Multi-Objetive Optimization

Deision making in real-life is often subjet to multiple objetives to be

met. Inmany senarios,satisfyingone objetive istypially inonit with

satisfyingtheother. TheeldofMulti-CriterionDeisionMaking (MCDM)

aims at developing mehanisms for supporting the deision making proess

when treating multiple objetives. The idea is to study the nature of the

trade-o between thevariousobjetives,to seekagoodompromise, and to

avoida lose-losesenario.

Naturally, we are interested inthe optimization perspetive of MCDM,

andespeiallyinevolutionarymulti-objetiveoptimizationalgorithms(EMOA).

The latterhasdeveloped inthelast two deades, andhas beome a eldof

intense researh.

Next,webrieyreviewhereformallythebasioneptsofMulti-Objetive

Optimization.

5.1.1 Formulation

Givenanoptimizationproblemwith

m

objetives,weonsiderits

m

-dimensional objetivespae,alsoreferredtoasthesolutionspae. Bydenition,theve-

torof objetives isin

R

m

:

~

Weassumethatallobjetivesaretobeminimized. Apartialorderisdened

onthesolutionspae,

F

=f~(X)

,bymeansoftheParetodominationonept for vetorsin

R

m

,inthefollowing manner:

Denition 5.1.1. Given any

f~

(1)

_∈Rm

and

f~

(2)

_∈

_Rm

, we state that

f~

(1)

stritly Paretodominates

f~

(2)

,notedas

~

f(1)≺f~(2),

ifand onlyifthefollowing holds:

∀i_{∈ {}1, . . . m_}:f_i(1)_≤f_i(2)

_{∧ ∃}i_{∈ {}1, . . . , m_}:f_i(1)< f_i(2)

(5.2) Note,that inthebi-riteria asethis denitionis reduedto:

~

f(1)≺f~(2)

:⇔f₁(1)

< f₁(2)∧f₂(1)≤f₂(2)∨f₁(1)≤f₁(2)∧f₂(1)< f₂(2)

(5.3) Inadditionto thestritdomination

≺

,wedene further omparisonopera- tors:

~

f(1)f~(2)_⇐⇒f~(1)_≺f~(2)_∨f~(1)

=f~(2)

(5.4) Moreover, we state that

~

f(1)

is inomparable to

~

f(2)

,notedas

~

f(1)_||f~(2),

ifand onlyif

~

f(1)_f~(2)_∧f~(2)

_f~(1)

(5.5)

The ruial laim isthat for any ompat subset of

R

m

, say

F

, there exists a non-empty set of minimal elements with respet to the

partial order

(see,e.g., [97℄,pp.29).

We an nowdene non-dominated pointsasfollows:

Denition 5.1.2. Non-dominated points are the set of minimal elements

withrespetto the partialorder

:

FN

={f~∈ F|∄f~′

∈ F

:f~′

≺f~}

(5.6) whereasubsript

N

willdenotefromnowonanon-dominated setinthe ontext of multi-objetive optimization.

Having dened the non-dominated set and the onept of Pareto domi-

nation for general sets of vetors in

R

m

, we are now ina position to relate

it to the optimization mission. Theaim of Pareto optimization is to obtain

the non-dominated set for

F

=f~(X)

and its pre-image in

X

, the so-alled Pareto optimal set, also referred to as the eient set. We may then de-

Inmanypratialappliationswearealsosatisedwithasetofsolutions

whoseimageunder

f~

yieldsagoodapproximationtothenon-dominated set, thougha denition of whatis a good approximationis problemdependent.

Often, it is desired to ahieve a uniform distribution on the Pareto front

and agoodonvergene of allpointsinthe approximationset to some non-

dominated solution.

For notational onveniene, we shalldene astrit pre-order on the de-

ision spae asfollows:

~x(1)

_≺~x(2)

_⇐⇒f(~x(1))_≺f(~x(2))

(5.7) Aordingly,we dene thepre-order

~x(1)~x(2)

_⇐⇒f(~x(1))f(~x2)

(5.8) Note, that this is not a partial order, as the antisymmetry axiom does not

haveto besatised. Thisstemsfromthefat,thattwodistint vetorsmay

have the same funtion value. For the same reasons, itis also possible that

the eient set omprisesmore members than thePareto front.

5.1.2 The NSGA-II Algorithm

Due to their robustness and exibility, Evolutionary Multi-Objetive Op-

timization Algorithms (EMOA) have reently reeived inreased attention

as problem solvers for diult simulator-based optimization problems [98 ,

99, 100 ℄. Among these methods, the NSGA-II method is one of the most

popular,and ithasbeen suessfullyapplied to manyreal-world problems.

The NSGA-II algorithm has been proposed by Deb [98 ℄. It aims at

obtainingawelldistributedapproximationsetofpointsthatareloseto the

Pareto front. It is a

(µ+λ)

-EA (see Algorithm 1), whih employs spei variationoperators(fordetailswereferthereaderto[98℄),aswellasaunique

seletion operator. We hoose to desribe the latterindetail.

The NSGA-II seletion onsists of two phases, that orrespond to pri-

mary versus seondary seletion riteria. At rst, a proedure alled non-

dominated sortingisapplied,thatobtainsperfetorderonthesetofdeision

vetors. Next,the solutionswhih sharethesame rankaresortedbymeans

of the rowding distane riterion. Expliitly,non-dominated sortingworks

as follows: Given a population

R

, its non-dominated subset

R1

=

RN

is extrated. This set forms the best ranked solutions (rank=1). Given the

set

R−RN

,the non-dominated subset

R2

= (R−RN)N

is thenextrated, and so on. This is repeated until the set of solutions is empty. The sets

R1, . . . , Ri, . . . , Rℓ

arealled thenon-dominated sets of rank

i

,

i= 1, . . . , ℓ

. Sine these sets an possibly ontain more than one member, a seond ri-

Figure 5.1: Non-dominated sorting. Figure ourtesy of Mihael Emmerih

[101 ℄.

alled the rowding distane: Given a solution

~x

(i)

_∈Rn

, we determine the

orresponding

f~=f(~x)

inthesolutionspae, and thenevaluate

d(f~) =

n

X

k=1

h

min

{fk(j)|j∈{1,...,|R|}−{i}∧f(k)≤f(i)}

f_k(i)−f_k(j)+

min

{f_k(j)|j∈{1,...,|R|}−{i}∧f(k)_≥f(i)_}f

(j)

k

−f

(i)

k

i

(5.9)

Foravisualizationofthenon-dominatedsortingproedureandtherowding

distanealulationonabi-riteriaoptimizationproblemwerefertoFigures

5.1and 5.2, respetively.

A omprehensive overview on the NSGA-II and other EMO algorithms

an be found in [98, 99℄. Reently, an interesting method alled the SMS-

EMOA [100 ℄ was proposed, and was shown to outperform theNSGA-IIal-

gorithmonstandard benhmarks. However, theNSGA-IIanbeonsidered

Figure5.2: Crowding distane. Figureourtesyof Mihael Emmerih [101 ℄.

In document Niching in derandomized evolution strategies and its applications in quantum control (Page 97-101)