A new Taxonomy of Continuous Global Optimization Algorithms

CIplus
Band 4/2018
A new Taxonomy o Con inuous Global
Op imiza ion Algo i hms
Jö g S o k, A.E. Eiben, and Thomas Ba z-Beiels ein
A new Taxonomy o Con inuous Global Op imiza ion
Algo i hms
Jö g S o k1, A.E. Eiben2, and Thomas Ba z-Beiels ein1
1TH Köln, S einmüelle allee 1, 51648 Gumme sbach, Ge many
2VU Ams e dam, De Boelelaan 1105, 1081 HV Ams e dam, Ne he lands
joe [email p o ec ed], [email p o ec ed], [email p o ec ed]
Abs ac
Su oga e-based op imiza ion and na u e-inspi ed me aheu is ics ha e become he s a e
o he a in sol ing eal-wo ld op imiza ion p oblems. S ill, i is di icul o beginne s
and e en expe s o ge an o e iew ha explains hei ad an ages in compa ison o he
la ge numbe o a ailable me hods in he scope o con inuous op imiza ion. A ailable ax-
onomies lack he in eg a ion o su oga e-based app oaches and hus hei embedding in
he la ge con ex o his b oad ield. This a icle p esen s a axonomy o he ield, which
u he ma ches he idea o na u e-inspi ed algo i hms, as i is based on he human beha io
in pa h inding. In ui i e analogies make i easy o concei e he mos basic p inciples o he
sea ch algo i hms, e en o beginne s and non-expe s in his a ea o esea ch. Howe e ,
his scheme does no o e simpli y he high complexi y o he di e en algo i hms, as he
class iden i ie only de ines a desc ip i e me a-le el o he algo i hm sea ch s a egies. The
axonomy was es ablished by explo ing and ma ching algo i hm schemes, ex ac ing simi-
la i ies and di e ences, and c ea ing a se o classi ica ion indica o s o dis inguish be ween
i e dis inc classes. In p ac ice, his axonomy allows ecommenda ions o he applicabil-
i y o he co esponding algo i hms and helps de elope s ying o c ea e o imp o e hei
own algo i hms.
1 In oduc ion
Con inuous global op imiza ion (CGO) ackles a ious di icul p oblems eme ging om he
con ex o complex physical o chemical p ocesses. Sol ing op imiza ion p oblems o his kind
necessa ily elies on pe o ming eal-wo ld expe imen s o on using compu e simula ions,
which a e equen ly employed in black-box ashions. A undamen al challenge in such sys-
ems is he high cos o unc ion e alua ions. Whe he we a e p obing he eal physical sys em
o que ying he simula o , he ime needed o ecei e an objec i e unc ion alue is ypically
e y high and can ange om hou s o mon hs. CGO me hods o such p oblems hus need
o ul ill a ce ain se o equi emen s. They need o wo k wi h black-box s yle p obes only, so
wi hou any u he in o ma ion on he s uc u e o he p oblem. Fu he hey mus app oach
he icini y o he global op imum wi h a limi ed numbe o unc ion e alua ions.
The imp o emen o compu a ional powe in he las decades has been in luencing he de-
elopmen o algo i hms. A massi e amoun o compu a ional powe became a ailable o
1
esea che s wo ldwide h ough mul i-co e desk op machines, pa allel compu ing, and high-
pe o mance compu ing clus e s. This has imp o ed he ollowing ields o esea ch: Fi s ly, he
de elopmen o mo e complex, na u e-inspi ed, and gene ally applicable heu is ics, so called
me aheu is ics. Secondly, majo ad ances in he ield o accu a e, da a-d i en app oxima ion
models, so-called su oga e models and hei embodimen in an op imiza ion p ocess.
Nowadays, CGO di e s la gely om ea ly app oaches. Fo example, mul i-s aged me hods
e alua e objec i e unc ions no di ec ly on he p oblem. They u ilize a combina ion o su -
oga e modeling wi h classical o me aheu is ic op imiza ion me hods o maximize he use o
a ailable p oblem in o ma ion ins ead. These amewo ks, such as sequen ial pa ame e op i-
miza ion (Ba z-Beiels ein, Lasa czyk, and P euß, 2005) o he su oga e managemen amewo k
(Booke , Dennis J , F ank, Se a ini, To czon, and T osse , 1999; Se a ini, 1999), de ine a new
class o algo i hms ha a e no well in eg a ed in p e ious axonomies.
In his a icle we will p opose a new axonomy on basis o algo i hm ea u es and display
plausible desc ip ions ounded on he na u al human beha io in pa h inding. To es ablish a
comp ehensi e axonomy, we ocus on iden i ying key elemen s o algo i hm design and u i-
lizing hese o de ine a clea sepa a ion be ween a small numbe o algo i hm classes. Al hough
abs ac ion is necessa y o de eloping ou esul s, we will p esen esul s ha a e use ul o
p ac i ione s. The u ilized abs ac ion allows us o p esen simply comp ehensible ideas on
how he indi idual classes di e and mo eo e , how he espec i e algo i hms pe o m hei
sea ch. Fo his pu pose, we di ide CGO algo i hms in o i e in ui i e classes: Exac , Wande e ,
Guide, Ca og aphe , and Hyb id. This a icle pa icula ly add esses di e en kinds o ead-
e s: beginne s will ind an in ui i e axonomy o CGO algo i hms, especially wi h ega d o
common me aheu is ics and newe de elopmen s in he ield o su oga e-based op imiza ion.
Fo ad anced eade s, we also discuss he sui abili y o ce ain classes o speci ic p oblem
p ope ies o p o ide basic knowledge o easonable algo i hm selec ion. An ex ensi e lis
o e e ences is p o ided o he expe ienced use s. The axonomy can be used o c ea e eal-
is ic compa isons and benchma ks o he di e en classes o algo i hms. I u he p o ides
insigh s o use s, who aim o de elop new sea ch s a egies, ope a o s and algo i hms.
The goal o global op imiza ion is o ind he o e all bes solu ion, i.e., o he common ask
o minimiza ion, o disco e decision a iable alues which minimize he objec i e unc ion
alue. We deno e he global sea ch space as compac se S={x|xl≤x≤xu}wi h xl,xu∈
R
n
being he explici , ini e lowe and uppe bounds on x.
Gi en a eal alued objec i e unc ion :
R
n→
R
wi h eal alued inpu ec o s xwe a emp
o ind he loca ion x∈
R
nwhich minimizes he unc ion: a g min (x),x∈ S.
Finding he global op imum is always he ul ima e goal and as such desi able, bu o many
p ac ical p oblems a solu ion imp o ing he cu en bes solu ion in a gi en budge o e alua-
ions o ime will s ill be a success. Pa icula in CGO he global op imum commonly canno be
iden i ied exac ly, hus mode n heu is ics a e designed o spend hei esou ces as e icien ly as
possible o app oxima e nea -bes solu ions, while inding he global op imum is ne e gua an-
eed. The emainde o his a icle is o ganized as ollows: Sec ion 2 p esen s he de elopmen
o op imiza ion algo i hms and hei co e concep s. Sec ion 3 mo i a es a new axonomy by
e iewing he his o y o a ailable CGO axonomies, illus a es algo i hm design aspec s and
p esen s ex ac ed classi ica ion ea u es. Sec ions 4 o 8 in oduce he i e di e en classes o
he new axonomy wi h examples and sugges ions ega ding hei applicabili y. Sec ion 9 sum-
ma izes and concludes he a icle wi h he ecen ends and challenges in CGO and cu en ly
impo an esea ch ields.
2
2 E olu ion o Op imiza ion Algo i hms
In o de o de elop a axonomy, i is necessa y o unde s and he me hodology and de elop-
men his o y o he co esponding algo i hms. Be o e p esen ing he equi emen s o he new
axonomy in Sec ion 3, we will desc ibe he undamen al p inciples o mode n sea ch algo-
i hms, pa icula he elemen s and backg ounds o su oga e-based op imiza ion.
2.1 Heu is ics and Me aheu is ics
In mode n compu e -aided op imiza ion, heu is ics and me aheu is ics a e well es ablished
solu ion echniques. Al hough p esen ing solu ions which a e no gua an eed o be op imal
o pe ec , hei gene al applicabili y and abili y o p esen ing as su icien solu ions make
hem e y a ac i e o applied op imiza ion, pa icula o indus ial p oblems. They a e buil
upon he p inciple o ial and e o , whe e solu ion candida es a e e alua ed and ewa ded
wi h a i ness. The e m i ness has i s o igins in e olu iona y compu a ion (Eiben and Smi h,
2015), whe e he i ness desc ibes he compe i i e abili y o an indi idual in he ep oduc ion
p ocess. The i ness is in i s simples o m he objec i e unc ion alue y= (x)in ela ion
o he op imiza ion goal, e.g., in a minimiza ion p oblem smalle alues ha e a highe i ness.
Mo eo e , i can be pa o he sea ch s a egy, e.g., scaled o adjus ed by addi ional unc ions,
pa icula o mul i-objec i e o cons ained op imiza ion.
Heu is ics can be de ined as p oblem-dependen algo i hms, which a e de eloped o adap ed
o he pa icula i ies o a speci ic op imiza ion p oblem o p oblem ins ance (Pea l, 1985). Typ-
ically, heu is ics pe o m e alua ions in a sys ema ic manne , al hough u ilizing s ochas ic el-
emen s. Heu is ics use his p inciple o p o ide as , no necessa ily exac (i.e., no op imal)
nume ical solu ions o op imiza ion p oblems. Mo eo e , heu is ics a e o en g eedy o p o-
ide as solu ions, bu ge apped in local op ima and ail o ind he global op imum.
In hei s a ing days in he 1960s, heu is ics we e no conside ed as eliable p oblem sol e s,
because mos esea che s in academia p e e ed classical ma hema ical app oaches and only
some p ac i ione s used heu is ics o ge as , possibly inaccu a e solu ions (Zanakis and E ans,
1981). This si ua ion changed in he 1970s. Heu is ic op imiza ion became a majo pa o
academic esea ch. Possible easons o his change migh be:
•The need o sol e mo e sophis ica ed nonde e minis ic p oblems, which had a polyno-
mial un ime and hus could no be sol ed e icien ly wi h exac algo i hms (Fomin and
Kaski, 2013).
•The a ailabili y and easy access o academics o mo e compu a ional powe .
Fu he ad an ages o heu is ics we e summa ized by Zanakis and E ans (1981). The mos
impo an a e:
•Simplici y o he algo i hm.
•Accu acy, i.e., small e o o inal solu ion.
•Robus ness, i.e., good solu ions wi hin easonable ime o di e en p oblems.
•Speed, i.e., du a ion o he compu a ion.
While heu is ics a e de eloped and op imized o e icien ly sol e a ce ain p oblem, he im-
p o ed a ailabili y o compu e esou ces ga e ise o highe -le el heu is ics, he me aheu is-
ics. Me aheu is ics can be de ined as p oblem independen , gene al pu pose op imiza ion al-
go i hms. They a e applicable o a wide ange o p oblems and p oblem ins ances. The e m
me a desc ibes he highe -le el gene al me hodology, which is u ilized o guide he unde lying
heu is ic s a egy (Talbi, 2009).
3

They sha e he ollowing cha ac e is ics (Boussaïd, Lepagno , and Sia y, 2013):
•The algo i hms a e na u e-inspi ed; hey ollow ce ain p inciples om na u al phenom-
ena o beha io s (e.g., biological e olu ion, physics, social beha io ).
•The sea ch p ocess in ol es s ochas ic pa s; i is based on p obabili y dis ibu ions and
andom p ocesses.
•They do no use he g adien o Hessian o he objec i e unc ion o ely on in o ma ion
o he p ocess which is a ailable be o e he s a o he op imiza ion un, so-called a p io i
in o ma ion.
•As hey a e mean o be gene al applicable sol e s, hey include a se o con ol pa ame-
e s o adjus he sea ch s a egy.
Du ing he emainde o his a icle we will ocus on heu is ic, espec i ely me aheu is ic algo-
i hms.
2.2 Mode n Op imiza ion Algo i hms
Based on he undamen als o heu is ics and me aheu is ics, we a e able o iden i y a simila i y
in he design o mode n op imiza ion algo i hms, which a ge s a la ge class o p oblems. Im-
po an ly, we ha e o conside he No F ee Lunch Theo em (Wolpe and Mac eady, 1997), which
s a es ha he e is no op imiza ion algo i hm ha is supe io o all o he s i hei pe o mance
is a e aged o e all possible p oblems. Consequen ly, any algo i hm needs o be adap ed o
he s uc u e o he p oblem a hand o achie e op imal pe o mance. This can be conside ed
du ing he cons uc ion o an algo i hm, be o e he op imiza ion by pa ame e uning o du ing
he un by pa ame e con ol (Ba z-Beiels ein e al., 2005; Eiben, Hin e ding, and Michalewicz,
1999). Tö n and Zilinskas (1989) men ion h ee p inciples o he cons uc ion o an op imiza-
ion algo i hm:
1. An algo i hm u ilizing all a ailable a p io i in o ma ion will ou pe o m a me hod using
less in o ma ion.
2. I no a p io i in o ma ion is a ailable, he in o ma ion is comple ely based on e alua ed
candida e poin s and hei objec i e alues.
3. Gi en a ixed numbe o e alua ed poin s, op imiza ion algo i hms will only di e om
each o he in he dis ibu ion o candida e poin s.
As mos mode n algo i hms ocus on handling p oblems whe e li le o no a p io i in o ma ion
is gi en, he p inciples displayed abo e lead o he conclusion ha he mos c ucial design
aspec o any algo i hm is o ind a s a egy o dis ibu e he ini ial candida es in he sea ch
space and o gene a e new candida es based on a ia ion o solu ions. These wo p ocedu es
de ine he sea ch s a egy, which needs o ollow he wo compe ing goals o explo a ion and
exploi a ion. In gene al, he main goal o any me hod is o each hei a ge wi h high e iciency,
i.e., o disco e op ima as and accu a e wi h as li le esou ces as possible. Mo eo e , he goal
is no manda o y inding he global op imum, which is a demanding and expensi e ask o
many p oblems, bu o iden i y a aluable local op imum o o imp o e he cu en a ailable
solu ion. We will explici ly discuss he design o mode n op imiza ion algo i hms in Sec ion
3.3.
4
2.3 Exac Algo i hms
Exac algo i hms, also e e ed o as comple e algo i hms (Neumaie , 2004), a e a special class
o de e minis ic, sys ema ic and non-heu is ic op imiza ion algo i hms. I su icien a p io i in o -
ma ion abou he objec i e unc ion is a ailable, hey ha e a gua an ee o inding he global
op imum wi hin using a p edic able amoun o esou ces, such as unc ion e alua ions o com-
pu a ion ime (Fomin and Kaski, 2013). I hey a e applicable o he p oblem, hese algo i hms
a e mo e eliable han heu is ics, as hey allow con e gence p oo s o inding he global op i-
mum. Wi hou a ailable a p io i in o ma ion, he s opping c i e ium needs o be de ined by a
heu is ic app oach, which so ens he gua an ee o inding he op imum. Mo eo e , i is he-
o e ical possible o apply hese algo i hms o he class o black-box p oblems wi h he gi en
abili y o ind he global op imum wi h ce ain y a e ini e ime. Howe e , hey will need
exponen ial compu a ion ime due o an expensi e, dense sea ch. This ende s hem no ap-
plicable o many esou ce-limi ed applica ions. The exac class, p esen ed in Sec ion 4, con ains
he ela ed algo i hms.
2.4 Su oga e-based Op imiza ion Algo i hms
Su oga e-based op imiza ion algo i hms a e designed o p ocess expensi e and complex p ob-
lems, which a ise om eal-wo ld applica ions and sophis ica ed compu a ional models. Real-
wo ld p oblems a e commonly black-box, which means ha hey only p o ide e y spa se
domain knowledge. Consequen ly, p oblem in o ma ion needs o be exploi ed by expe imen s
o unc ion e alua ions.
Su oga e-based op imiza ion is de eloped o op imally exhaus he a ailable in o ma ion by
u ilizing a su oga e model. A su oga e model is an app oxima ion which subs i u es he o ig-
inal expensi e objec i e unc ion, eal-wo ld p ocess, physical simula ion, o compu a ional
p ocess du ing he op imiza ion. In gene al, su oga es a e ei he simpli ied physical o nu-
me ical models based on knowledge abou he physical sys em, o empi ical unc ional models
based on knowledge acqui ed om e alua ions and spa se sampling o he pa ame e space
(Sønde gaa d, Madsen, and Nielsen, 2003). In his wo k, we ocus on he la e . The e ms su -
oga e model,me a-model, esponse su ace model and pos e io dis ibu ion a e used synonymous
in he common li e a u e (Mockus, 1974; Jones, 2001; Ba z-Beiels ein and Zae e e , 2017). We
will b ie ly e e o a su oga e model as su oga e. Fu he mo e, we assume ha i is c ucial
o dis inguish be ween he use o an explici su oga e o he objec i e unc ion and gene al
model-based op imiza ion (Zlochin, Bi a a i, Meuleau, and Do igo, 2004), which addi ionally
e e s o me hods, whe e a s a is ical model is used o gene a e new candida e solu ions (c .
Sec ion 3.3). As hese wo de ini ions o model-based op imiza ion a e equen ly used in a
non-consis en manne , we will clea ly dis inguish be ween he wo di e en e ms su oga e-
based and model-based o a oid con usions. Ano he e m p esen in he li e a u e is su oga e-
assis ed op imiza ion, which mos ly e e s o he applica ion o su oga es in popula ion-based
e olu iona y compu a ion (Jin, 2011), whe e e olu iona y op imiza ion and su oga e-based
op imiza ion a e applied in an hyb id app oach (see Sec ion 8).
Impo an publica ions ea u ing o e iews o su eys on su oga es and su oga e-based op-
imiza ion we e p esen ed by Sacks, Welch, Mi chell, and Wynn (1989), Jones (2001), Queipo,
Ha ka, Shyy, Goel, Vaidyana han, and Tucke (2005), Fo es e and Keane (2009). Su oga e-
based op imiza ion is commonly applied, bu no limi ed o he case o complex eal-wo ld
op imiza ion applica ions, whe e wo ypical p oblem laye s and a su oga e laye can be de-
5
(op ional)
uning
p ocedu e
op imiza ion algo i hm
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s(x)
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su oga e model
1(x)
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1(x)
<la exi sha1_base64="58e JI4/e+23 IDakVSEuBI2PGA=">AAAC0XichVFLT8JAEB7qC/CFe TSCCZ4IS0XPZL4iBcTjPJIkJB WbCh 7QLEYmJ8e Nq/4x/S0e/HY JkoM22xn9p p3ZsULXiYVh Ke0hcWl5ZV0J u6 6xmd a s BMLJ5zQ7cIGpaLOau4/OacITLm2HEmWe5 GENjmW8MeJR7AT+ RiH O2x u/0HJsJQM1C 2MW7w4KnVzeKBlq6bOOmTh5SlY1yH3QDXUpIJuG5BEnnwR8lxjF+FpkkkEhsDZNgEXwHBXn9EBZ5A7B4mAwoAP8+zi1E THWW GK GLS52hEyd9 HPlKIF yVw49hP7H Fdb/94aJUpYVjmE KGaU4gVwQbdgzM 0Eua0l mZsi BPTpS3TioL1SI7NP+0TlBJAI2UBGdThWzDw1LnUd4AR+2hg kK08VdNVxF5Ypy5WKnygy6EWw8 VRD8Zs/h3q FM l0yjZF6W8xUzGXiadmmPipjqIVXonKqoQ07zhV7pTb Sx qj9 RN1VJJzg79W zF9SgkMs=</la exi >
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physical eal wo ld p ocess
con ol pa ame e s op imized con ol
pa ame e s
decision
a iables
op imized
a iables
p ocess pa ame e s,
es ima ed ou pu
simula ed
inpu
inpu ou pu
candida e solu ions p edic ed i ness
app oxima ion and e i ica ion su oga e-based op imiza ion
2(x)
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2(x)
<la exi sha1_base64="OYhAsR8zTgd/F7lNWP Ba 3g+Sg=">AAAC0XichVFLT8JAEB7qC/CFe TSCCZ4IS0XPZL4iBcTjPJIkJB WbCh 7QLEYmJ8e Nq/4x/S0e/HY JkoM22xn9p p3ZsULXiYVh Ke0hcWl5ZV0J u6 6xmd a s BMLJ5zQ7cIGpaLOau4/OacITLm2HEmWe5 GENjmW8MeJR7AT+ RiH O2x u/0HJsJQM1C 1Mu3h0UO m8UTLU0mcdM3HylKxqkPugG+pSQDYNySNOPgn4LjGK8bXIJINCYG2aAI gOS O6YGyyB2CxcFgQA 493FqJaiPs9SMVbaNW1zsCJk67WO KUULbHk hx/D mL K6z/7w0TpSw HMNaUMwoxQ ggm7BmJ pJcxpL MzZVeCenSkunFQX6gQ2a 9o3OCSARsoCI6nSpmHxqWOo/wAj5sDRXIV54q6K jLixTlisVP1Fk0I g5eujHozZ/D UWadeLplGybws5y mM A07dIeFTHVQ6 QOVVRh5zmC73Sm3aljbVH7embqqWSnB36 bTnL9cKkMw=</la exi >
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<la exi sha1_base64="OYhAsR8zTgd/F7lNWP Ba 3g+Sg=">AAAC0XichVFLT8JAEB7qC/CFe TSCCZ4IS0XPZL4iBcTjPJIkJB WbCh 7QLEYmJ8e Nq/4x/S0e/HY JkoM22xn9p p3ZsULXiYVh Ke0hcWl5ZV0J u6 6xmd a s BMLJ5zQ7cIGpaLOau4/OacITLm2HEmWe5 GENjmW8MeJR7AT+ RiH O2x u/0HJsJQM1C 1Mu3h0UO m8UTLU0mcdM3HylKxqkPugG+pSQDYNySNOPgn4LjGK8bXIJINCYG2aAI gOS O6YGyyB2CxcFgQA 493FqJaiPs9SMVbaNW1zsCJk67WO KUULbHk hx/D mL K6z/7w0TpSw HMNaUMwoxQ ggm7BmJ pJcxpL MzZVeCenSkunFQX6gQ2a 9o3OCSARsoCI6nSpmHxqWOo/wAj5sDRXIV54q6K jLixTlisVP1Fk0I g5eujHozZ/D UWadeLplGybws5y mM A07dIeFTHVQ6 QOVVRh5zmC73Sm3aljbVH7embqqWSnB36 bTnL9cKkMw=</la exi >
<la exi sha1_base64="OYhAsR8zTgd/F7lNWP Ba 3g+Sg=">AAAC0XichVFLT8JAEB7qC/CFe TSCCZ4IS0XPZL4iBcTjPJIkJB WbCh 7QLEYmJ8e Nq/4x/S0e/HY JkoM22xn9p p3ZsULXiYVh Ke0hcWl5ZV0J u6 6xmd a s BMLJ5zQ7cIGpaLOau4/OacITLm2HEmWe5 GENjmW8MeJR7AT+ RiH O2x u/0HJsJQM1C 1Mu3h0UO m8UTLU0mcdM3HylKxqkPugG+pSQDYNySNOPgn4LjGK8bXIJINCYG2aAI gOS O6YGyyB2CxcFgQA 493FqJaiPs9SMVbaNW1zsCJk67WO KUULbHk hx/D mL K6z/7w0TpSw HMNaUMwoxQ ggm7BmJ pJcxpL MzZVeCenSkunFQX6gQ2a 9o3OCSARsoCI6nSpmHxqWOo/wAj5sDRXIV54q6K jLixTlisVP1Fk0I g5eujHozZ/D UWadeLplGybws5y mM A07dIeFTHVQ6 QOVVRh5zmC73Sm3aljbVH7embqqWSnB36 bTnL9cKkMw=</la exi >
compu a ional model
(op ional) uning o pa ame e s (op ional) di ec op imiza ion
Figu e 1: A su oga e-based op imiza ion p ocess o a eal-wo ld p ocess wi h he di e en objec i e unc ion
laye s and ou lined inpu s and ou pu s; he complexi y o he objec i e unc ions is isible by he dec easing size
o hei boxes. The ull g ey a ows illus a e he app oxima ion and e i ica ion pa hs, he yellow dashed and ed
do ed a ows indica e he su oga e-based op imiza ion and op ional di ec op imiza ion. The blue dashed a ows
show he op ional pa ame e uning o he op imiza ion algo i hm o su oga e modeling p ocess.
6
ined. The de ined laye s can be ans e ed o di e en compu a ional p oblems wi h expen-
si e unc ion e alua ions, such as complex algo i hms o machine lea ning asks. Each laye
can be he a ge o an op imiza ion o used o e ie e in o ma ion o guide he op imiza ion
p ocess. Figu e 1 illus a es he di e en laye s o objec i e unc ions and he su oga e-based
op imiza ion p ocess o eal-wo ld p oblems. In his case, he objec i e unc ion laye s, om
he bo om up, a e:
L1 The eal-wo ld applica ion 1(x), gi en by he physical p ocess i sel o a physical model.
Di ec op imiza ion is o en expensi e o e en impossible, due o e alua ions in ol ing
esou ce demanding p o o ype building o e en haza dous expe imen s.
L2 The compu a ional model 2(x), gi en by a simula ion o he physical p ocess o a complex
compu a ional model , e.g., a compu a ional luid dynamics model o s uc u al dynamics
model. A single compu a ion may ake minu es, hou s, o e en weeks o compu e.
L3 The su oga e s(x), gi en by a da a-d i en eg ession model. The accu acy hea ily depends
on he unde lying su oga e ype and numbe o a ailable in o ma ion (i.e., unc ion e al-
ua ions). The op imiza ion is, compa ed o he o he laye s, ypically cheap. Su oga es a e
cons uc ed ei he o he eal-wo ld applica ion 1(x)o he compu a ional model 2(x).
Fu he mo e, he su oga e-based op imiza ion cycle includes he op imiza ion p ocess i sel ,
which is gi en by an adequa e op imiza ion algo i hm o he selec ed objec i e unc ion laye .
No su oga e-based op imiza ion is pe o med, i he op imiza ion is di ec ly applied o 1(x)
o 2(x). The su oga e-based op imiza ion uses 1(x)o 2(x) o e i ica ion o p omising
solu ion candida es. Mo eo e , he con ol pa ame e s o he op imiza ion algo i hm o e en
he comple e op imiza ion cycle including he su oga e modeling p ocess can be uned (Ba z-
Beiels ein e al., 2005).
Each laye imposes di e en e alua ion cos s and solu ion accu acies: he eal-wo ld p oblem
is he mos expensi e o e alua e, while he su oga e is he cheapes o e alua e. The main
bene i o using su oga es is hus he educ ion o needed expensi e unc ion e alua ions on
he objec i e unc ion 1(x)o 2(x)du ing he op imiza ion. The s udies by Loshchilo , Schoe-
naue , and Sebag (2012), Ma sden, Wang, Dennis J , and Moin (2004), Ong, Nai , Keane, and
Wong (2005) and Won and Ray (2004) ea u e benchma k compa isons o su oga e-based op-
imiza ion. Ne e heless, he model cons uc ion and upda ing o he su oga es also equi es
compu a ional esou ces, as well e alua ions o e i ica ion on he mo e expensi e unc ion
laye s. An ad an age o su oga e-based op imiza ion is he a ailabili y o he su oga e model,
which can be u ilized o gain u he global insigh in o he p oblem, which is pa icula ly alu-
able o black-box p oblems. Fo ins ance, he su oga e can be u ilized o iden i y impo an
decision a iables o isualize he na u e o he p oblem, i.e., he i ness landscape.
A common op imiza ion p ocess using su oga es is ou lined by he ollowing s eps:
1. Sampling he objec i e unc ion a kposi ions wi h yi= 1(xi)o yi= 2(xi),1≤i≤k
o gene a e a se o obse a ions D ={(xi,yi),1≤i≤k}. The sampling design plan is
commonly selec ed acco ding o he su oga e.
2. Selec ing a sui able su oga e. The selec ion o he co ec su oga e ype can be a com-
pu a ional demanding s ep in he op imiza ion p ocess, as o en no p io in o ma ion
indica ing he bes ype is a ailable. Common ypes o su oga es will be p esen ed in
Sec ion 7.
7
pa icle swa m op imiza ion (Kennedy and Ebe ha , 1995; Shi and Ebe ha , 1998) uses he
mo emen o bi d locks as a ole model; an colony op imiza ion (Do igo, Bi a a i, and S u -
zle, 2006) mimics, as he name sugges s, he ingenious pa h inding and ood sea ch p inciples
o an popula ions. These wo examples indica e ha analogies a e use ul o inspi e de elope s
o c ea e new sea ch s a egies. They a e also help ul o explain he beha io o hese sea ch
algo i hms, which makes hem aluable o be used in a comp ehensible axonomy.
3.3 The Fi e Elemen s o Algo i hm Design
Any mode n op imiza ion algo i hm, as de ined in Sec ion 2.2, can be educed o he ou key
sea ch s a egy elemen s Ini ializa ion,Va ia ion,E alua ion and Selec ion. All hese key elemen s
a e con olled by a i h elemen : he con ol pa ame e s o he di e en unc ions and ope a o s
in each elemen . Algo i hm 3.1 displays he key elemen s and he abs ac ed undamen al
s uc u e o op imiza ion algo i hms (Ba z-Beiels ein and Zae e e , 2017). These s uc u e and
elemen s can be mapped o any mode n op imiza ion algo i hm. E en i he sea ch s a egy is
inhe en ly di e en o elemen s do no ollow he illus a ed o de o appea mul iple imes
pe i e a ion.
Algo i hm 3.1: Gene al Op imiza ion Algo i hm
1se ini ial con ol pa ame e s
2begin
3 = 0
4ini ialize candida e(s)
5e alua e ini ial candida e(s)
6while no e mina ion-condi ion do
7 = + 1
8 a ia e solu ions o ge new candida e(s)
9e alua e new candida e(s)
10 selec solu ion(s) o nex i e a ion
11 op ional: upda e con ol pa ame e s
12 end
13 end
The ini ializa ion o he sea ch de ines s a ing loca ions o a schema o he ini ial candida e
solu ions. Two common s a egies exis :
1. I he e is no a p io i knowledge abou he p oblem and i s sea ch space, he bes op ion
is o use s a egically andomized s a ing poin s. Pa icula ly in e es ing o su oga e-
based op imiza ion a e sys ema ic ini ializa ion schemes by me hods om he ield o
design o expe imen s.
2. I domain knowledge o o he a p io i in o ma ion is a ailable, such as in o ma ion om
he p ocess o da a om p e ious op imiza ion uns, he algo i hm should be ini ialized
u ilizing his in o ma ion o ull ex end, e.g., by using a selec ion o hese solu ions, such
as hese wi h bes i ness. In su oga e-based op imiza ion a ailable da a can be used o
he ini ial modeling.
14

The ini ial candida es ha e a la ge impac on he balance be ween explo a ion and exploi a-
ion. Space- illing designs wi h la ge amoun s o andom candida es o sophis ica ed design
o expe imen s me hods will lead o a ini ial explo a ion o he sea ch space. S a ing wi h a
single candida e will p esumably lead o an exploi a ion o he neighbo hood o he selec ed
candida e loca ion. Hence, algo i hms using he i s scheme a e in gene al mo e obus , while
he la e a e sensi i e o he selec ion o he s a ing candida e, pa icula in mul i-modal land-
scapes. The obus ness can be u he inc eased by mul i-s a s a egies, which a e pa icula
common o single-candida e algo i hms, and also equen ly ecommended o popula ion-
based algo i hms (Hansen, Auge , Ros, Finck, and Pošík, 2010a).
The a ia ion du ing he sea ch p ocess de ines he me hods o gene a ing new candida es,
wi h special ega d on how a ailable o ob ained in o ma ion abou he objec i e unc ion is
used. A s anda d app oach is he a ia ion o exis ing obse a ions, as i u ilizes, and o a
ce ain ex end p ese es, he in o ma ion o p e ious i e a ions. E en he simples wande e
class algo i hms (Sec 5.1), which do no equi e any global in o ma ion o s o ed knowledge o
o me i e a ions, u ilize he las ob ained solu ion o gene a e new candida e(s). Sophis ica ed
algo i hms gene a e new candida es on he basis o exploi ed and s o ed global knowledge
abou he objec i e unc ion and i ness landscape. This can be conduc ed explici ly by ei he
keeping an a chi e o all a ailable o selec ed obse a ions, o implici ly by using dis ibu-
ion o da a models o a ailable obse a ions. Ano he op ion o gene a e new candida es is
combining in o ma ion o mul iple candida es by dedica ed unc ions o ope a o s, pa icula
p esen in he guide class (Sec 6.1). The exac ope a o s o gene a ion and a ia ion o can-
dida e solu ions a e a ious and a key aspec o keeping he balance be ween explo a ion and
exploi a ion in a sea ch s a egy.
The e alua ion de ines how he i ness o he candida es is compu ed and which objec i e unc-
ion is u ilized. The e alua ion is he key aspec o any algo i hm, as i de ines he basis o any
in o ma ion gain and has a huge in luence on he sea ch s a egy. Fo black-box p oblems, he
e alua ion o candida es is he only op ion o exploi any p oblem in o ma ion. We di e en i-
a e be ween a di ec e alua ion o he objec i e and indi ec e alua ion by using he p edic ed
i ness p o ided by a su oga e. How e alua ion is pe o med depends mainly on he unde ly-
ing p oblem and is la gely in luenced by he design o he objec i e unc ion. Impo an aspec s
in eal wo ld p oblems a e noise,cons ain s and mul iple objec i es.
While mos compu e expe imen s can be seen as de e minis ic, i.e., i e a ions using he same
alue se o he associa ed decision a iables should deli e he same esul s, eal-wo ld p ob-
lems a e o en non-de e minis ic. They include non-obse able dis u bance a iables and s ochas-
ic noise. Typical noise handling echniques include mul iple e alua ions o solu ions o educe
he s anda d de ia ion and special sampling echniques. The in e es ed eade can ind a su -
ey on noise handling by A nold and Beye (2003).
Mo eo e , many eal-wo ld p oblems equen ly include di e en cons ain s, which need o
be conside ed du ing he op imiza ion p ocess. Cons ain handling echniques can be di ec ly
pa o he op imiza ion algo i hm, bu mos algo i hms a e designed o minimize he objec i e
unc ion and cons ain handling is added on op. Thus, i is o en in eg a ed by adjus ing he
i ness, e.g., by penal y e ms. Di e en echniques o cons ain handling a e discussed by
Coello (2002) and A nold and Hansen (2012).
The e alua ion o mul iple objec i es can include se e al co ela ed objec i e unc ions and
usually deli e s a se o non-domina ed solu ions, a so-called Pa e o-se (Naujoks, Beume, and
Emme ich, 2005). In his case, a so-called decision make is u ilized o compu e he i ness o a
15
solu ion and selec solu ions om he Pa e o-se (Fonseca, Fleming, e al., 1993).
The selec ion de ines he p inciple o choosing he solu ions which will be used in he nex
i e a ion. We use he e m selec ion, which has i ’s o igins in e olu iona y compu a ion. Beside
he simples s a egy o choosing he solu ion(s) wi h he bes i ness, ad anced selec ion s a e-
gies ha e eme ged, which a e pa icula p esen in me aheu is ics (Boussaïd e al., 2013). These
selec ion s a egy a e pa icula common in algo i hms wi h se e al candida es pe a ia ion
s ep, hus he mos sophis ica ed selec ion me hods we e in oduced in he scope o e olu-
iona y compu a ion (Eiben and Smi h, 2015). A common s a egy is based on ela i e i ness
compa isons, so-called anked selec ion. De ailed examples o selec ion s a egies a e gi en in
Sec ion 6.2.
Con ol pa ame e s de e mine how he sea ch can be adap ed and imp o ed by con olling
he abo e men ioned key elemen s. We dis inguish be ween in e nal and ex e nal pa ame e s:
Ex e nal pa ame e s, also known as o line pa ame e s, can be adjus ed by he use and need
o be se a p io i o he op imiza ion un. Typical ex e nal pa ame e s include he numbe
o candida es and se ings in luencing he abo e men ioned key elemen s. Beside common
heo y-based de aul s (Schwe el, 1993), hey a e usually se by ei he u ilizing a ailable domain
knowledge, ex ensi e a p io i benchma k expe imen s (Gämpe le, Mülle , and Koumou sakos,
2002), o educa ed guessing. Sophis ica ed uning me hods we e de eloped o exploi good
pa ame e se ings in an au oma ed ashion. Well known examples a e sequen ial pa ame e
uning (Ba z-Beiels ein e al., 2005), i e a ed acing o au oma ic algo i hm uning (López-
Ibáñez, Dubois-Lacos e, Cáce es, Bi a a i, and S ü zle, 2016), bonesa (Smi and Eiben, 2011) o
SMAC (Hu e , Hoos, and Ley on-B own, 2011).
In di e ence, in e nal pa ame e s a e no mean o be changed by he use . They a e ei he
ixed o a ce ain alue, which is usually based on physical cons an s o ex ensi e es ing by
he au ho s o he algo i hm, o a e sel -adap i e. Sel -adap ion o online con ol changes he
pa ame e s online du ing he sea ch p ocess on basis o he ga he ed knowledge o exploi ed
p oblem in o ma ion (Eiben e al., 1999) wi hou use in luence. Algo i hms using sel -adap i e
schemes hus end o gain ou s anding gene aliza ion abili ies and a e especially in e es ing
o black-box p oblems, whe e no in o ma ion abou he objec i e unc ion p ope ies is a ail-
able (Hansen, Mülle , and Koumou sakos, 2003). In gene al, he se ings o algo i hm con ol
pa ame e s di ec ly a ec he balance be ween explo a ion and exploi a ion du ing he sea ch
and a e c ucial o he pe o mance.
3.4 Fea u es o an In ui i e Taxonomy
Taxonomies a e o en based on he subjec i e au ho ’s expe ience and o me de ini ions, as
well as mo e impa ial ea u es and simila i ies. P io o es ablishing ou new axonomy, we
de ined a se i e essen ial classi ica ion ea u es (CF), which a e based on o me a ailable ax-
onomies as well as signi ican ea u es and simila i ies o CGO algo i hms. They a e in ended
o gi e a good unde s anding on how we sepa a ed ou classes o c ea e a dis inc axonomy,
which s ill emains comp ehensible and in ui i e. P io o each class desc ip ion we will ou -
line he hei ea u es in a ex box as exempli ied in Figu e 3. We will e e o ou classi ica ion
ea u es as CF-I o CF-V:
16
CF I) Use o In o ma ion:
The in o ma ion ea u e has ou possible ca ego ies: The i s ca ego y is memo yless. The
e m desc ibes algo i hms, which only use he a ailable in o ma ion o he p io i e a ion
(o ini ializa ion).
The second is explici memo y. I de ines hose algo i hms, which in o ma ion o p io
i e a ions in a di ec ashion, e.g., by main aining an a chi e o all obse a ions.
Thi d is implici memo y; hese algo i hms combine in o ma ion o se e al i e a ions and
solu ions by ope a o s, unc ions o models.
Finally, algo i hms which equi e a p io i in o ma ion abou he objec i e unc ion, such
as he alue o he op imum.
CF II) Candida e E alua ion:
The candida e e alua ion ea u e de ines i he objec i e unc ion alue is only di ec ly
calcula ed o also indi ec ly app oxima ed. The app oxima ion o he i ness du ing he
candida e a ia ion phase can g ea ly lowe he necessa y amoun o objec i e unc ion
e alua ions, bu no op imiza ion p ocess can be eliable and success ul wi hou e i ica-
ion o hese candida es wi h he objec i e unc ion
CF III) Type o Candida e:
This ea u e e e s o he numbe and ype o candida e solu ions used in he a ia ion and
main ained in each i e a ion. I has h ee ca ego ies:
The i s is single and implies ha he a ia ion is based on a single candida e solu ion.
Mo eo e , hese algo i hms main ain only a single solu ion o hei nex i e a ion.
The second ype is popula ion, whe e in each i e a ion he a ia ion is based on se e al
candida e solu ions and mo eo e , se e al solu ions a e s o ed o he nex i e a ion.
The mos sophis ica ed ype a e model-based algo i hms, which u ilize a candida e dis-
ibu ion model o he a ia ion, which is s o ed and adap ed in each i e a ion. The
candida e e alua ion is no a ec ed and emains di ec .
CF IV) Region o Sea ch:
This ea u e desc ibes he e ec i e sea ch egion o an algo i hm.
Local algo i hms ha e no ope a o s o unc ions o explo a ion and a e hus no capable
o escaping he so-called egion o a ac ion o an op imum.
Global algo i hms ha e he abili y o ind op ima in mul i-modal landscapes by in oduc-
ing ope a o s o unc ions o balance explo a ion and exploi a ion.
CF V) P oblem P ope ies:
This ea u e is a collec ion o objec i e unc ion p ope ies. Algo i hms which a e e icien
in sol ing a p oblem wi h he gi en p ope y a e assigned wi h he ela ed ea u e.
•Domain Knowledge: This ea u e desc ibes p oblems wi h known unc ion p ope -
ies, such as a ma hema ical p oblem o mula ion o in o ma ion abou he numbe
and objec i e unc ion alue o op ima. This knowledge can be exploi ed and used
o a e icien o exac sea ch p ocess.
•Unimodal: This e m desc ibes objec i e unc ions wi h a single op imum in a linea
o con ex sea ch space.
17
•Mul imodal: P oblems a e called mul imodal when hey ha e se e al local and/o
global op ima.
•Black-Box: P oblems a e called black-box i hey do no p o ide any domain knowl-
edge and all in o ma ion needs o be ga he ed by objec i e unc ion e alua ions.
Many eal-wo ld p oblems can be associa ed wi h his cha ac e is ic.
•Discon inui ies: Func ions who ha e jumps in hei objec i e unc ion alues a e
discon inuous and no di e en iable.
•Noisy: Noisy objec i e unc ion do no e u n de e minis ic unc ion alues. Mul-
iple e alua ions o he same candida e on a noisy unc ion can lead o di e en
esul s.
•Mul i-Objec i e: Mul i-Objec i e p oblems ha e se e al (compe ing) objec i es.
•Expensi e: Expensi e p oblems ha e a high cos o each unc ion e alua ion in
e ms o ei he physical esou ces o compu a ion ime.
Classi ica ion Fea u es:
I Use o In o ma ion: a p io i, memo yless, explici /implici memo y
II Candida e E alua ion: di ec , indi ec
III Type o Candida e: single, popula ion, model
IV Sea ch Space: local, global
V P oblem P ope ies: unimodal, mul imodal, black-box, ...
Figu e 3: Fundamen al ea u es o algo i hm classi ica ion
Fo an o e iew o all classes and he connec ed classi ica ion ea u es CF I-IV, we u ilize a
decision ee, illus a ed in Figu e 4. In his igu e, we u ilized he ea u es I o IV as classi i-
ca ion nodes which conclude in ou new axonomy wi h he in oduced classes exac ,wande e ,
guide and ca og aphe . Mo eo e , he associa ed p oblem p ope ies a e displayed. The igu e is
in ended o p o ide a as o e iew on how he new axonomy wo ks. New algo i hms could
be easily included by assigning he abo e lis ed ea u es and hen using he illus a ed decision
ee. We know ha no all algo i hms will i in o ou axonomy, as hey ha e p ope y com-
bina ions, which a e no displayed in ou scheme. They may also belong o he hyb id class,
which ep esen combina ions o me hods om he displayed classes. The hyb id class is no
shown in he igu e, as i has no dis inc p ope ies.
18
Wande e
IV: Region o
Sea ch
global
Global Me hods
Simula ed
Annealing
local
Local Me hods
Line Sea ch,
Quasi-New on,
I e a i e Hill Climbe
unimodal, mul imodal
memo yless
no memo y o p io
obse a ions
Exac Guide Ca og aphe
I: Use o
In o ma ion
Exac Me hods
G id Sea ch
B anch and Bound
explici o implici
memo y o p io
obse a ions
II: Candida e
E alua ion
di ec e alua ion;
u ilizing objec i e
unc ion
III: Type o
Candida e
indi ec e alua ion;
u ilizing su oga e o
objec i e unc ion
Su oga e-based
Efficien Global
Op imiza ion,
Bayesian
Op imiza ion,
Sequen ial Pa ame e
Op imiza ion
model
Model-based
Es ima ion o
Dis ibu ion
Algo i hms,
Co a iance Ma ix
Adap ion- E olu ion
S a egy, An
Colony Op imiza ion
single
Single-solu ion
Tabu-Sea ch,
Va iable
Neighbo hood
Sea ch
popula ion
Popula ion-based
E olu iona y
Algo i hms, Pa icle
Swa m Op imiza ion
domain
knowledge
black-box, noisy, discon inui ies, mul i-objec i e expensi e
Objec i e
Func ion
Class Subclass
a p io i
in o ma ion abou
p oblem p ope ies
Algo i hm complexi y inc eases, objec i e unc ion cha ac e is ics add up
Figu e 4: Algo i hmic classi ica ion including he classes exac , wande e , guide, and ca og aphe . The de ined
classi ica ion ea u es I o IV a e used as nodes o a decision ee wi h he subclasses and ela ed example algo-
i hms as he inal lea es. Below he subclass he co esponding main classes and he associa ed objec i e unc ion
cha ac e is ics a e illus a ed. The objec i e unc ion cha ac e is ics a e displayed bo om-up, i.e., he displayed
cha ac e is ics a e adding up and ge ing mo e sophis ica ed om le o igh .
19

4 The Exac Class
Exac Class:
I Use o In o ma ion: a p io i
II Candida e E alua ion: di ec
III Type o Candida e: single
IV Sea ch Space: global
V P oblem P ope ies: unimodal,mul imodal,domain knowledge
The gene al desc ip ion o exac algo i hms was p esen ed in Sec ion 2.3. Fo an e icien sea ch,
exac algo i hms need a p io i in o ma ion abou he objec i e unc ion. The e o e, hey a e
only sui able o a limi ed class o p oblems whe e his in o ma ion is a ailable. Mo eo e , he
applica ion o exac algo i hms is always a adeo decision be ween compu a ion ime and
p ecision.
Fo example: Gi en a nonde e minis ic polynomial (NP)-ha d p oblem, usually no exac algo-
i hm exis s ha is able o ind he bes solu ion in polynomial ime. The a eling salesman
p oblem is a common combina o ial NP-ha d p oblem, whe e he goal is o ind he minimal
leng h ou h ough a ixed numbe o ci ies. An exac algo i hm could sol e his p oblem by
calcula ing e e y possible ou and selec ing he bes . Despi e p o iding he bes solu ions, his
s a egy would use a lo o compu a ion ime. Pa icula i he numbe o ci ies ge s la ge, his
p oblem canno be sol ed exac in easonable ime.
I should be no ed ha e en i exac algo i hms a e no e icien o NP-ha d and black-box
p oblems, hey a e commonly pa o su oga e-based op imiza ion amewo ks, because he
models p o ide he equi ed in o ma ion o an exac sea ch o a e compu a ionally e y cheap
o e alua e (see Sec ion 2.4 and Figu e 1). Fo example, e icien global op imiza ion (Sec ion 7.2.1
)and bayesian op imiza ion (Sec ion 7.2.2) can use exac algo i hms.
Two common me hods om he amily o exac me hods a e g id sea ch and b anch-and-bound.
G id sea ch combines a mul i-s a local op imiza ion wi h an inc easingly sub le sampling
g id o s a ing poin s. B anch-and-bound op imiza ion is conduc ed by spli ing he o igi-
nal p oblem ecu si ely in o subp oblems wi h he goal o excluding o sol ing hem, un il
i is gua an eed ha no subp oblem can lead o a be e solu ion. I is known as b anch-and-
bound (Lawle and Wood, 1966) as lowe bounds on he objec i e unc ion a e compu ed. Mo e
complex exac sea ch me hods combine b anching and local op imiza ion, Lipschi zian op i-
miza ion, con exi y o in e al analysis (Neumaie , 2004; Floudas, 2013; Hansen and Wals e ,
2003; Ho s and Tuy, 2013).
Example 4.1 (Di iding Rec angles).An example o an exac algo i hm is di iding ec angles
(DIRECT), ini ially p oposed by Jones, Pe unen, and S uckman (1993) as a modi ica ion o
Lipschi zian op imiza ion. While assuming ha he objec i e unc ion is Lipschi z con inuous,
he algo i hm does no need a speci ica ion o he Lipschi z cons an , as i is sampled du ing
he op imiza ion un. The algo i hm uses hype cubes o di ide he sea ch space. The cen e
cio each hype cube is sampled and gi en a i ness, based on he objec i e unc ion alue and
he size o he associa ed hype cube. Based on his i ness, he hype cube which is mos likely
o inhe i he op imum is selec ed and u he di ided in o smalle hype cubes. Then hei
i ness is sampled and he p ocess epea s, un il a s opping c i e ion is me o he algo i hm has
con e ged.
20
5 The Wande e Class
The wande e class encompasses algo i hms wi h non-complex sea ch s a egies which gene -
a e and main ain a single candida e pe i e a ion. New candida es a e gene a ed in he icini y
o he cu en solu ion by a s ochas ic p ocess which is independen o p e ious sea ch s eps.
The consecu i e candida es desc ibe a ajec o y in he sea ch space ha o ms in he ideal case
a di ec line o he op imum. In hei sea ch, hey only use he local objec i e unc ion in o ma-
ion abou he p io solu ion. Thus, hese algo i hms do no use global in o ma ion abou he
p oblem in he a ia ion o selec ion s eps.
Analogy 1 (The Wande e ).
The in ui i e desc ip ion o a wande e is a single indi idual who wande s h ough he land-
scape o ind he mos a ac i e place in a gi en a ea. Du ing i s sea ch i only u ilizes local
in o ma ion abou i s cu en posi ion o ind he bes di ec ion. I he goal o his indi idual is
o ind he highes moun ain, i will likely ollow he ascending way, because i di ec ly sa is ies
he cu en objec i e. I does no memo ize ga he ed in o ma ion, so ha he e is a chance ha
i will ci cle a posi ion, e isi a place o ge s comple ely los .
We sepa a e be ween he local and global subclass o he wande e : While he local wande e
keeps a g eedy selec ion, he global wande e also allows he accep ance o non-imp o ing can-
dida es.
5.1 Local Wande e
Local Wande e :
I Use o In o ma ion: memo yless
II Candida e E alua ion: di ec
III Type o Candida e: single
IV Sea ch Space: local
V P oblem P ope ies: unimodal
The local wande e subclass consis s o basic local op imiza ion algo i hms, which include clas-
sical g adien -based algo i hms as well as de e minis ic o s ochas ic hill-climbing algo i hms.
These algo i hms a e designed o as con e gence o a local op imum si ua ed in a egion o
a ac ion A ⊆ S and ha e no explici s a egy o explo a ion.
G adien -based me hods, auch as quasi-New on Me hods (Shanno, 1970), di ec ly compu e o
app oxima e he g adien s o he objec i e unc ion o ind he s eepes di ec ion o he op i-
mum. Di ec -Sea ch me hods pe o m an i e a i e and g adien - ee sea ch by using a minimal
amoun o in o ma ion abou he objec i e unc ion. O e iews o di ec sea ch me hods we e
p esen ed by Lewis, To czon, and T osse (2000) and Kolda e al. (2003). Mo eo e , he (1+1)
e olu ion s a egy wi a basic selec ion ope a o ( u he explained in Sec ion 6.2) can be associ-
a ed wi h his class.
Example 5.1 (I e a ed S ochas ic Hill-Climbe ).The i e a ed s ochas ic hill-climbe (Michalewicz
and Fogel, 2013) is a ypical example o he local wande e subclass. I has a elemen a y al-
go i hm design, whe e he sea ch a ia ion is s ochas ic and he selec ion ypically g eedy. In
each i e a ion o he algo i hm a new candida e x is c ea ed by sampling om a p obabili y
dis ibu ion Da ound he p io solu ion x −1. The a iance o he dis ibu ion, which is o en
21
uni o m o Gaussian, de ines he so-called s ep size o he a ia ion and is he mos impo an
con ol pa ame e o his algo i hm. The g eedy selec ion wo ks as ollows: I he new candida e
x has a be e i ness alue han he p io solu ion x −1, i is accep ed as new solu ion and he
i e a ion is epea ed. I he i ness is no imp o ed, he p io solu ion is kep o he nex i e -
a ion. This g eedy selec ion scheme is based on a compa a i e, anking-based selec ion o he
candida es wi h no in luence o he absolu e di e ence in he objec i e unc ion alues. This
implies in a iance o linea ans o ma ions and scaling o he objec i e unc ion. As he basic
scheme includes no ope a o o escape om a local op imum o explo a ion, his algo i hm will
mos likely con e ge o a local op imum. The e mina ion condi ion is usually se o a numbe
o non-imp o ing i e a ions, a e which is assumed ha an op imum has been ound. The
gene alized design o he i e a ed hill-climbe is ou lined in Algo i hm 5.1.
Algo i hm 5.1: I e a ed Hill-Climbe
1begin
2 = 0
3ini ialize x wi h ( andom) candida e x∈ S
4e alua e x
5while no e mina ion-condi ion do
6 = + 1
7sample new candida e x om p obabili y dis ibu ion Da ound he cu en
solu ion x −1
8e alua e candida e x
9i new solu ion imp o es he i ness y <y −1 hen
10 accep new solu ion
11 else
12 keep old solu ion x =x −1
13 adjus a ia ion s ep size/ a iance o p obabili y dis ibu ion (op ional)
14 end
15 end

As he name implies, he local wande e is in he i s place sui able o unimodal unc ions o
o exploi local op ima. I can be applied o global op imiza ion o mul imodal landscapes i
an adequa e mul i-s a s a egy is used. These mul i-s a s a egies ypically demand a high
numbe o unc ion e alua ions and a e only easonable o be used o p oblems wi h ela i ely
cheap objec i e unc ions, such as in su oga e-based op imiza ion. Theo e ically, a local wan-
de e could es ablish a global sea ch by sampling he candida es om a dis ibu ion wi h a
dispe sion o e he comple e sea ch space, which is equi alen o a e y la ge s ep size o he
a ia ion. This s a egy would be ine icien as i leads o andom sampling o candida es wi h-
ou conside ing any local in o ma ion and di ec ion o sea ch. Thus, i is common o limi he
dispe sion o he p obabili y dis ibu ion o he icini y o he cu en solu ion, o o m a small
neighbo hood which is e y small compa ed o he comple e sea ch space. This leads o he
ou lined hill-climbing sea ch s a egy which pe o ms a ajec o y o small, i ness-imp o ing
s eps. The maximal s ep size is consequen ly an impo an con ol pa ame e , which is o en
designed as being adap i e in many e sions o he algo i hm (c . mu a ion ope a o in Sec ion
22
6.2). In gene al, local wande e s a e o en pa o sophis ica ed algo i hms as as con e ging
local sea ch s a egy.
5.2 Global Wande e
Global Wande e :
I Use o In o ma ion: memo yless
II Candida e E alua ion: di ec
III Type o Candida e: single
IV Sea ch Space: global
V P oblem P ope ies: mul imodal
The global wande e subclass encompasses algo i hms which implemen ope a o s o balance
explo a ion and exploi a ion in he selec ion p ocess o candida es. They di e om local wan-
de e s by hei explo a i e sea ch s a egies, which u he enable global op imiza ion. Simi-
la o he local e sion, he global wande e u ilizes a s ochas ic a ia ion o candida es in he
neighbo hood o he cu en solu ion wi hou conside ing s o ed o modeled global in o ma-
ion. Explo a ion is achie ed by in oducing ope a o s o unc ions which allow o expand he
sea ch space and escape he egion o a ac ion o a local op imum. The well-known simula ed
annealing (SANN) will be used o exempli ying his app oach.
Example 5.2 (Simula ed Annealing).Ki kpa ick e al. (1983) in oduced SANN as a sea ch p o-
cedu e o global combina o ial op imiza ion. I is known o be a signi ican con ibu ion o he
ield o me aheu is ic sea ch algo i hms. The basic sea ch p ocedu e o e e y SANN me hod
is inspi ed by annealing in me allu gy, whe e a ma e ial is i s hea ed, hen cooled. Following
his analogy, he mos impo an elemen o con ol he sea ch is an adap i e pa ame e : he
empe a u e (T). Analogous o he he modynamic ene gy in a hea ed ma e ial, i inc eases he
possible mo emen o he candida es.
The con inuous e sion (Go e, Fe ie , and Roge s, 1994; Sia y, Be hiau, Du din, and Haussy,
1997; Van G oenigen and S ein, 1998) o he SANN algo i hm basically ex ends he i e a ed
s ochas ic hill climbe . I includes a new elemen o allow global sea ch, he so-called accep ance
unc ion P(x −1,x , T). The accep ance unc ion is used du ing he selec ion and de e mines he
p obabili y o accep ing an in e io candida e as solu ion by u ilizing T as a pa ame e . This
dynamic selec ion allows o escape local op ima s eps by accep ing mo emen in he opposi e
di ec ion o imp o emen , which is he undamen al di e ence o a hill climbe . A common
example o an accep ance unc ion is he so-called Me opolis unc ion:
P(x −1,x , T) = min 1,exp − (x )− (x −1)
T (1)
The Me opolis unc ion always accep s i ness imp o ing s eps owa ds he minimum ( (x )−
(x −1)≤0) and mo eo e , has a p obabili y o accep ascending ( (x )− (x −1)>0) s eps
based on T. Highe T alues hus inc ease he p obabili y o accep an in e io candida e. In
con as o ank-based selec ion schemes, his unc ion is scale-based and u ilizes absolu e di -
e ences in he i ness, which ende s i sensi i e o linea ans o ma ions o he objec i e unc-
ion, e.g., a mul iplica ion by a scala .
A he end o each i e a ion, a so-called cooling ope a o Cadap s T. This ope a o can be used
o balance explo a ion and exploi a ion (Hende son, Jacobson, and Johnson, 2003). A common
23
and λo sp ing candida es acco dingly o hei anked i ness, whe eby he op µsolu ions a e
kep o he nex gene a ion. In he (µ, λ)-s a egy all pa en s µa e disca ded and he su i o s
a e selec ed only om he o sp ing λacco ding o hei anked i ness. This equi es o c ea e
λ≥µcandida es in he ecombina ion s ep o p e en ex inc ion o he popula ion. Fu he , in
age-based selec ion s a egies, each solu ion only su i es a de ined numbe o i e a ions be o e
i ge s disca ded. This is o en combined wi h i ness-based selec ion, as in he (µ, λ)-s a egy,
whe e each solu ion su i es only one i e a ion. Aging adds a handicap o old domina ing so-
lu ions which su i ed se e al gene a ions. This depic s an explo a i e s a egy o escape local
op ima and inc ease he di e si y o he popula ion. 
EAs a e e y lexible in hei implemen a ion and adap able by uning. They a e obus and
sui able o sol e a la ge class o p oblems, including mul imodal, mul i-objec i e, dynamic
and black-box p oblems, e en wi h noise o discon inui ies in he i ness landscape (Jin and
B anke, 2005; Ma le and A o a, 2004). Fu he , hey ha e success ully been applied o a la ge
numbe o di e en indus ial p oblems (Fleming and Pu shouse, 2002) bu ypically equi e
a ela i ely la ge numbe o unc ion e alua ions o con e ge. This makes hem no he i s
choice o expensi e p oblems whe e he numbe o e alua ions is s ongly limi ed.
The lexibili y and obus ness o EAs is caused by di e en mechanisms and s a egies o con-
olling he balance be ween explo a ion and exploi a ion. A good o e iew is p esen ed in
he su ey by ˇ
C epinšek, Liu, and Me nik (2013). The de ailed su ey classi ies he di e en
a ailable e olu iona y app oaches and p esen s an in ensi e discussion which mechanisms
in luence explo a ion and exploi a ion. Theo e ical aspec s o e olu iona y ope a o s a e dis-
cussed by Beye (2013). The pe o mance o EAs is in luenced by pa ame e uning and con ol,
e.g., he se ing o popula ion size, mu a ion s eng h and selec ion p obabili y. An ex ensi e
o e iew o he di e en on- and o line uning app oaches o pa ame e con ol in EAs was
published by Eiben e al. (1999).
Fu he common s a egies o con olling explo a ion and exploi a ion and mul imodal op i-
miza ion a e so-called niching s a egies, which u ilize sub-popula ions o main ain he di e -
si y o he popula ion, in es iga e se e al egions o he sea ch space in pa allel o conduc
de ined asks o explo ing and exploi ing (Shi and Bäck, 2005; Filipiak and Lipinski, 2014).
6.3 Model-based Guide
Model-based Guide
I Use o In o ma ion: implici memo y by he model/dis ibu ion
II Candida e E alua ion: di ec
III Type o Candida e: model/dis ibu ion
IV Sea ch Space: global
V P oblem P ope ies: mul imodal, black-box, noisy, discon inui ies, mul i-objec i e
The model-based guide class encompasses algo i hms, which explici ly u ilize ma hema ical o
s a is ical models. The dis inc ion o su oga e-based op imiza ion is based on he assump ion,
ha hese models do no depic di ec app oxima ions o he unde lying objec i e unc ion and
o no a ge o model he comple e i ness landscape. Ins ead, hey a e specialized local models
which a e used in he a ia ion . They a e no explici ly sea ched o op ima, al hough o en he
mean o he dis ibu ion is u ilized as p edic ed op imal solu ion.
30

A common class o model-based algo i hms a e es ima ion o dis ibu ion algo i hms (EDA) (La -
añaga and Lozano, 2001). They gene ally belong o he la ge ield o EAs (Sec ion 6.2). The
main di e ence o EAs is ha he a ia ion ope a o s, such as ecombina ion o mu a ion, a e
no di ec ly applied o he candida es, bu o dis ibu ion models. The dis ibu ion models a e
build using in o ma ion o p io e alua ed popula ions. Di e en dis ibu ion models can be
u ilized, e.g., Bayesian ne wo ks o mul i a ia e Gaussian dis ibu ions. Fu he common ex-
amples o model-based algo i hms a e he co a iance ma ix adap ion - e olu ion s a egy (CMA-
ES) (Hansen e al., 2003) and an colony op imiza ion o con inuous domains ACOR(Do igo e al.,
2006; Socha and Do igo, 2008). ACORis based on he social beha io o an colonies and hei
communica ion ia phe omones. An s sea ch o ood a ound hei nes in a andom manne
and lea e ails o phe omones on hei way o ma k and enable o he an s o ollow hese
ails. The idea o hese phe omone ails a e ans e ed o use special dis ibu ion models
and a ia ion ope a o s. As he gene al idea o model-based algo i hms is simila , we will ou -
line hei mechanisms on he example o a gene alized EDA and gi e u he insigh on he
pa icula i ies o ACORand he CMA-ES.
Example 6.3 (Es ima ion o Dis ibu ion Algo i hms).The gene al idea behind he dis ibu-
ion model is ha i is bene icial o lea n he s uc u al in o ma ion o he unde lying popula-
ion (La añaga and Lozano, 2001; Hauschild and Pelikan, 2011). The s uc u al in o ma ion
allows o acqui e quali y knowledge abou he dependences among he a iables. Mo eo e ,
his in o ma ion is used o gene a e new candida es and hus, o guide he sea ch o he op i-
mum. A gene al EDA is ou lined in Algo i hm 6.3.
Algo i hm 6.3: Es ima ion o Dis ibu ion Algo i hm
1begin
2 = 1
3ini ialize µ andom popula ion P ={x ,1:µ}⊆S
4e alua e µ andom candida es P
5while no e mina ion-condi ion do
6selec pa en s om popula ion P∗
⊆ P
7es ima e dis ibu ion pa ame e s o pa en s P∗
, such as mean, a iance and
co a iance
8mu a e o adap pa ame e s o dis ibu ion G
9sample dis ibu ion unc ion G o ge λo sp ing O={x∗
,1:λ}
10 e alua e o sp ing
11 selec su i o s om P ∪ O o nex gene a ion P +1
12 = + 1
13 end
14 end
The algo i hm s a s wi h popula ion o µsolu ions, ei he ini ialized a andom o based on
exposed p oblem knowledge. Simila o an EA, pa en s a e selec ed, ypically based on anking
selec ion. I is common o use he i es solu ions as pa en s (c . selec ion in EAs, Sec 6.2).
Ins ead o a di ec ecombina ion o mu a ion, he pa en s a e used o cons uc a dis ibu ion
model. Fo example, ypical pa ame e s o a mul i a ia e Gaussian dis ibu ion, such as mean,
a iance and co a iance o he selec ed pa en popula ion a e compu ed. In ACORGaussian
31
ke nel unc ions o each dimension o he decision a iable space a e cons uc ed. The dis i-
bu ion pa ame e s a e hen a ge o he a ia ion ope a o s. This a ia ion is he c ucial aspec
o he sea ch s a egy, simila o he ecombina ion and mu a ion in he basic EA. Fo exam-
ple, he CMA-ES adap s he pa ame e s du ing each i e a ion ollowing he his o y o p io
success ul i e a ions, he so-called e olu ion pa hs. These e olu ion pa hs a e basically exponen-
ially smoo hed sums o each dis ibu ion pa ame e o e he consecu i e p io i e a i e s eps
(c . adap i e mu a ion s eng h in Sec ion 6.2). They hus u ilize he in o ma ion o se e al
success ul sea ch s eps, which is in ended o quickly app oach an op imum.
In he nex s ep, a numbe o λo sp ing candida es a e gene a ed by sampling he adap ed
dis ibu ion. A e wa ds he candida es a e e alua ed and su i o s o he nex gene a ion
a e selec ed, again based on su i o selec ion ope a o s, as explained in (Sec ion 6.2). The
CMA-ES uses only ank-based selec ion schemes, which makes i insensi i e o scaling o he
objec i e unc ion. 
In gene al, model-based guides y o combine he bene i s o s a is ical models and hei ca-
pabili y o s o ing and p ocessing in o ma ion wi h popula ion-based sea ch ope a o s. They
a e high le el me aheu is ics and ad anced EAs which in ended o be lexible, obus and ap-
plicable o a la ge class o p oblems, pa icula hose wi h unknown unc ion p ope ies. This
makes hem e y success ul in popula black-box benchma ks (Hansen e al., 2010a). Fo ex-
ample, he design o he CMA-ES is seeks o make he algo i hm pe o mance obus and no
dependen on he objec i e unc ion o uning. The a ious con ol pa ame e s o he algo i hm
we e p e-de ined on basis o heo e ical aspec s and p ac ical benchma ks.
7 The Ca og aphe Class
Ca og aphe Class:
I Use o In o ma ion: implici /explici memo y by su oga e and solu ion a chi e
II Candida e E alua ion: indi ec
III Type o Candida e: single
IV Sea ch Space: global
V P oblem P ope ies: mul imodal, black-box, noisy, expensi e
Ca og aphe algo i hms di e om all o he de ined classes in hei ocus on acqui ing, ga he -
ing and u ilizing global in o ma ion abou he i ness landscape. They u ilize p io e alua ed
candida es and model he acqui ed in o ma ion o p edic he i ness o new candida es. These
models a e hen used o an e icien indi ec sea ch in which ypically a single new candida e
is p oposed in each i e a ion, ins ead o pe o ming mul iple, di ec and localized sea ch s eps.
Analogy 3 (The Ca og aphe ).
The in ui i e idea o he ca og aphe is a specialis who sys ema ically measu es a landscape
by aking samples o he heigh o c ea e a opological map. This map esembles he eal land-
scape wi h a gi en app oxima ion accu acy and is ypically exac a he sampled loca ions (i
he measu emen s a e wi hou a iance) and models he emaining landscape by eg ession.
I can hen be examined and u ilized by any o he indi idual, such as a wande e o guide, o
ind a desi ed loca ion. One could hink o a guide using a pape map o na iga ion sys em o
ind he place o in e es .
32
As illus a ed in Sec ion 2.4, he su oga es s(x)depic he maps o he i ness landscape o an
objec i e unc ion ( 1(x)o 2(x)) in an algo i hmic amewo k. In his Sec ion, we will i s
gi e a b ie in oduc ion o common su oga e models and hen ou line ypical ca og aphe
amewo ks and hei sea ch s a egies.
7.1 Su oga e Models
The su oga e is he co e elemen o any su oga e-based op imiza ion and essen ial o hei
pe o mance. A pe ec su oga e p o ides an excellen i o p io obse a ions, whils ideally
possessing supe io in e pola ion and ex apola ion abili ies. Howe e , he la ge numbe o
a ailable su oga e models all ha e signi ican di e ing cha ac e is ics, ad an ages and disad-
an ages. Model selec ion is hus a complica ed and di icul ask. I no domain knowledge is
a ailable, such as in eal black box op imiza ion, i is o en ine i able o es di e en su oga es
o hei applicabili y.
Su oga es a e buil on basis o p io obse a ions, which p o ide in o ma ion abou he i -
ness landscape o he p oblem. Thus, he ini ial candida es a e commonly selec ed ollowing
di e en in o ma ion c i e ia and sui able expe imen al design. Fo example, linea eg ession
models can be build wi h ac o ial designs, while Gaussian p ocess models a e bes coupled
wi h space- illing designs, such as La in hype cube sampling (Mon gome y, Mon gome y, and
Mon gome y, 1984; Sacks e al., 1989).
Common models a e: linea , quad a ic o polynomial eg ession, Gaussian p ocesses (also
known as K iging) (Sacks e al., 1989; Fo es e , Sobes e , and Keane, 2008), eg ession ees
(B eiman, F iedman, S one, and Olshen, 1984), a i icial neu al ne wo ks and adial basis unc-
ion ne wo ks (Haykin, 2004; Ho nik, S inchcombe, and Whi e, 1989) including deep lea n-
ing ne wo ks (Collobe and Wes on, 2008; Hin on, Deng, Yu, Dahl, Mohamed, Jai ly, Senio ,
Vanhoucke, Nguyen, Saina h, e al., 2012; Hin on, Osinde o, and Teh, 2006) and symbolic e-
g ession models (Augus o and Ba bosa, 2000; Flasch, Me smann, and Ba z-Beiels ein, 2010;
McKay, Willis, and Ba on, 1995), which a e usually op imized by gene ic p og amming (Koza,
1992).
Fu he , a lo o e o in cu en s udies is o esea ch he bene i s o model ensembles, which
combine se e al dis inc models (Goel, Ha ka, Shyy, and Queipo, 2007; Mülle and Shoemake ,
2014; F iese, Ba z-Beiels ein, and Emme ich, 2016). The goal is o c ea e a sophis ica ed p e-
dic o ha su passes he pe o mance o a single model. A well-known example a e andom
o es s (F eund and Schapi e, 1997), which use bagging o i a la ge numbe o decision ees
(B eiman, 2001). We ega d ensemble modeling as s a e-o - he-a o cu en esea ch, as hey
a e able o combine he ad an ages o di e en models o gene a e ou s anding esul s in bo h
classi ica ion and eg ession. The d awback o hese ensemble me hodologies is ha hey a e
compu a ional expensi e and pose a demanding p oblem in ega d o e icien model selec ion,
e alua ion and combina ion.
7.2 Ca og aphe Algo i hms
Ca og aphe algo i hms a e su oga e-based op imiza ion me hodologies, which explici ly
use a su oga e in hei op imiza ion cycle, ollowing he gene al p inciple ou lined in Sec ion
2.4. They a e ei he ixed algo i hms designed a ound a ce ain model, such as K iging in he
well-known e icien global op imiza ion (EGO) (Jones e al., 1998), o amewo ks wi h a choice o
33
possible su oga es and op imiza ion me hods. We p esen wo common amewo ks and dis-
cuss hei pa icula i ies: gene al bayesian op imiza ion (Mockus, 1974) and sequen ial pa ame e
op imiza ion (Ba z-Beiels ein e al., 2005; Ba z-Beiels ein, 2010).
Fo es e and Keane (2009) and Ba z-Beiels ein and Zae e e (2017) gi e o e iews o su oga e-
based op imiza ion, di e en su oga e models and in ill c i e ia. Mo eo e , hey ma ch su o-
ga es o p oblem classes and gi e hin s abou hei indi idual applicabili y. In gene al, he se-
lec ion o an adequa e model, expe imen al design and op imize equi es bo h domain knowl-
edge and expe ise. We will ocus on he abo e-men ioned amewo ks as hey deli e a good,
ye no comple e, iew o he su oga e-based sea ch s a egy.
7.2.1 E icien Global Op imiza ion
EGO (Jones e al., 1998) was mo i a ed by he u ge o de elop a me hodology o op imize
expensi e black-box unc ions. I u ilizes K iging su oga es and mo i a es he use expec ed
imp o emen as in ill c i e ia. In gene al, he algo i hm consis s o wo phases: i s , he ini ial-
iza ion by La in hype cube sampling and he cons uc ion o a K iging su oga e; second, he
i e a i e imp o emen o he bes solu ion u ilizing he su oga e. Be o e s a ing he second
phase, Jones e al. (1998) sugges s o analyze he su oga e model i . I he i is no sa is ac o y,
i can be ied o imp o e i by a uning o he model pa ame e s o ans o ma ion o he da a.
Algo i hm 7.1: E icien Global Op imiza ion
1begin
// phase 1: ini ial su oga e building
2 = 1
3ini ialize kcandida e solu ions {x1:k}by la in hype cube sampling
4e alua e hem on objec i e unc ion yi= 1(xi)o yi= 2(xi),1≤i≤k;
5build ini ial K iging su oga e s wi h ini ial obse a ions D ={(xi,yi),1≤i≤k}
6analyze and imp o e model i (op ional)
// phase 2: use and upda e su oga e
7while no e mina ion-condi ion do
8i >1 hen
9upda e he K iging su oga e s wi h he se o all obse a ions D
10 end
11 calcula e expec ed imp o emen in ill c i e ia on su oga e s
12 op imize EI o maximum by b anch and bound; use op imum as candida e x
13 e alua e x on he objec i e unc ion y = 1(x )o y = 2(x )
14 add new solu ions o he se o all obse a ions D +1 ={D ,(x ,y )}
15 = + 1
16 end
17 end
The second phase s a s he i e a i e op imiza ion p ocess, as desc ibed in Sec ion 2.4. Du -
ing he a ia ion, a new candida e is sea ched by op imizing he expec ed imp o emen in ill
c i e ia o he K iging su oga e. Fo his op imiza ion, he exac b anch and bound me hod is
used (c . exac class, Sec ion 4). Expec ed imp o emen is mo i a ed as in ill c i e ia, because i
gua an ees a balance o explo a ion and exploi a ion by u ilizing bo h he p edic ed bes mean
34
alue o he model, as well as he model unce ain y. An example o he comple e me hod-
ology is ou lined in Algo i hm 7.1. The sea ch s a egy o EGO is a undamen al example o
mos su oga e-based op imiza ion which a e applicable o expensi e op imiza ion p oblems.
Howe e , we sugges o use ad anced amewo ks based on his base e sion o EGO. These
amewo ks a e mo e lexible and applicable o a la ge class o p oblem.
7.2.2 Bayesian Op imiza ion
The e m Bayesian op imiza ion (BO) was in oduced by Mockus (1974, 1994, 2012) and de-
sc ibes no a single, bu a scheme o algo i hms, which we ega d as su oga e-based, pa ic-
ula hese based on Gaussian p ocesses. While he gene al BO scheme hus emains simila
o he ou lined algo i hm in Sec ion 2.4, BO di e s in he unde lying e minology: In BO, he
uses selec s and ini ial, so-called p io dis ibu ion, which should suppo he a p io i belie s
abou he unde lying unknown objec i e unc ion. Gaussian dis ibu ions a e sugges ed and a
common choice. Algo i hm 7.2 displays a gene al BO algo i hm.
Algo i hm 7.2: Bayesian Op imiza ion
1begin
// phase 1: ini ializa ion
2 = 1
3selec p io dis ibu ion s −1(x)
4ini ialize kcandida e solu ions {x1:k}
5e alua e hem on objec i e unc ion yi= 1(xi)o yi= 2(xi),1≤i≤k
6upda e p io wi h D ={(xi,yi),1≤i≤k} o ge ini ial pos e io dis ibu ion s (x);
// phase 2: op imiza ion
7while no e mina ion-condi ion do
8i >1 hen
9upda e pos e io dis ibu ion s wi h he se o all obse a ions D
10 end
11 calcula e in ill unc ion on pos e io dis ibu ion s (x)
12 op imize in ill c i e ion; use op imum as candida e x
13 e alua e x on he objec i e unc ion y = 1(x )o y = 2(x )
14 add new solu ions o he se o all obse a ions D +1 ={D ,(x ,y )}
15 = + 1
16 end
17 end
This p io dis ibu ion is upda ed by sampled obse a ions o acqui e he pos e io dis ibu-
ion. The op imiza ion cycle includes he op imiza ion o he acquisi ion (o in ill) unc ion o
maximize u ili y o minimize isk ( i ness). Typical choices include he p obabili y o imp o e-
men (Kushne , 1964), expec ed imp o emen (Jones e al., 1998) and con idence bounds (Cox
and John, 1997). Algo i hms such as EGO can be seen as applied a ian s o BO. I is widely
applicable o di e en applica ions, including expensi e op imiza ion p oblems (Lizo e, 2008;
Khan, Goldbe g, and Pelikan, 2002) and machine lea ning (Snoek, La ochelle, and Adams,
2012; Swe sky, Snoek, and Adams, 2013). B ochu, Co a, and De F ei as (2010) gi e a u o ial on
BO wi h di e en applica ion examples.
35

7.2.3 Sequen ial Pa ame e Op imiza ion Toolbox
The sequen ial pa ame e op imiza ion oolbox (SPOT), de eloped by Ba z-Beiels ein (2010), is
a dynamic su oga e-based op imiza ion amewo k, which was ini ially in ended o o line
uning o algo i hm con ol pa ame e s. Va ious me hods o ini ial sampling designs, di e en
models and op imiza ion echniques a e included.
SPOT is s ongly in luenced by s a is ical me hods om design o expe imen s, whe e i is
a emp ed o p o e a ce ain s a is ical hypo hesis on basis o es ing. He eby, he a ailable
budge o expe imen s (i.e., unc ion e alua ions) is used sequen ially o imp o e a solu ion
and upda e he su oga e. This is done un il su icien knowledge abou he sea ch space is
a ailable o accep o ejec he ini ial s a ed hypo hesis.
The o e all design was dedica ed o algo i hm uning and ollows wo goals: one was imp o -
ing he e iciency o an algo i hm, i.e., disco e ing he algo i hm pa ame e s o sol e a de ined
p oblem ins ance as as as possible. The o he goal was imp o ing he obus ness o an iden-
i ied pa ame e se up, i.e., o sol ing di e en p oblem ins ances which o example di e
in hei egion o in e es o sea ch space dimensionali y. To une s ochas ic algo i hms, SPOT
in eg a es noise handling echniques by dynamic e-sampling o solu ions. This design can be
ans e ed o gene al su oga e-based op imiza ion, as he me hods ackle he p esen balanc-
ing p oblem o explo a ion ( obus ness) and exploi a ion (e iciency).
The gene al amewo k o SPOT is simila o he gene al su oga e-based op imiza ion algo-
i hms EGO and BO, which a e di ided in ini ializa ion and i e a i e op imiza ion phases.
SPOT also explici ly includes a p io pa ame iza ion phase, whe e he uses has o choose he
su oga e and ini ial sampling design. SPOT de ines a lexible amewo k and is hus appli-
cable o a la ge ange o p oblems, such as he men ioned algo i hm uning and indus ial
op imiza ion.
8 The Hyb id Class
Hyb id Class:
I Use o In o ma ion:implici /explici memo y by su oga e o candida e a chi e
II Candida e E alua ion: di ec /indi ec
III Type o Candida e: single ,popula ion,dis ibu ion model
IV Sea ch Space: global
V P oblem P ope ies: mul imodal, black-box, noisy, discon inui ies, mul i-objec i e, ex-
pensi e
The hyb id class depic s combina ions o algo i hms om he p e iously men ioned classes.
Hyb id algo i hms we e de eloped as a s a egy o imp o e o ackle indi idual algo i hm
weaknesses. The algo i hms a e o en gi en dis inc i e oles o explo a ion and exploi a ion,
as hey a e combina ions o an explo a i e global sea ch me hod pai ed wi h a local sea ch
algo i hm. Fo example, popula ion-based algo i hms wi h ema kable explo a ion abili ies
can be pai ed wi h local algo i hms wi h as con e gence. This app oach gi es some bene i s,
as he combined algo i hms can be adap ed o uned o ul ill hei dis inc asks. Mo eo e ,
he concep s can be easily adap ed o pa allel amewo ks.
One o he mos success ul ype o hyb ids a e he su oga e-assis ed e olu iona y algo i hms (Em-
me ich, Giannakoglou, and Naujoks, 2006; Lim, Jin, Ong, and Sendho , 2010). An o e iew
36
o su oga e-assis ed op imiza ion is gi en by Jin (2011), including se e al examples o eal-
wo ld applica ions. An example is ou lined in Algo i hm 8.1.
Algo i hm 8.1: Su oga e-Assis ed E olu iona y Algo i hm
1begin
2 =0
3ini ialize µ andom popula ion P ={x ,1:µ}⊆S
4e alua e popula ion P
5while no e mina ion-condi ion do
6 un e olu iona y algo i hm o ind λo sp ing O ={xi, µ < i ≤µ+λ}
7build su oga e s (x)( wi h cu en obse a ions D ={(xi,yi),1≤i≤k})
8p edic i ness o o sp ing O using su oga e s (x)
9selec λ∗bes o sp ing O∗
based on p edic ed i ness
10 e alua e selec ed o sp ing O∗
wi h objec i e unc ion
11 un local op imize om x∈ O∗
as s a ing solu ion o ge e ined solu ions O∗∗
(op ional)
12 selec su i o s om P ∪ O∗
∪ O∗∗
o nex gene a ion P +1
13 = + 1
14 end
15 end
Example 8.1 (Su oga e-Assis ed E olu iona y Algo i hm).In his hyb id sea ch s a egy, a
local su oga e is buil upon he cu en pa en popula ion and u ilized o p edic he i ness o
a numbe o λo sp ing candida es. The selec ion is hen based on he p edic ed i ness o he
su oga e. Op ionally, a local op imize can be used o u he e ine he compu ed solu ions.
Ex ensi e use o local sea ch leads o a as con e gence o local op ima. This hyb id s a egy
can be al e ed by using he su oga e only o a pa o he gene a ed o sp ing, while he o he
pa is e alua ed wi h he eal i ness unc ion.
Addi ional examples o hyb ids can be ound in he li e a u e, co e ing all possible class com-
bina ions:
Meme ic algo i hms, as de ined by Mosca o e al. (1989), a e a class o sea ch me hods which
combine popula ion-based guides wi h a local wande e . An ex ensi e o e iew o meme ic
algo i hms is gi en by Molina, Lozano, Ga cía-Ma ínez, and He e a (2010). They desc ibe
how di e en hyb id algo i hms can be cons uc ed by looking a sui able local sea ch algo-
i hms wi h special ega d o hei con e gence abili ies.
Ba z-Beiels ein, P euss, and Rudolph (2006) desc ibe a hyb id app oach an e olu ion s a egy
o explo a ion wi h quasi-New on me hod o exploi a ion. The algo i hm uns he ES and he
local sea ch in a consecu i e way and he budge o e alua ions o each me hod is con olled
by a con ol pa ame e . They pe o med expe imen s in which hey a ied he budge pa am-
e e o es i his hyb id app oach can be supe io o unning bo h me hods indi idually. They
came o he conclusion, ha hyb idiza ion can be bene icial o di icul objec i e unc ions, as
he ES p o ides in o ma ion abou in e es ing egions whe e he local sea ch is hen applied.
The su oga e managemen amewo k (Booke e al., 1999; Se a ini, 1999) u ilizes a combina ion
o a global su oga e-based algo i hm wi h an exac local pa e n sea ch. Mo eo e , i uses
i ness space ans o ma ions om con inuous o combina o ial by in oducing a ini e mesh
37
o possible solu ions. The key concep is o lowe he op imiza ion cos s by educing he eal
unc ion e alua ions using a su oga e, while a he same ime e aining he bene i s o a com-
bina o ial sea ch space and pa e n sea ch, i.e., he obus con e gence beha io . The algo i hm
u ilizes wo dis inc global and local sea ch phases, which a e execu ed du ing he sequen ial
op imiza ion. In he global sea ch s ep he selec ed in ill c i e ion (e.g. expec ed imp o emen )
is op imized in con inuous space and he nea es mesh poin is selec ed. In he local poll s ep,
a se o candida es si ua ed on he mesh a ound he cu en bes solu ion is e alua ed di ec ly
on he objec i e.
Taddy, Lee, G ay, and G i in (2009) combine su oga e-based op imiza ion based on eed Gaus-
sian p ocesses(TGP) wi h exac asynch onous pa allel pa e n sea ch (APPS) in a pa allel sea ch
amewo k. The algo i hm s a s wi h a space illing ini ial sampling using a la in hype cube
design, hen uns TGP and APPS in pa allel. In his case, TGP is used o p edic a anked lis
o a ixed numbe o new candida es, while APPS is used o pe o m local op imiza ion uns.
The budge o e alua ion and compu a ion is spli be ween hese wo componen s and all
obse a ions a e s o ed in an sha ed a chi e.
Hyb id algo i hms a e applicable o a la ge class o p oblems, de ined by which class hei
algo i hms o igina e om. Thei downside is hei la ge complexi y and he isk, ha hei
highe complexi y does no lead o imp o ed pe o mance, due o he di icul balancing and
equi ed uning o he dis inc algo i hms. Thei complex sea ch s a egies wi h a la ge numbe
o con ol pa ame e can make hem di icul o une. The algo i hm i sel becomes a black-box,
as he unde lying sea ch s a egy and he con e gence beha io is in luenced by nume ous
ope a o s and di icul o comp ehend.
9 Concluding Rema ks
In his wo k, we p esen ed an o e iew o con inous global op imiza ion algo i hms wi h ocus
on explaining hei sea ch s a egies using a new in ui i e axonomy. We de ined a se o i e
classes: exac , wande e , guide, ca og aphe , and hyb id in he Sec ions 4 o 8 and ou lined hei
indi idual p ope ies and exempli ied algo i hms o each o he p oposed classes.
To sho ly ecapi ula e: he exac class u ilizes (a p io i) p oblem in o ma ion o sol e a p ob-
lem wi h a gua an ee o inding he op imum. The heu is ic sea ch s a egies o he class o
wande e algo i hms a e sui able o as con e gence in a unimodal sea ch space and o en pa
o o he algo i hms. The well-known me aheu is ics om he guide class we e de eloped o
gene al applicabili y, pa icula o mul i-modal p oblems wi h unknown p ope ies. The ca -
og aphe class ocuses on su oga e-based algo i hms and ega ded amewo ks o p oblems
wi h expensi e unc ion e alua ions. Las bu no leas , we ook a look a hyb id class algo-
i hms, which y o combine he s eng hs o di e en algo i hms o o e come hei indi idual
weaknesses.
In gene al, i is bene icial o each use o iden i y i an op imiza ion algo i hm is sui able o
hei p oblem be o e applying hem. To suppo use s in selec ing a sui able algo i hm, we
poin ed ou he p os and cons o he di e en sea ch s a egies, he indi idual algo i hm ea-
u es and ypical cha ac e is ics o CGO p oblems hey a e able o handle.
A his poin , we also wan o highligh a new p omising esea ch a ea in he ield o algo i hms,
which is au oma ed algo i hm selec ion and pa icula au oma ic algo i hm con igu a ion. Bo h ideas
ackle he p oblem o selec ing he co ec sea ch s a egy o a gi en p oblem. Au oma ed
algo i hm selec ion ies o ind he mos sui able algo i hm o a ce ain p oblem based on
38
machine lea ning and exploi ed p oblem in o ma ion, such as explo a i e landscape analysis.
This me hod o algo i hm selec ion has shown o be able o ou pe o m a single algo i hm on
a se o benchma k unc ions (Ke schke and T au mann, 2017). An e en mo e p omising e-
sul was p esen ed by an Rijn, Wang, an S ein, and Bäck (2017), whe e algo i hm con igu a ion
was used o selec algo i hmic componen s o c ea ing a sea ch s a egy ou pe o ming a ail-
able algo i hms. The pa icula in e es ing idea is he e, ha sea ch ope a o s o algo i hms a e
iden i ied, ex ac ed and hen again combined o a new sea ch s a egy. The whole p ocedu e
also shows he s ong connec ions be ween di e en named algo i hms, pa icula in he a ea
o bio-inspi ed me aheu is ics.
In e es ing challenges o u u e algo i hm design a ise om p oblems in enginee ing applica-
ions, whe e he da a a ailable is es ic ed o ce ain condi ions, such as s eaming and online
da a and dynamic p oblems. The need o new op imiza ion app oaches eme ges om apid
de elopmen o communica ing senso s and machines in he ield o enginee ing, also known
as in e ne o hings (A zo i, Ie a, and Mo abi o, 2010). Sui able op imiza ion algo i hms need o
be di ec ly included in he p oduc ion cycle, adap ing o gene a e obus solu ions in challeng-
ing dynamic en i onmen s wi h mo ing op ima. Dynamic, su oga e-based online lea ning,
whe e a complex s a ic su oga e is cons uc ed and combined wi h ime- a ying modeling, is
s ill an open issue (Jin and B anke, 2005). Nowadays, cloud compu ing and high-pe o mance
compu ing clus e s a e a ailable o a wide ange o use s. Many op imiza ion algo i hms a e
no ye i ed o he needs o pa allel compu a ion and need o be adap ed Rehbach, Zae e e ,
S o k, and Ba z-Beiels ein (2018). The la ge and success ul ield o deep lea ning ne wo ks (Le-
Cun, Bengio, and Hin on, 2015; Schmidhube , 2015) decla es a comple e new ield om which
e y complex and di icul op imiza ion p oblems a ise. The ex ension o su oga e-based op i-
miza ion o hese ields, e.g., pa allel amewo ks and deep lea ning, is an in e es ing esea ch
opic. Fu he , we iden i y a lack in he ield o ealis ic benchma ks, which a e based on eal-
wo ld da a se s, which would allow a ealis ic compa ison o di e en algo i hmic app oaches.
10 Acknowledgemen s
Thanks o all ou colleagues o help ul p oo eading, ema ks and w i ing sugges ions. This
wo k is pa o a p ojec ha has ecei ed unding om he Eu opean Union’s Ho izon 2020
esea ch and inno a ion p og am unde g an ag eemen no. 692286.
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P o . D . Thomas Ba z-Beiels ein,
P o . D . Wol gang Konen,
P o . D . Bo is Naujoks,
P o . D . Ho s S enzel
Ins i u e o Compu e Science,
Facul y o Compu e Science and Enginee ing Science,
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P o . D . Thomas Ba z-Beiels ein,
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TH Köln,
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ISSN (online) 2194-2870