On Diversity in Multi-Objetive Optimization

Reently it has been pointed out that not only high diversity of solutions

inthe objetivespae but alsohighdiversity ofsolutions intheeient set

an be ofinterestfor the deisionmaker[68 ,102 ℄. For instane, ifaspei

pointon thePareto front isseletedbythedeisionmaker,itmight alsobe

interesting to onsider dierent possible realizations to this solution in the

deision spae. Hene, if there aretwo dierent pre-images of the seleted

point on the Pareto front in the eient set, both of them are of potential

interest for thedeision maker. Thissituation is illustrated inFigure 5.3.

More preisely, the dierene between the lassial seletion priniple to

our proposed approah an be formalized as follows. Let

A

denote an ap- proximation set on whih we would like to apply ranking, and let

~xA

and

~xB

be two solutions in

A

. In the lassial seletion method, as employed bythe NSGA-IIor SMS-EMOA algorithms, a solution

~xA

is preferred to a solution

~xB

if

~xA

hasa better dominane rank than

~xB

in

A

,with respet tonon-dominated sorting. Giventhat

~xA

and

~xB

sharethesamedominane rank in

A

, then

~xA

is preferred to

~xB

, if and only if

~xA

ontributes more

Decision Space

Objective Space

Figure5.3: Diversityfordeisionmaking: Illustrativeexampleforasenario

where two adjaent pointson the Pareto front are mapped onto two points

in two ompletely dierent regions in the deision spae. Unitsand sales

arearbitrary.

to the diversityofthe approximation setintheobjetive spaethan

~xB

. In the proposed seletion priniple,

~xA

remains preferable to

~xB

, if

~xA

has a betterdominanerank than

~xB

in

A

. However, given that

~xA

and

~xB

share the same dominane rank in

A

, then

~xA

is preferred to

~xB

, if and only if it ontributes moreto the diversity intheaggregated spae (i.e.,inboth

objetiveanddeisionspaes). Thisprinipleanbeinstantiated indierent

ways,dependingon thediversitymeasuredened on theaggregated spae.

Multi-objetive optimization methods aim at maintaining diversity, by

their denition, and indeed, one of the popular mehanisms for diversity

maintenane is therowding onept [67 ℄, whih is also applied, yet dier-

ently, as a single-objetive nihing tehnique. Thus, the important ompo-

nent ofdiversity is the linkingelement between theelds ofmulti-objetive

andmulti-modaloptimization. However,inmulti-objetiveoptimizationthe

diversitymaintenane istypiallysought intheobjetivespae, for thesake

ofobtainingafairoverageoftheParetofront,whilenottakenintoaount

for the Paretooptimal setinthedeision spae.

5.2.1 Related Work

Severaldierent studies treated relatedtopisto thework presentedinthis

hapter. We reviewthem hereshortly.

NihingforMOEA: TheNPGA Nihingtehniqueshavebeenalready

multi-objetiveoptimization,knownastheNihed-ParetoGA(NPGA).The

algorithm wasa variant of thetness sharing nihing method, whereas the

nihing distane metri wasset toonsiderthe objetive spae only. These-

letionwasbasedontheso-alledParetodominationtournaments or onthe

minimalniheount,otherwise. TheNPGAwasalassialexampleofusing

an existingsingle-objetive nihing tehnique, ina straightforward manner,

for multi-objetive optimization - only by redening the nihing distane

measure and the seletion mehanism. However, its kernel was the simple

GA, whih typially suers from limited performane in high-dimensional

ontinuous landsapes,and itlaked anyself-adaptation mehanism.

TheOmni-Optimizer Deb'sso-alledOmni-Optimizer[68 ℄isonsidered

to be one of the rst and only attempts of introduing a generi optimiza-

tion routine whih aims at overing the four ategories of funtion opti-

mization: Single-objetive uni-global, single-objetive multi-global, multi-

objetive uni-global, and multi-objetive multi-global problems. Also, it is

one of the rst attempts to take diversityinthe deisionspae into onsid-

eration.

In priniple, this algorithm extends the NSGA-II by onsidering addi-

tionallythediversityinthedeisionspae. Thisisimplementedbymeansof

therowdingdistanealulationinthedeisionspaeforalltheindividuals.

The assignedrowding distane isdened asfollows:

if rowd_dist_obj

(i)

>

avg_rowd_dist_obj or rowd_dist_de

(i)

>

avg_rowd_dist_de

then rowd_dist

(i) =

max

(

rowd_dist_obj

(i),

rowd_dist_de

(i))

else rowd_dist

(i) =

min

(

rowd_dist_obj

(i),

rowd_dist_de

(i))

i.e., ifthe individual hasabove-the-average rowding distane, eitherin the

deision or objetive spae, the larger of them is assigned to it, otherwise

thesmallerofthetwodistanes isassigned. Thisriterionis rathergeneral,

andstrongly reliesonuniform distributionofpeaks aswell asontheir equal

tness values. Also,thesalabilityofthetwodierent spaes isnottreated.

We would like to speulate that it is expeted to experiene diulties on

non-uniform multi-modal landsapes, for instane. From the pratial per-

spetive,thealgorithm wasreportedin[68 ℄tobetestedonlyonasingletest

funtion, onstruted by Deb for this purpose, with uniformly-distributed

equi-tness minima landsape. Weshall revisitthis test-funtion inour ex-

perimental proedure.

Deision-Spae Diversity as an Independent Objetive Toolo and

Benini [104 ℄ also promoted the issue of geneti diversity in multi-objetive

diversity of trial solutions in the deision spae, quantied by means of a

overage funtion,asanindependentobjetive,subjet to maximization, in

the ongoingmulti-objetive searh. This GA-based approah wasshown to

outperform the NSGA on a set of

30D

bi-riteria minimization problems introdued byZitzler etal.[105 ℄.

Self-AdaptationinMulti-ObjetiveOptimization Self-adaptationof

strategy parameters [106 ℄ hasbeome afundamental omponent intheevo-

lutionary optimization routine. Moreover, the self-adaptation of the muta-

tionstrategy parameters hasbeenshown tobeneessaryforeient single-

objetiveoptimization within ES[106℄.

Self-adaptation is expeted to fail inthe lassial multi-objetive optimiza-

tion routine. This is due to the fat that given oniting objetives, a

suessfulmutation toward one objetive isnot neessarilya suessfulmu-

tation toward theothers andhene shouldnot beseleted.

Bühe, Müller and Koumoutsakos [107 ℄ onduted a pioneering study of

self-adaptation in multi-objetive optimization. They onsidered three dif-

ferent lasses of multi-objetive algorithms - independent sampling, ooper-

ative populationsearh withdominane riterion and ooperative population

searh without dominane riterion. Three representatives - CMEA, SPEA

andSDM-mathingthelassesrespetively,weretestedonamulti-objetive

generalizationofthespheremodel,andomparedwithrespettoeahother.

Self-adaptation had been plugged-in into theevolutionaryoremehanisms

of the algorithms, in a limited way (rotation angles, for instane, were not

always adapted). The onlusion was thatself-adaptation didnot work for

ooperativepopulationsearheswhihusethedominaneriterion inthet-

ness assignment (SPEA), andthis resultwasreassuredbytestingmorerep-

resentativesfromthatlassofalgorithms, suhastheNSGA-IIandSPEA2.

However, self-adaptationouldworkfor theCMEAand SDM,whihdo not

usedominane, butrather onsiderasingleobjetive foroptimization while

the otherobjetivesaretreated asonstraints. Theonludingmessagewas

lear self-adaptation doesnot work inits lassial denitionupononsid-

ering multiple objetives ashadbeen speulated.

Reently, the self-adaptation obstale was treated suessfully by using

the so-alled hyper-volume indiator (also known as S-metri) [99 ℄ as a se-

letionriterion,similarto[100 ℄,intheMulti-Objetive CMA-ES[33 ℄,tobe

disussed next. Asimilar approah,yetemploying asimpler ESkernel,was

also reportedreently in[108 ℄.

CMA-ES forMulti-Objetive Optimization Analgorithm for multi-

objetiveoptimizationwithaCMAkernelwasintrodued reently [33 ℄,em-

riterion, followedbythe maximizationof thePareto front hyper-volume as

a seondaryriterion. Crowding distanewasalsoonsidered asan alterna-

tive seondary seletion riterion. In many ways, this algorithm resembles

our nihing framework. However, its diversity preservation stems from the

outome ofseletion withrespetto multiple riteria,rather than from the

spatialenforement ofspeiationbymeansofanihe denition. Itisimpor-

tant to noteinthis ontext,that thehyper-volume indiator is well-dened

asameasureofdiversityand solution-setqualityintheobjetivespae, but

annotbe applied asanindiator ofdiversity inthesearh spae.

In document Niching in derandomized evolution strategies and its applications in quantum control (Page 101-105)