A Gentle Introduction to Multi-Criteria Optimization with SPOT

Sch i en eihe CIplus, Band 1/2013 He ausgebe : T. Ba z-Beiels ein, W. Konen, H. S enzel, B. Naujoks A Gen le In oduc ion o Mul i-C i e ia Op imiza ion wi h SPOT Ma in Zae e e , Bo is Naujoks, Thomas Ba z-Beiels ein A Gen le In oduc ion o Mul i-C i e ia Op imiza ion wi h SPOT Ma in Zae e e , Bo is Naujoks, and Thomas Ba z-Beiels ein Facul y o Compu e and Enginee ing Sciences Cologne Uni e si y o Applied Sciences, 51643 Gumme sbach, Ge many i s name.las name@ h-koeln.de Sch i en eihe CIplus TR 1/2013. ISSN 2194-2870 Abs ac . Mul i-c i e ia op imiza ion has gained inc easing a en ion du ing he las decades. This a icle exempli ies mul i-c i e ia ea u es, which a e implemen ed in he s a is ical so wa e package SPOT. I de- sc ibes ela ed so wa e packages such as mco and emoa and gi es a comp ehensi e in oduc ion o simple mul i c i e ia op imiza ion asks. Se e al hands-on examples a e used o illus a ion. The a icle is well- sui ed as a s a ing poin o pe o ming mul i-c i e ia op imiza ion asks wi h SPOT. 1 Mul i C i e ia Op imiza ion: A Simple Example Op imiza ion in gene al is ully in eg a ed in o odays li e. We decide o he bes p ize when compa ing simila g oce ies o choose he bes quali y when compa ing di e en jacke s. Howe e , such decisions a e gene ally mo e di icul , mo e complex han jus deciding ei he o he bes p ice o he bes quali y. Some so o comp omise solu ion migh be p e e able. This is whe e mul i- c i e ia op imiza ion comes in o play. O cou se, he same holds o indus ial p ocesses. Inpu ma e ial has o be selec ed and he p ocess i sel can be s ee ed in di e en di ec ions ia con ol- ling some inpu a iables. All hese decisions will in luence he quali y o he inal p oduc , whe eas quali y can be exp essed in o m o di e en c i e ia o objec i es (e.g., p ice, obus ness, speed). As a consequence, a decision ega ding he abo e men ioned inpu pa ame e s can a ely be made wi hou conside ing mul iple c i e ia a once. The e a e some aspec s ha make such decisions mo e complica ed han classical single objec i e p oblems. The mos appa en one is he loss o o al o de . Compa ing wo p oduc s o p ocesses jus based on hei p ice is easy. The cheape is he be e . I quali y comes in o play, li e is much ha de . Se e al p oduc s migh exis which ep esen di e en comp omises be ween quali y and p ice. Fo example, le us conside he choice o selec ing a ca . Ca no. 1 may be he cheapes , bu also he one wi h he highes milage, whe eas ca no. 2 is he one wi h he leas milage bu also mos expensi e one. The e may exis se e al MCO wi h SPOT 3 mo e ca s which ep esen di e en comp omises be ween milage and p ice. An example o ha si ua ion is ep esen ed in Fig. 1. P ice [€] Milage [l / 100 km] 20.000 30.000 5 15 10 25000 Fig. 1: G aphical compa ison o 6 ca s, ega ding p ice and milage. These solu ions canno be so ed by hei ela i e quali y o each o he since he wo goals (milage and p ice) a e con lic ing. The o al o de o solu ions is los . Only he ca ma ked by he ed do is clea ly wo se, because he e exis ca s which a e be e wi h ega ds o bo h objec i es. O cou se, hings ge mo e di icul (and e en mo e compu a ionally expen- si e) i mo e han jus wo c i e ia ha e o be conside ed. Fo example, one could conside addi ional c i e ia like en i onmen al sus ainabili y o sa e y o he ca . A he same ime, his decision p ocess can be made ega ding di e en inpu pa ame e s. Fo he consume , he choice ha a ec s he di e en c i e ia is me ely ha o he six ca s a ailable. Fo he p oduce o he ca s, he ca s can be de ined by di e en pa ame e s like hei ma e ial, shape o ype o mo o . The ield o Mul i C i e ia Op imiza ion (MCO) deals wi h ha ype o p oblems. The Sequen ial Pa ame e Op imiza ion Toolbox o e s me hods om ha ield, which employ su oga e models o speed up he op imiza ion p og ess. The ollowing Sec. 2in oduces he ield o MCO as well as he nomencla u e. A e wa ds, Sec. 3p esen s he p og amming en i onmen R and some basic ools o MCO in ha en i onmen . To gi e an easy o use example, Sec. 4 in oduces a simple MCO es unc ion, and shows how o op imize his wi h 4 Zae e e e al. classical MCO algo i hms. The e y same es unc ions is op imized wi h he Sequen ial Pa ame e Op imiza ion Toolbox SPOT in Sec. 5. To gi e mo e de ails on he usage o SPOT addi ional ea u es a e shown in Sec. 6. Finally, he eade can es his unde s anding o he p esen ed in o ma ion in Sec. 7. 2 Sho o e iew on MCO Conside ing only one objec i e in applied op imiza ion is a simpli ica ion ha does no mi o he complexi y o he unde lying applica ion in mos (o almos all) cases. No mally, mo e han one and up o hund eds o housands o objec i es 1, . . . , nneed o be conside ed. The mos common way o deal wi h mul iple objec i es appea s o be agg ega ion, e.g., o a weigh ed sum (x) = Piwi i(x). In con as , mul i-c i e ia o mul iobjec i e op imiza ion (MCO) o e s a di e en way o handle mul iple objec i es in a mo e p incipled, maybe mo e e ec i e way. In MCO, a well-known concep is Pa e o dominance, i.e., a solu ion x om a decision space Adomina es ano he solu ion y∈Ai xis be e in a leas one dimension o he decision space and no wo se in all he o he s. Mo e o mally and conside ing minimiza ion, his eads (A⊂Rn,i, j ∈ {1, . . . , n}): x≺yi ∀i: i(x)≤ i(y)∧ ∃j: j(x)< j(y). I a solu ion xis no domina ed by any o he solu ion in he sea ch space (o gene a ed by he algo i hm), i is said o be nondomina ed, i.e. ∀y∈A:y6≺ x. This concep allows o anking se s o solu ions in he mul i-dimensional objec i e space. As a consequence, MCO algo i hms aim o a se A∗o solu ions wi h he p ope y ha e e y wo solu ions xand y om A∗a e mu ually non- domina ed, i.e. y6≺ x∧x6≺ y. A se {x∈A| 6 ∃y∈A:y≺x}o such solu ions is also called a Pa e o- on . Coming back o he ca example om abo e, all bu he ca ma ked in ed in Fig. 1a e non-domina ed, and hus ep esen he Pa e o- on . The ed poin is domina ed by wo o he do s. 2.1 E olu iona y Mul iobjec i e Op imiza ion Algo i hms Beside he shee numbe o objec i es, he e is a s uc u al change in he s ep om one o mo e objec i es. The s ic o de o solu ions in he single-objec i e objec i e space u ns o a pa ial o de in he mul i-objec i e objec i e space (wi h espec o Pa e o dominance). This is one o he easons why popula ion- based op imiza ion app oaches like EA a e e y success ul in ackling MCO MCO wi h SPOT 5 p oblems. This s uc u al change also implies ha besides Pa e o dominance a seconda y quali y indica o is equi ed o anking and hus o ank-based selec- ion in an EA. As a esul , e olu iona y mul iobjec i e op imiza ion algo i hms (EMOA) became e y popula o e he las decade(Deb,2001a;Coello Coello e al.,2007). Such algo i hms always keep a se o solu ions, also called a popula ion, du ing he op imiza ion un and hus p o ide a decision make wi h a se o compa ably good, non-domina ed solu ions, no jus one inal bes like in single- objec i e op imiza ion. As a consequence, he decision make has he chance and he bu den o choose he inal solu ion, he one o eally implemen , om a se o such solu ions. In ecen yea s, he hype olume indica o u ned om a equen ly used quali y indica o o a well-es ablished selec ion ope a o o EMOA (Zi zle & Thiele,1998;Zi zle ,1999). The hype olume o a Pa e o on A∗is de ined as he n-dimensional olume o he space spanned by he Pa e o on and a e e ence poin y e , which needs o be de ined by he use : Λ [ a∈A∗ {y0|a≺y0≺y e }! wi h Λ being he Lebesgue measu e o he gi en se . Maximiza ion o he hype olume co e ed by he popula ion esul s in max- imiza ion o possible ade-o s. Implici ly his co e s he adi ional goals o con e gence o he solu ion se o he op imal on as well as good solu ion sp ead. Mos p ominen ins ances o hype olume based selec ion MCO algo- i hms a e SMS-EMOA (Beume e al.,2007a), Hyp-E (Bade & Zi zle ,2011), as well as MO-CMA-ES (Igel e al.,2007). The (µ+ 1) selec ion mechanism in SMS-EMOA (c . Alg. 1) p o ides an elegan way o enla ge and diminish he popula ion size online. The e o e his algo i hm is used o he p esen s udy. Algo i hm 1: SMS-EMOA 1P0←ini () ; // Ini ialize µindi iduals andomly 2 ←0 ; 3 epea 4q +1 ←gene a e(P ) ; // Gene a e o sp ing by a ia ion 5P +1 ← educe(P ∪ {q +1}) ; // Selec new popula ion 6 ← + 1 7un il s opc i e ium eached; In Algo i hm 2 ∆S(s, RI) := S(RI)− S(RI {s}) (1) 6 Zae e e e al. Algo i hm 2: Reduce(Q) 1R ← as -nondomina ed-so (Q) ; // all Inon-domina ed on s o Q 2 ←a gmins∈RI[∆S(s, RI)] ; // elimina e elemen wi h lowes ∆S(s, RI) 3Q0←Q { } e u n Q0 calcula es he sole con ibu ion o a single solu ion s o he hype olume o a Pa e o on RI. MCO wi h SPOT 7 3 MCO in R: Language, Packages, Algo i hms In R, he open sou ce p og amming en i onmen o s a is ical compu ing, some packages implemen unc ionali y o MCO1. We in oduce h ee packages, which implemen se e al impo an unc ions and algo i hms. The e a e mo e MCO- ele an packages, which a e no co e ed he e. We encou age addi ional esea ch o in e es ed use s2. 3.1 The mco package The mco package p o ides se e al unc ions conce ning MCO wi h gene ic algo i hms. Mos impo an ly, i p o ides an implemen a ion o he NSGA2 algo- i hm. Se e al measu es o a Pa e o on s quali y a e a ailable as unc ions (e.g., Sp ead, Hype olume, Epsilon-Indica o ). Addi ionally, se e al ypical MCO es unc ions a e p o ided (e.g., ZDT1-3). 3.2 The emoa package The emoa package p o ides se e al use ul unc ions, including se e al mea- su es like hype olume con ibu ion o c owding dis ance. I also has se e al es unc ions ha a e no in he mco package. 3.3 The SPOT package The SPOT package p o ides mainly he means o do mul i objec i e su oga e model based op imiza ion. Addi ionally, i uses he emoa package o cons uc a e y basic SMS-EMOA implemen a ion. Mo e in o ma ion will be gi en in sec ion 5. 4 A simple Tes unc ion: ZDT2 The es unc ion used in he ollowing examples is pa o he ”mco” R- package3. In his example, i is expec ed ha his package is al eady ins alled. To ins all, uncommen he i s wo lines below. The mco package also con ains he NSGA2 algo i hm (Deb,2001b). SPOT should be loaded, oo, o he pu pose o op imiza ion. > #ins all.packages("mco") > #ins all.packages("SPOT") > equi e("mco") > equi e(SPOT) 1R, and all packages men ioned in his documen , a e a ailable om he CRAN pla o m: h p://c an. -p ojec .o g/ 2The h p://www. seek.o g/ home page migh be a i s s a poin o a sea ch on MCO in R. 3h p://c an. -p ojec .o g/web/packages/mco/ 8 Zae e e e al. The ZDT unc ions a e MCO es unc ions wi h scalable decision space di- mension n(Zi zle e al.,2000). To gua an ee a low numbe o unc ion e alua- ions as well as easy isualiza ion o esul s, bo h, he dimension o he decision and he objec i e space a e se o 2 in ollowing example. The unc ion ZDT2 R2→R2is de ined as: 1=x1 g= 1 + 9x2 2=g 1−x1 g2! 0≤x1≤1 0≤x2≤1 Bo h objec i es 1and 2ha e o be minimized. In he R-package mean(), he ZDT2 unc ion can be called as ollows. > x=c(0.5,0.4) #Inpu ec o > y=zd 2(x) > p in (y) [1] 0.500000 4.545652 The objec i e space o he ZDT2 unc ion can be isualized in sepa a e su ace plo s. The e o e, we will conside he ollowing unc ion: R2→R: (x1, x2)7→ 1(x1, x2). No e, we will use he apply(x) command o de ine an anonymous unc ion. > apply(x,1, ) will apply o ows o x, > apply(x,2, ) will apply o columns o x. > ## De ine unc ion ha calls ZDT2 in a ec o ized manne > ## bu e u ns only i s objec i e: > ZDT2obj1= unc ion(x)apply(x,1,zd 2)[1,] > ## Plo i s objec i e in gi en bounda ies > ## (lo o lowe , up o uppe ) > spo Su Con ou ( =ZDT2obj1,lo=c(0,0),up=c(1,1)) MCO wi h SPOT 9 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x1 x2 To isualize he impac o he second objec i e, we will conside he ollowing unc ion: R2→R: (x1, x2)7→ 2(x1, x2): > ## De ine unc ion ha calls ZDT2 in a ec o ized manne , > ## bu e u ns only second objec i e: > ZDT2obj2= unc ion(x)apply(x,1,zd 2)[2,] > ## Plo second objec i e in gi en bounda ies > ## (lo o lowe , up o uppe ) > spo Su Con ou ( =ZDT2obj2,lo=c(0,0),up=c(1,1)) 16 Zae e e e al. > spo (spo Con ig=append(lis ( epo . unc="spo Repo Con ou ", + epo .in e ac i e=FALSE), es2),spo Task=" ep") 0 2 4 6 8 10 ● ● ● ●●●●● ● ● ● ●● ● ●● ●●●● 0.0 0.4 0.8 0.0 0.2 0.4 0.6 0.8 1.0 p edic ed Model: Y.2 VARX1 VARX2 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ●●●●● ● ● ● ●● ● ●● ●●●● 0.0 0.4 0.8 0.0 0.2 0.4 0.6 0.8 1.0 p edic ed Model: Y.1 VARX1 VARX2 In hese con ou plo s, c osses indica e a poin sampled on he a ge unc ion and black do s (he e: e y close o he ho izon al axis) indica e poin s ha belong o he app oxima ed Pa e o on . Wi h he abo e se ing o epo .in e ac i e = FALSE, he su ace plo s will always show he beha io o he i s wo op imized pa ame e s. I mo e han wo a e op imized, he use can use a simple GUI (see Fig. 2) o choose wo pa ame e s o plo ing: > spo (spo Con ig=append(lis ( epo . unc="spo Repo Con ou ", + epo .in e ac i e=TRUE), es2),spo Task=" ep") Fig. 2: Twiddle in e ace He e, he use can choose which pa ame e s o plo , using he slide s aand b. Slide Cspeci ies which poin om he Pa e o on should be used o de e mine he emaining pa ame e s, which a e assumed o be cons an o he plo . When jus wo pa ame e s a e op imized, his choice has no in luence. 6.2 Di e en models o each objec i e In he abo e examples, each objec i e was modeled by he same chosen su o- ga e modeling echnique. I could howe e occu , ha he use wishes o model MCO wi h SPOT 17 he i s objec i e wi h a K iging model and he second wi h a Random Fo es model. This is possible, by using he ollowing con igu a ion: > con ig$seq.p edic ionModel. unc="spo P edic MCO" > con ig$mco.con igs=lis (lis (seq.p edic ionModel. unc="spo P edic Fo es e "), + lis (seq.p edic ionModel. unc="spo P edic RandomFo es ")) > es3<-spo (spo Con ig=con ig) spo .R::spo s a ed > es3$mco.h olume [1] 110.097 He e, he mco.con igs elemen is a me a-lis con aining se e al con igu a ion lis s. Any se ings no speci ied in mco.con igs will be aken om he main con ig- u a ion lis (he e: con ig) o se o de aul alues. Ins ead o selec ing di e en models, one can use he mco.con igs me a lis o choose di e en se ings o wo models o he same ype. 7 Tasks Imagine you ha e wo a ge unc ions, bo h wi h a sphe ical shape, bu wi h hei cen e a di e en poin s. Assume ha bo h unc ions a e de ined as: 1(x1, x2)=(x1−2)2+ (x2+ 4)2(2) 2(x1, x2)=(x1+ 1)2+ (x2−3)2(3) Toge he , hese unc ions o m a wo-objec i e op imiza ion p oblem, whe e bo h 1and 2a e minimized. Please y o sol e he ollowing asks: 1. Implemen he op imiza ion p oblem in R. 2. Plo he a ge unc ions sepa a ely. 3. Wi hou using he compu e any u he , wha do you expec he Pa e o on o look like? Desc ibe and/o d aw you idea. 4. Op imize he p oblem, using he NSGA2 Algo i hm in R. 5. Plo Pa e o on and Pa e o se o he NSGA2 esul s. 6. Op imize he p oblem using SPOT. 7. Plo he Pa e o on and se , compa e wi h NSGA2 esul s. Why do esul s di e ? Also, Compa e wi h expec a ion om ques ion 3. 8. Plo he su oga e models wi h SPOT and check whe he hey ep esen he ac ual a ge unc ions. I no , why? 9. In you opinion, which is he mos sui ed way o sol e his speci ic op imiza- ion p oblem, and why? 8 Solu ions Solu ions o he asks can be eques ed om he au ho s ia E-Mail. Bibliog aphy Bade , J. & Zi zle , E. (2011). HypE: An Algo i hm o Fas Hype olume-Based Many-Objec i e Op imiza ion. E olu iona y Compu a ion, 19(1), 45–76. Ba z-Beiels ein, T., Lasa czyk, C., & P euß, M. (2005). Sequen ial pa ame e op imiza ion. In B. 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Alle Ve ¨ o en lichungen diese Reihe k¨ onnen un e www.ciplus- esea ch.de ode un e h p://opus.bsz-bw.de/ hk/index.php?la=de abge u en we den. K¨ oln, Janua 2012 He ausgebe / Edi o ship P o . D . Thomas Ba z-Beiels ein, P o . D . Wol gang Konen, P o . D . Ho s S enzel, D . Bo is Naujoks Ins i u e o Compu e Science, Facul y o Compu e Science and Enginee ing Science, Cologne Uni e si y o Applied Sciences, S einm¨ ulle allee 1, 51643 Gumme sbach u l: www.ciplus- esea ch.de Sch i lei ung und Ansp echpa ne / Con ac edi o ’s o ice P o . D . Thomas Ba z-Beiels ein, Ins i u e o Compu e Science, Facul y o Compu e Science and Enginee ing Science, Cologne Uni e si y o Applied Sciences, S einm¨ ulle allee 1, 51643 Gumme sbach phone: +49 2261 8196 6391 u l: h p://www.gm. h-koeln.de/~ba z/ eMail: homas.ba z-beiels ein@ h-koeln.de ISSN (online) 2194-2870