View of On a class of fitness functions for genetic algorithms using proportional selection

it ""'-""ilrt

I'otlotos itnnt'eilícr,td1¡

tÌtal X*:

^V*.

aiuii,l

Ñnraton-I{cotttot'ouiclt' ctssunltttons,

Gy¡lg c,rxl T'apiu l4l

as weli ás otlrcts

t51-

t10l ^{proaiúed, th,e} J'o\totiitr,g ^Íto:rmcl Jo:r Neuloru's ^ntel'ho¡J

,t -

^t.,

uì- -

⁽¹

-

^(]2ln

^gz'-r _Jor all

^{n, >- o,}

1-ozo

wlticlt, cant¿ot be ,ím4troued,. ^Thc¡,t

is

tl¿e o'¡'cler o,f con"uergen'crt oJ Neutott')s ^'¡n'el'- äotl

is

^l,wo tulrcre (¿ri (4.13)) tlte Ítcttleu-trVernef nt'eth.ocl lms ot'iler th¡'ee.

t"i

Note utioi tiu.ct rue'lrc,ue shorun (by (a.13)) Lhe enisl'anca oJ infinitclu^

,nrrrry'irrllrocl,s

for''øelil,2)

^tuh,ere

^t¡'c

^up'per bou'tt'cls tt're Less

t¡un tlt'at

of Halley's 'ntetiJt'ocl

(u : L

tlrcn).

}IEIìI]RJìNCI'S

1, r\rg¡rfos, I. I\., ena<bQlfu crpuiioils antt tL¡:;pIítations lo Chrndtascithtu"s ^tut<i tdûlcdcqualions,

Íjtrll. r\rrstr:aì, lf aLlt. Soc., 3g ^(.t 9S5), 2-t5-292'

2, -, ^Ou tt cllss of ¡tottline¡rr it'tlcgral equirtiotts ctrisittcl irt ¡tt:nlrott !rttttsporl. Accluationcs l,IaLhcrn¿rtictrc, Jli ^{(1988), 99-1 11'}

3. Chcn, l)., Itlnloioniclt'-Osltóín:;1,i coilûaìgcn('c lhcorcnts and o¡'tlitnal u¡or ltotn<Is for,Iurntl's

itc't.atitie ntelhatls, lrtcru. ^.J. Coilrputcr J\talh., 3l ^{(3 -'r} 'l) ^{(1!90)' 221} --2:li:.

4. GIagg, \\¡. ll. ^{atrrl l'apia,} lì. t\.,0'ptintcl error boun<ls fun'lhe Netulott-l{ct¡tlorc¡oiclt l.luorcn, SJÀ;U.f. Nunrcr'.,\nal., lt I ^(1974), 10-líl'

5. I{atrttuovich, L' V' antl Ákilor', G' })" lìrulc/inrt¡tl

"lrlnlu'sis i¡t \tort¡t¿tl Spalcs' Pcrgarrotr Prcss, Ncrv YoìÌi, 196!ì.

(ì. OsL'orvsiii, A. ìI., Sglrrliott o[ ]Ìqttcrlions nrrd ,S1,'s/rnts of lÌc1tt<tliotts, Âcarlcnric Ptcss, ^New

Yolk, ^19(i(ì.

7. I)otr.a, ¡i.4. ¿¡rl l,ialt, \.., ^shru,p trror bctuntls for N¿¡D¿on,.r ¡nelht¡tl, Nurrct'. \lath., ^lÌ4,

(.1 9{i()), 63-72.

g. Wcr.ncr.,'\V,, Snrne intprcnentcnls ol'clttssìtul ìleì'(liúc ntdltotls fctr ilic søIulions of nottlitttar eclttrrliotts. l,cctÙ-r'e llOtes in I,Iat.li., Nrrrnci i¿:rù solttlíott of rtortlitteat cquali<ttrs. Ìtroccttlittgs, Illetttcn, ^{878, (19tì0), 'l'2'1} _-'1'lO'

g. y¿¡lliarnoto, 'l'., À cnrttiir.eetu:c lhect¡e¡tt for Netuk;rt-lilr'c ntclJtotls in Bantu'lt .spcccs, Nunrct' l\'Iath., 5l ^{(19t17), 545-:iir7.}

10. Zabr.ejl<o, lr. lr. rn<l Ngìlell, ,t. 1;., ^',l'Ìtt nrtjorturl ¡tttlltod in lhe lhtotu of -'\rurlon-linttlotopítlt trppt'ot:intulÌ<tns r¿ntl lltc Ptnlr r,r;ril eslitt¡¡tles. Nrtrutlr. lìtrtlct. ,\rirl . atrd Optinriz , ^{f) (lf)ij7i,}

l!71 -(:ill4'

tRÌìvulì Ð'AN/\t,ysE lrlllrfÉIltQuE

n'r

ÐIì T-IrÉ()Itììì IJH r,'ÀppÌt0xlur\T.lolrj

X'omc 9Í1, Iiìo

I,

^IÐÐ4, ¡rp. t5-p3

ON A CLASS OF FITNESS FUNCTIONS FOR GENETIC AL GORITHMS USING

PROPORTIONAL SELECTION

RIIÁìì,ION-IjRNö BAI,ÁZS

(Cìu j-Napoca)

1. tN'rnonllf.'iloN

74 l. l(. Àrsvros, ìf ^. À. 'fabat¿rbai, l). ^(ìltctt

is

inaertittle ct"ìLd .frctnt, the estriltùGte

P(X*) - ^P(r'r¡ -

^P',(Y'r

^{+ (X'r} -

Y'F))d(x',¡

- ^Y*),

lìeceilcd B X ^19U2

^ ^{-[n [2]}

Grefenstette

and Bakel

cliscussed

the impact of the

fitrress

ftlirctions

on

the

behavioul of genel,ic algorithms.

lhe¡i

shorved situations rn'hele Hollancl's _schema

theoremlsae

_¡ã1¡ cloes

not

have a cleal iritelpr,e-

tation

ancl su_ggested,

for

a

lather

large class of genetic algor,itìrrns a

iirn-

ple

but

useful chalacterization of

the implicit

pãrallelìsrnl

- ^rn

^{the present pâpel we_ s1,udy a clais} of fitness functions

for

genetic algorithms using

proportionâl

seie¿tion.

'Ihe first

sectjon contains

îhe

re- hrns using a monotonic fjtness func-

2. GNI,N[I'¡'IC AI,GOIII';'IINS USINç A NTONÛ'¡'(}NIc FI'|¡{IìSS IìUNC'I'TON

^ß*

r) A tf ON OTONIC S ll ¡,tr(ìTION

^LG(XrI'tìÉIlU

,

Selgcti_on is probably

the

rnost

irnpoltant

step

in

a ^gen beca,nse

it

clel,err'ines _r.vhich

inrlividrials rvill cdltribuie

amonnt)

to

l,he creatjon

of a

^lrer.v

population. ¡ls jn [2]

^rve

selection

to

be par,titioned

into twô

steps :

eàch

cÌ'ea-

rentlale on the selectjon algolithrn,

algolithrn

is

siven.

it'lius ive sìra,li

rdividual as .i,yell &s the effect of such er,planes.

ilhis

latter,

rvjll lie

cìr¿lr,ac_

morrent f by the targel, sam,pliu,g rale t,sr'(H,r)

: I _,.y ^Yy+,

,n(H, l)

14

i ⁾ t ^p il ! ù1 r:!i I ^ct [ ^],[ ut hetnt.tI itrLI Scicltcrs, {:Qtlt¿l ott U ttioersilg

Lontlott, Oli 73505, Ii.S.A.

))r.ptttlmettÍ ol Ìlalh,5ci¿rlc¿s

l-lrt i¡r¿'¡.silil ^o I ^{Ã t )i<i ti} -<tt s

I¡ttUtllct¡iIl<, ^A Il 7 2'10 1, ^rJ'S.A

16 i\I. E. llalrizs

\'r'heÌe

n(Htt)

denotcs

the nurnlrcÌ of

repleseul,ati\¡es

of

h¡'pet'plane

Il ⁱⁿ

the

population

at

rnonìerìt

I

(clenntecl b,1. P(f)).

rr'ol a

problem forrnulat'ed

in terms of an

objectir.e

functiotl

_{,f, the}

target

sarnpling rate is girren

b'l'

cornposing trvo functions

:

a filn,ess ,frunc-

tion

u,

and a

selention alç¡ot'ith,nt, s,

that

is

tu'(n,t) :

s('u(r, f), ú).

For the ¡est of

this

paper rve sllall consiclel

that

the fitness does nt¡t d.êltencl

olt

¿,

i.e.

lsr(e;,

Í) :

s(t¿(æ), l).

In

sil,ual,ions s'hen the

olljectivc

function is to be minirnizecl or when

it

can take on negativc l'a,lues the fitness frurction js _obtainecl b1'a trans-

lornration of tìle objective function

such

that

u,(*)

:

/r(./(er)).

llowcver

these ate

not the

onl¡r ¡¿¿ootls

for

using' f itness functjons. _{Às we} shall ^see ,

thev greatly

influence

the behavionr of

genetic algolithnrs.

llre

t¡est hltou'n sclcction

algolithm

is proportional selection definccl

by

úsr(er,

f) -

^tt'(n)_il'(t)

u'ltete,i7(l) denol,es

the

average fitncss of the indivjduals ^ir-L tltc.¡ropulation al, rnornent

f.

The

filst chalacterization of

gcnetic algorithnts,

given by Hollancl [3] ^is

^basecl

^on propoltional

selection.

In _[2]

Grefenstette and.

Ilaket

shol'ecl

tbat

evcn

fol verv

simplc fitness functions (linear ones) the

interpletation

of Hollancl 's Sclterna 'Ihe- orern is

not

clcar. Accolding

to

them

this

^problem

^js

^clue

to the fact

that,

the

l,heotern refers

to

the fitness

function,

lvhich

is

a desìgn pararnetel of

the

genetic

algolithm, instead of gir.ing a

chalacterizabion

in terrns

of

the obiective function. they

suggest

that

charactelizations

of

^genel,ic

alg-olithms shoulcl

st,ate((hor.the

space dcfined

b]'the

objcctive funci,ion

is

sealched Jrv

thc

genctio algolithrn').

In

the rest of

this

sectiorr ure gir.e a shcllt plesental,ion of

thc

l'csults

in _{[2]. flo}

makc cliscussion sirnple

in

the follorvings s'e shall c,onsider' (rvit-

hout

loss of genelalit¡r)

that /

^is

^to

bc maximized.

Dot¡rtitttloN 1.1.

A

_f,itness Ju,nction,

u is

r¡'¿onotoníc iJ' thc follotoi,n'g condition, ltolds:

T¿(ø)

(

u(u)

i.fÎ /(r) ^<

^/(y).

Note 1.1.

As

shown

in _{[2] this}

class

of

rnonotonic Ïii,ness functions inclucle many

freqlentl-v

used fitness ïunctions, tlrus

it

constitutes a rta-

tural subject of

sturl¡..

DnrrnnroN

1.2.

A

selection algoritltm,

is

^{n't o n' rs} t ^o n

i c iJ it

ussigns

u

target sunpLin,g rate to each'

ittdiaidual,

^at,

alty

momeltt ^s'uch tllo,t

tsr(n,t) (

/sr(r¡,

t) ifJ

u,(n)

<

^u(y).

Note l.2..Plopor'1,icural

selcctìon,

selection b-v rattliìnp¡,

as rvoll

^as

nrân.\'

other

l<norvn sclecbion

algorithms ale

monotonic.

l.he.results

pr.eserrtecl

in the

follor.l,ings

rvill

concern genetic algo_

ritltm,s usllg a

monotonic selection

algoritrin and a

rnonolonic fitn"ess f-u19i]9n' 1.o

give the central

char-actefization

of tl'ris sectio¡

one rnoïe

definition is

^neecled.

Dn¡'rNrrron

l.B.

Ghen, the p.tpsttlntion,

p(t)

_{Loe say r,ltnt} the lryperltlane

H,

i.s d,

o,t i,,

tt,t c

cl

by the hy1te.íptane

Hr,

ìtenoting

it by A, {

^o,,H,

6

max {/(r)lr

^e

tt, n ^p(r)} ç ^rnin{/(r) lr

^{e EI,}

ⁿ

^p(ú)}.

Non

$'e can gir.e

the following lesult (t2l)

:

Trrnonnu 1.1

. rn

cr,ry1 ç¡enetic argotitrrn't, using

a

motroto,nic selection a,Igorithm and

u

mt¡norc¡nic fitricss Juctctiott,,

for

^any "ltyperltl,a,nes, EIr,

Í1,

^,in,

P(t)

E[, 4 ø,,H2+

Lsr(IIr, ü)

ç

úsr(Er, t)

t'he ploof of Theoren

1.1

is

basecl on

the rlefinitions

and

is

straightfor- wal'd.

s a

x.eaker characterization

of

the

nn¡'rx*rox

r.4.4,rtrlyterprcnt,e Fr,

is

c

on,¡tr eterq

d ont,,ino,t,ed,

b1y cr,ttotlret ltEperplane

Hr,

ãcnàtelt by

H, _< ;Hrii.f

^it

tnax{./(ø)ln, e

Hr,\ (

rnin{./(rr,-) ln e

trIrl.

Tire follou'itrg

cor,olla,r.y holds :

conolr,¡.nv r. r.

rn, cLny gener,i,c ctrgoritrnn u,s,itttl

u

tnonotonic serection algorithnt cut,ll o, ntonotottic

filnels

functioit,, Jor ^cnty h.yþe,rltlarnes

H,

rnd, n{,

IIr < Hz

=+ V(f)tsr(I{r, ú)

( fsr(ãr,

t).

lhis lesult

_states

that

unc{er,

the

gir.en conditions -H, glorvs

nt

rãu*t as_fasb as- .Hl does

in

arr.\¡ g^euerartion,

in

any genebic aigoriihm

of the

con- siderecl class.

fn the

encl of llre,ir paper, Grefenstette ancl

Bal

search

fol'

corrclitions n,hich allow chalactelizations _of

fhe s

lection

algolithms.

X'he

next

sect,ion prcscnts _soure

of oul

dilec_

tion.

3. GüL\Iì'I'IÍ .1\Lf,iOIüTIIiU,S USING t'Ii{)I,OtÈ'i,[ONl\L S]ìf,ntìl,ION

. ^{trn the ìrlevious}

^sect,ion

^a clmractcrization of

genet,ic algorithms using

a

Dronotoriic selection

algolitlun and a

monobonic fibness

ïunction

was given.

This

covers

a

bloacì class

of

genctic algorithms

"sc,i i"

^p.àc_

2

-

^{c. 1080}

On Genctic Algor.itlrrrrs t7

tice. Ilol,ever

^{these l'esulls can}

^lleither gile

bouncls on

the glorvth

^{of the}

¡epreserrt,a|irres

of

h¡'pelplanes,

llol' captlr'e sensitiyit¡'

^aspects

tlÏ

^genetrc

algolithns.

we

claiur tha1, one possiblc l'eàson

for

these cleficiellcjes

is thal' in the

^clttss co.sicLeÌecl

soÌle;clection

aly,^olitlt'lS

nla¡' "\'oì'ì(

.-tgàillst" ^Solne c1¡alities

of celtain

^fitncss

fullctions.

'l'o

illustlate this

lct, us collñiclel tne

fitness function

æo

:

log(J(ø))

- ^1,

^e

^<

^J(t':)

{

^a¿

fol all

^{possìble o}

(rvhich

js sirlilar

19

¿¿:

^Ò

- ^tog(/(r))

co¡sitlelecl

i1

_l2.l) rvhele

/ ^{is to}

^be

rnaxirnizecl.

this

litness

fu¡ci,io1

has soure tlice pr-operties

I'hich

^rnake

it

^aplleáù-

Iing,

such as

a. il,

^r.ecluccs

the

clanger ^o1 pl',eÌra,trU'e Con\¡el'getlce ll.-v clatnping out cliffererrces l:ets'celr

lalge

I'alucrs of .

b.

^mahes

^

ts^t"*1, ãifference

be

^J'(r)-'

Iìor a

^{g^enetic}

^lgnt:itht

anal

the

fitness fùnc1,ion ^{?l,o 1\¡e}

ca

^mo-

notonic

sclecbion

ntg'oilttrins I ut

Let for

^jnstance

tbc

^tar:ge1, satnpline

lale

be clefinetl

ìly

üsro(r:'

q: !!-,

MÁT)

rvlre}o

][r(t) : ruar{./(r)ln

^e P(t,)}.

s[bstjtul,jug

^?¿o

ftÌ'

^¿¿

\\t¡

o.btain

¿rrn.tl

.il ^*.¡

lst'o(;1,

,, :

*

rr,,

: jl(Ð '

îhe

genetic algor,ithur using ^tsi'o aurl ^ir,o uses â

linear

selectjon rvhich laoks

the

a,bove-nen1'jonctl pl'ollerties.

For

^{1,he l,càson} lireniionecl ¿ncl illusl,ratecl above rve

shall

^clisouss

genetrc'-alg'ot'i1,hms

s,hich

uses

plopoli,ional

selection. Obviously

this

se-

iu.6or-, aìgit1i1,ì¡n

is

rnonoton;c ãnd

in

atlclitioarr

it

has

the

follorvillg pro-

On lìclrctic Âlgoli [ìrnrs 19

Note 3.1. Oþrriousl¡r

all

1,he lesults

in the

plevions section are v¿licl

for ^{l'SGA-s using}

^;u rnotlotonic fit'ness

futlction'

r\s shou,lt;ñ

¡21, r'esults

sirnilal to

those prese,ntecl

in thc

previous

soctjot

^{can be}

o'lrtaiñcci

ronotonic fitñess functions ancl

strictly

rnonotonic selectr'on

al ^he

follorvings

by

rnonotonic fitness iuncl,ion u,c shall

rnean

^t¿'¿c ones. OllviorLsl¡t pl'opoltion¿ì'l ^Se-

lection js

sl,r'jtl,lv

rloltotonic.

Tret

¡s

norv tr'.y

to give

a chalactel'jzation

for llSGÄ-s

usitlg ^ar lnollo-

tonicfitness function.lüi.h

propeltjcs sirnilal to those of tt'o a'bove.

fn

ol'dLer

to

^c1o

t}is

^rve neecl sorne rnol'e ilefinitiolts.

J)nnrrqrrroN

3.\.

lVe ^{søu thctt} tt

is

^rc cottlec) _Jilness .furtctio'tt,

iJ for

^øn'y

#t¡ fiz

ancl n,

[./(ø,),

l(nr), :f@');

^I'L)

>

^o

na-here

ln, ^y, z; h,l

^denoles lhe secontl r,trtler clíticlect difference' ^{aJ t'he} tun'ction

It in, nr,

fizt ïs.

Tlrc _fit'ness Junctiott, ^t¿ i,s ^su,ícl, Io be cottcctue

if

Jr'tr

aN!

tr1, n, and n"

l/(rr),

J(.t:r),

J(rr); øl <

^0.

the followins

trvo theoÏerus

give

a

plopelty

of PSG-{-s usìng a mo-

notonic

ancl convex, r,espectivcly .-(]oncave

fitness

furtction,

rvhich ^mahes them jnteresting frotn the

point

of vierv of s1.¡cly

of

sensitiveness.

Tlrnonn¡vr

3.I'

^{ß'ot' r¿}

^tr'SGÄ

^trsirtrt

tt

ntrtttolottic att'd conuet;^Jilness .frr,tr,ct'ion

u,

_Jor ^tt'rx'y

qi r,

^ccn'cl

r"

^ítt'

P(ti

^{suclt' tlmt}

/(¿i) < _/(r:r) <./(rt)

JTør)

- ^l(*r)

^{> l@n)}

-/(rr) - ^tst'(n,t) - ^tsr(r,f) )

^{fsr'( üz,t)}

-

^Úsr('rt,f)

'I'he col'r'esponding

result

Tol: conc¿ut'e fitness

f¡nctions is

^{given loy}

Trrnottnu 3.2. For

rl,

PSGÄ

l;s'i"tl(l (c nt'ott'ottttt'ic tt'n'd concaue Jitness fu,ttction ^u,, Jor ^{tutuy ç)1)}

t;,

ctntl

r,

^'i,tt

P(ti

^stt'clt, lltat

T(*t) <

_J@")

< _f(nt)

l(r") _-J(ør)

^<

^î\rr) - ^T(sr) - ^lsr(r'

^t')

- ^{tsr(n^f)} (

fsr'(ør,

t) - ^tsr(trnt\

Since

the

_lrroofs

of the trvo

theolems at'c absolutely analogous we

shall only ^give

^1,he

^proof for tlrc lattel

^one.

ProoJ. (lìheorem

3.2.)

Since

u js

conc¿ù\re'

ï'e

^ha,\'e

llØ;'), !(n"), Jþ;'); øl ^<

^o,

r,vhich

by the

clefinitión

of the

clividecl cliffelenee ^is u,(n")

- tt(ur)

^u'(ur)

-

^u(nr)

.f(uo)

- ^{l(*,) l(a,\} - ^{lþ') <0.}

l@) -

^l@,)

18 4

perty

lI. l.l. ì:lal hz-s

For

any lr1'perpklne

II , rú ^any

^{rnoment f}

/sr'(e;.

fl

tsr(

lI.l,\ - ç _fr¡1

_{n(H, t)}

-

^ru(r)

_s. - _Èn

uQ)n(H,t)

u'(H, t,)

tr(t)

rvhere

u(Hrt) is the

^¿ùvela,ge

^fitness of ^the

lepresetrtat,ives

of

-È1

in P(fl'

lYe

shali cád such

geneticäIgoûthms

Proporl,ional Selection ^Gonetic

Al-

gorithrns

(PSGA).

2D

llf . E. Ilalázs _{6 I}

Using the

rr¡-pothesis

that l(*rl < I(nr) </(nr.)

we oLrtain u(n")

-

^nt('T,\

l@"\ - ^l@,)

wìrich, sinc¿ ø

is

monotonic,

is

eç¡uir.¿i,laut

to

a nrcrrrbcr _ttll

p(t)

_{snclr tll¿1,} f/.1.n¡in\

_

2'

j-,

_{Lrrt_t}

b¡'

.¡,'l'i'

o ,',n,rlìrLí';i

_þ(ú)

n ])(l)Ì

for,

i : l, 2. B. '- -

\v,'

cle

of the

co¡rclusiolr,

of the

ilreoreru

¡,vlriclr

is the

sa,rne

a,s l(u?^") *

'f(ø1*^")

(

2/(ei't"), ./(æi

-")

-

,/(e,à",") < ./(ø1,,,,,)

_

.f(,,1.,n_*).

tsJ'

the

coneavitv

of

tJre fjtness

functjon

¿¿

ilrjs

inrpplies t.t(nl'"^)

- u,(r|",) {

¿(r1,,r,,)

_

_{¿( ø.1,n*),}

u'hich is

equivalent

to

O¡l Genctic

^l^{golitht¡ s}

tt(r)

, tt([I r, y)

27

nr(nrl

- ^{tr,(rr) ç þt(nrl} - ^u,{*r\l fl.ì_Jt.¿

_.

Í(r,t -

^J@,1

Tlre conclition

/(ør) -

^T@r)

<,/(rr) _ l(*r)

^means

ilrat l(nrl -

^.l(n,l

l(n,l -

^l@,1

by

rvhich the previous eclualit¡. l:ecornes

u(nrl -,u(ør) ç u(rrl - ^u,{rrl.

Divicling

bhis inecluality b), ^,r?(f) _rve

olttain

tst'(rr,

t) _

_tsr.(rn

_f) (

fsr(e;¿,

tl _

_lsr(xr,

_tl,

s.an

intuil;ivc

cxptr..r,rratiorr rvhy sca_

in

sclection

pr.actice. ¹

^"

vex

ancl concal¡e selecl,ion

algolithr is that in oul future l,olh ive int

seusili,r'if¡'

of

¡nclrLlcls of a cìass of g

{ilsl.

sto¡r

in tìlis

tlil,cc,l iort n orrlrl l¡e, l-,r¡'

llrc lif

ncss f tulcLjt¡u [or, 1 ltc sarrrc

otonic and concave fitncss function,

the

con.r,erx case bcirrg siruilar.

'if ,i,::;'ií:"i;"Y(Tf ,i:;;"':r:;J'p,'z

max

{/(r)ln

^e

il, 0 ^p(r)} f ^max{/(ø lln

^e

H, n

I_,(r)} <

(

2

rnin{/(r)ln

e

fI, n

^J?(ú)}

+

+ tsr(ÍIr, t) _

_{tsr(Hr, ú)}

(

tsr,(

tr _,, t) _ _tsr(Iil.t)

u(IIr,t) _ _{u,(IIr,t) <} u(H,t) _,(Ht,t),

wlrich dtivi¡teet

by û,(t)()

_{0) gives}

tsr(fIy t) _

_tst(Ilz,

_{f) <}

_{ts{}7 ,,}

_t) _ _tsr(II[

_t).

riTo"

o';

'i;(íili.ïi" ,fii,ï,,íi,,!,1

_, !;,,"îi,:, t

t4H1,t)

\-

./_)

i: I

;Ur't)'

^t.r(øi'n')

Sincc

r¿

is

monot,onic ^-w,e havc

that

is

,Ð,^iffi-,p,,, ffi* "\,

lsr(Hr,

t) ) I

^{-l- ,r,f" irr}

rrr, ..f ttt)

,tt(II,

_t)

u'(æ)

-

(.l( e,l"'')

- .l(øi"'))

lsr(IIr,, _t),

f vDìin -_tnAr,

tt¡here

tn',,'t

^t _J

I

l(a) - J(,ti,")

^{J ø e} ferfrin,;¡lÌ'ax) : Illtll

rt(ri)")

I Ol¡ Cìeuctic

^li3-orillulì.s

4. {:O\(:f,TtSIONsi ÀNI) IrU't'UIIIì \tOltIi

c.) l{ . E. Ilalázs

'I¡rnonnlr 3.5. Ior

^¿r

PSGA

lrsitr{J

a

?nt)motonic ^(x,tr{Í, concalie filness funct'ion ^{xL ctlxd,}

for

lryTterpianes

IL

&n(t ^"Hz i.n

P(t)

^sualt tlre,t I-11(r,.r

Ilrllte

Jollowing in'ec1uctl'ity

is

^{trnLe :}

f

^{^"}

('!t"'

_'lto*l tsr'(I1.,

t)> _l1 ¡

^1!t" .

-- - '

(./(oi"")

-,/(n'Ì'^*))ltsr(I1.,

^t),

I

^rr(

^II,, f)

J

,

(,jt"t, ,!'**l

-

^{rrri rr}

J l(ri"") -

^,r¿(ø)

_lø

e (a;f,r,,,

fi,,-] _Ì

^.

u)tte?'?

tltu

{ ¡*i*,¡ 1¡") I'

Note 3.7. Since the l':ìhtcs of

/alc

consitlclcd

to

be l¡ountlcct artd ^{t¿ is} rnonotonic

both of

i,hc consiclelecl

minitnurns

exisb.

\Ye give 1,he proof

fol

^Tlteorem 3.5,

for thc

cotrespontllrLg one

rvith

convex

fitness function the proof is

arialogous.

Prctof. ('l'heolem

lì.5.) Sincc

r¿

is a

cìonc&\re

fitness function,

using

the sane

r"Lcltation

as

aì:or.e 'vve have [.f(ci"""),./(rt!""'),

Í(n); u] (

0

for

^nl ^n:l(ø)

^e (/(a.,ï''"), "f(¿]"n')1,

that

is

. fl'hc

main poìnt

of tJris paper' ìs

that thcle is a rather

large class of genetìc algorithr-ns

for-n'hich

some

sensitivilr- plopelties ca¡ lié

estaSlis-

hed

cle.pending on

the natulc of lhe

Lrnclelil'ìirg'^fitness

frnctio¡. As in orll

rnairì r'efelence

[2]

^lire

lcsulls

are folrnulaicci'in _{ter,ms of}

the

objegtiye function.

llhe intclpletatìon of thc

mentioned

resulls shol. that

fclr I,SGA-s usitrg nlonotonÌc fitness

functions tle lattels

shoulcl tre chosen

to lte

con-

opclat

conr¡ex

in tlie

aclr.anced ones. This

useful

tness

fu¡ctiol

clependitrg on

nehow

ogress oT sc¿lrch. Conceriring

¿rt

least l-hich

mar' :¡r.ise :

a. tfotY to

clesign

a

(parametrìzetl) fitness

function to

e,xploit the preserrtecl resulis ?

oultl

the_

palametcr l'liich dirccts scalirc' be

ancl þorr'

shou

cluring 1,hé operaLion of

the

genetic algor.ithrn?

lialler,\\'e ^sh¿rll

tlv to

give sonre possìblè anss,els

to

t,fuese

tï,o

IÌIjIì]JIìENCìJS

1. Iìalázs, \'I. li., Ott an l.):t¡terinrcrtl for tjsirtq (]enctic AlooLitlLnts for ^Soluin(t ]iqLLnliorts, subrnit- tccl to Slrrdia lirir'. llaìrcs-tìol¡.li (19()l _).

2. (ìr'cIcrrstcLtc, .I. J. anrl lìul<ct., J,li., Itotu Genctic A|got i\tnts ll:orl¡ ; A Ot ilical l.ool: ul IntpIi_

cil Pcu'rtLLcIisnr, l)r'occcrìings of I(jÇ;\,g9, lirl. Scìralfcr. (J ggg).

3. F{olland, J.Il., AdopLctliott in nalural ancl utlil'icial srTslcnrs, rinn .ilbor,: I'nir.cr.sit.r. I\Iic¡iga¡

l:,r'css (197õ).

23

u(n) -

tl(r12""')

l@) -

"f(ø'j'"')

_

,u(rä"'")

-

^ø(ø'f'o*)

,/(r|"") - .f(r'i"") (0

for any # ^: f(rc) ^e (/(øÏ""),"f(r'i'-.)

Í(n) -"f(øi''.)

î'his is tlte

sarnc

lvith

1r}rrt, r|.exl

tn'(H",1) > ^tsr(Hþl) f

^{,ITI U (./(} øä""')

-

^./( rìi'""))

l'ìcccivotl 27 \ III ^199:i llrtlttl-JlrLl qri L-rtItcr silg Dtpru Ltttcttl r;l llulltentaLics rrt:ti

IttIotntulics 51r. .l/. Iioqrrlttirctttttr 7,

3400 Clnj-Ìt-a¡trsta, lìo¡ttìn ia

'l','hich

is the

same

u'ith

/-lrllll,.lll&fr

.tsr(I{r,t) >

^L

+t,li,'

^_'-"' _'(/(r,1,,^)

_./(øi.-)).

fsr(

flr' 1) tr(ll

_t, ^l)

.l'his

latter

incqtLal.ib¡. is equivalent to the oüc ì\'c ltatt

to

proYe, i.e.

_ (.'Ðt"t,"tjto*l

tsr.(t1,,,,,

It * ""*

(,ftn.i"") --.f(,rÏ'o-)) Itsr'

(//,,

l),

Thus

fol

I,SGA-s u'hich uses ^¿ù rnonotonic antl cither' ¿ìi oonvcx

or

^a

cronca\¡e litness

function

n'e coulcl give il' bound of l,ltc

l'olative glowth

of hyperplzr,nes

in

^P(ú).