ea ha ing
only a e y slim chance o escape. S a ing he algo i hm nea a known good solu ion
migh help. Also in e ac i e e olu ion could be a e y powe ul echnique as he use
can in e ac i ely esol e some issues i he algo i hm ge s s uck.
16
Michalewicz & Schoenaue (1996) su ey se e al EA based app oaches o cons ained
pa ame e op imiza ion p oblems. Coello (2013a) has compiled a lis o mo e han
1,000 e e ences on cons ain -handling echniques used wi h EAs.
Expensi e Func ion E alua ions
In indus ial applica ions, supe io solu ions ha e been ound needing a minimum num-
be o e.g. 150 i ness unc ion e alua ions only. Two app oaches, which ackle his
p oblem, a e conside ed nex : (i) pa alleliza ion and (ii) me a-modeling.
Pa allel e alua ion o se e al indi iduals can speed up he algo i hm o e en inc ease
he p obabili y o inding be e solu ions h ough u ilizing mul iple pa allel popula-
ions ha communica e spo adically h ough mig an s. A discussion o hese pa -
alleliza ion concep s goes a beyond he scope o his a icle, he eade is e e ed
o Can ´
u-Paz (2001) o an elemen a y in oduc ion. Alba & Tomassini (2002) in es-
iga e pa allelism and EAs. Hu e al. (2010) analyze he e ec o a iable popula ion
size on accele a ing e olu ion in he con ex o a pa allel EA. Can ´
u-Paz (2007) e iews
pa ame e se ings in pa allel GAs.
Me a-modeling can be a e y powe ul ool in case he e alua ion o a i ness unc ion
is oo expensi e. Me a-modeling is a echnique ha eplaces an expensi e ma he-
ma ical model (such as FEM o CFD) o a complex physical expe imen by a o en
c ude bu e y quick o e alua e model. S a is ical app oaches such as Design o Ex-
pe imen s (DoE) om Taguchi ype models o sophis ica ed K iging a e ypical. Em-
me ich (2005) desc ibes he de elopmen o obus algo i hms o op imiza ion wi h
ime-consuming e alua ions. The main wo king p inciple o hese echniques is o
combine spa ial in e pola ion echniques wi h EAs. Cu en esul s om hese demand-
ing eal-wo ld applica ions a e p esen ed du ing GECCO’s e olu iona y compu a ion in
p ac ice ace, see, e.g., h p://www.sige o.o g/gecco-2013/ecp.h ml.
Analyzing he Resul s
Because he ou pu o an EA un is s ochas ic, a solu ion he algo i hm inds in one un
may sligh ly o some imes qui e signi ican ly di e om ano he un. A ho ough anal-
ysis o EA esul s needs s a is ical analysis. Any c i ical EA analysis should con ain a
leas box-and-whiske s plo s o illus a e he s a is ical sp ead o he esul s as any EA
un will esul ypically in a dis ibu ion o solu ions a ound any (local) op imum he
algo i hm inds. Ve y help ul in indus ial applica ions could be a look a he pa ame e
se ings o a solu ion. A solu ion close o a cons ain could indica e ha he op i-
miza ion p oblem migh be o e -cons ain o he e is a po en ial be e solu ion in he
eal-wo ld when a cons ain can be elaxed. Also he pa e n o he pa ame e se ings
in he pa ame e space can help unde s anding he unde lying p oblem s uc u e. A
andom walk s uc u e o example may indica e local pla eau a eas. In case solu ions
can be isualized h ough CAD i can some imes be help ul o iew he ac ual e olu-
ion o he solu ion as a ideo. This can help imp o ing pa ame e se ings ( inding he
17
na ow e olu iona y window, i.e., he pa ame e window whe e he EA con e ges bes
owa ds o be e solu ions) o de e mining ealis ic s opping c i e ia o he algo i hm.
I should be highligh ed ha obus ness o he solu ion is o en a e y impo an c i-
e ion in indus ial p ac ice. The bes solu ion may be con ained in a e y na ow se
in e als ha when le he solu ions de e io a e quickly. Robus solu ion in his sense
would allow some a ia ion in he pa ame e se ings. Any obus ness analysis o a p o-
posed solu ion (e.g., using ANOVA) gi es ex a con idence in a esul ha is supposed
o be applied in a c i ical indus ial applica ion.
Limi a ions
Some EAs ace limi a ions when i comes o budge ed i ness e alua ions. The s ochas-
ic na u e o EA may also be a limi ing ac o when i comes o sa e y c i ical ap-
plica ions whe e epea abili y is impo an . Typically in hese cases, EA a e used o
imp o e he pa ame e se ings o de e minis ic algo i hms. The explana ion o a solu-
ion can also be di icul as he way how a solu ion was deduced is based on a complex
s ochas ic sea ch a he han on a de e minis ic one-s ep-a -a- ime app oach. A p oo
ha a solu ion is op imal will no be p o ided by any cu en EA, no e en how close a
solu ion migh be o an op imal solu ion.
Ko don (2010) gi es a li ely desc ip ion how o apply EAs in indus y and desc ibes
se e al pi alls. Filipiˇ
c & Tuˇ
sa (2013) p esen wo case s udies o applying op imiza-
ion me hodology in indus y, one in ol ing nume ical op imiza ion based on simu-
la ion models, and he o he combina o ial op imiza ion wi h speci ic cons ain s and
objec i es. They iden i y some o he challenges equen ly me by solu ion p o ide s
o indus ial op imiza ion p oblems.
Da a Mining and Knowledge Disco e y
EAs ha e been success ully applied in a la ge a ie y o eal-wo ld p oblem a eas.
Gi en he ocus o he jou nal, i seems sensible o b ie ly discuss a pa icula applica-
ion a ea o EAs: Da a mining and knowledge disco e y. In a sense, da a mining is
abou inding good and meaning ul models and ules, and hus essen ially an op imiza-
ion ask. So i is no su p ising ha EAs may be help ul also in his a ea.
They ha e been p oposed o a a ie y o da a-mining ela ed asks, including ea u e
selec ion and ea u e cons uc ion, ins ance selec ion, o ule ex ac ion, Ghosh & Jain
(2005) p o ide a numbe o examples. F ei as (2002a,b, 2008) p esen s a comp ehen-
si e in oduc ion o da a mining and EAs and in oduces he e m e olu iona y da a
mining o subsume any da a mining using EAs.
Mo e speci ic, Tan e al. (2005) p esen a dis ibu ed coe olu iona y classi ie o ex-
ac ing comp ehensible ules in da a mining. Vladisla le a e al. (2013) o ecas he
ene gy ou pu o wind a ms using GP and epo on he co ela ion o he di e en a i-
ables o he ene gy ou pu . No e ha GP is able o sea ch o symbolic ep esen a ions
18
ha a e in e p e able (Vladisla le a, 2008). Schmid & Lipson (2008) use GP ech-
niques o au oma ically e e se enginee ing symbolic analy ical models o dynamical
sys ems di ec ly om expe imen al obse a ions. EAs ha e been used indi ec ly o
une pa ame e s o da a-mining algo i hms (Lessmann e al., 2005) o eplace machine
lea ning algo i hms such as clus e ing (Handl & Knowles, 2007).
E en he au oma ed design o new da a mining algo i hms has been p oposed (Pappa
& F ei as, 2010). Las , bu no leas , Schmid & Lipson (2009) has o be men ioned.
The au ho s apply sophis ica ed GP echniques o he ’iden i ica ion o non i iali y’.
Mo ion- acking da a cap u ed om a ious physical sys ems, e.g, ha monic oscilla-
o s and chao ic double-pendula, is used o e-disco e Hamil onians, Lag angians, and
o he laws o geome ic and momen um conse a ion. In e es ingly, no p io knowl-
edge abou physics, kinema ics, o geome y, is used by he algo i hm.
So wa e
A a ie y o so wa e amewo ks o GP is a ailable: Da aModele is a so wa e pack-
age ha is de eloped wi hin he con ex o indus ial da a analysis (E ol ed Analy ics LLC,
2010). I implemen s se e al non-linea modeling echniques such as Pa e o-symbolic
eg ession, s a is ical lea ning heo y, and non-linea a iable selec ion. The GP-based
so wa e Discipulus is applied o Da a Mining as well as o p oblems equi ing p e-
dic i e Analy ics and Classi ica ion (F ancone, 2010). Eu eqa is a so wa e ool o
de ec ing equa ions and hidden ma hema ical ela ionships in da a (Dubˇ
c´
ako ´
a, 2011;
Aus em, 2012). GPTIPS is a ee gene ic p og amming (GP) and p edic i e modeling
oolbox o MATLAB (Sea son e al., 2010).
MATLAB’s Global Op imiza ion Toolbox has gene ic algo i hms o single and mul i
objec i e unc ions (Ma hwo ks, 2011). Implemen a ions o he CMA-ES and links o
lib a ies ha con ain such implemen a ions can be ound on he au ho ’s web page3(Hansen
e al., 1995). The Ja a E olu iona y Compu a ion Toolki (ECJ) is a eewa e e olu-
iona y compu a ion esea ch sys em w i en in Ja a. I implemen s se e al EA ech-
niques, e.g., GAs, GP, and ES (Luke, 2013). The MOEA F amewo k is an open-sou ce
e olu iona y compu a ion lib a y o Ja a ha specializes in mul i-objec i e op imiza-
ion (Hadka, 2012).
S a e-o - he-a so wa e packages o pa ame e uning and algo i hm con igu a ion
such as Bonesa (Smi & Eiben, 2011), i ace (L´
opez-Ib´
anez e al., 2011), Pa amILS (Hu e
e al., 2010), and SPOT (Ba z-Beiels ein & Zae e e , 2011) a e eely a ailable om
he au ho s’ web pages.
Alcal´
a-Fdez e al. (2009) de elop KEEL, an open sou ce Ja a so wa e ool o assess
e olu iona y algo i hms o da a mining p oblems. Miku & Reischl (2011) discuss
he his o ical de elopmen and p esen a ange o exis ing s a e-o - he-a da a mining
and ela ed ools. They p o ide a lis o da a mining ools, which includes EA based
3h ps://www.l i. /˜hansen/cmaes_inma lab.h ml
19
app oaches, oo. Weka and RapidMine , which p o ide se e al machine lea ning al-
go i hms o sol ing eal-wo ld da a mining p oblems, con ain a couple o EA-based
sea ch me hods (Wi en & F ank, 2005; Rapid-I, 2010). Finally, he s a is ical so -
wa e R(R Co e Team, 2005) should be men ioned. Se e al EAs, e.g., emoa,GA, o
cmaes a e a ailable as R packages, see h p://c an. -p ojec .o g/web/
packages.
Conclusion
E olu iona y algo i hms a e es ablished s ochas ic di ec sea ch algo i hms.
The e olu iona y cycle can be seen as he common g ound o EA. They a e
ying o each op imal s a es by successi e imp o emen s. Imp o emen s oc-
cu by a ia ion (mu a ion, ecombina ion). Se e al p oblem speci ic selec ion
me hods enable o cope wi h di e en si ua ions, e.g., noisy and dynamically
changing en i onmen s. Since hey a e popula ion-based sea ch algo i hms,
EAs a e well sui ed o sol e mul i-objec i e op imiza ion p oblems. They a e
also lexible ools o da a-mining p oblems. By modi ying he e olu iona y cy-
cle, new membe s o he EA amily a e gene a ed. Bio-inspi ed algo i hms
such as pa icle swa m op imiza ion (Ebe ha & Kennedy, 1995) o an colony
algo i hms (Do igo, 1992) en ich he EA amily wi h new p oblem speci ic op i-
miza ion echniques.
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