C Multiparent Recombination

AE Eiben

Leiden University

Abstract

In this section we survey recombination op erators that can utilize more than two

parents to create ospring Some multiparent recombination op erators are dened

for a xed number of parents eg havearity three while in some op erators the

number of parents is a random number that might b e greater than two and in yet

other op erators the arity is a parameter that can b e set to an arbitrary integer We

pay sp ecial attention to this latter typ e of op erators and summarize results on the

eect of op erator arityonevolutionary algorithm p erformance

C Intro duction

To make the coming survey unambiguous we have to start with setting some conventions on

terminology The term population will b e used for a multiset of individuals that undergo es selection

and repro duction This terminology is maintained in genetic algorithms ev olutionary programming

and genetic programming but in evolution strategies all individuals in a or strategy

are called parents We however use the term parents only for those individuals that are selected

to undergo recombination In other words parents are those individuals that are actually used as

inputs for a recombination op erator the arity of a recombination operator is the numb er of parents

it uses The next notion is that of a donor b eing a parent that actually contributes to at least one

of the alleles of the children created by the recombination op erator This contribution can be

for instance the delivery of an allele as in uniform crossover in canonical GAs or the participation

in an averaging op eration as in intermediate recombination in ES As an illustration consider a

steadystate GA where individuals form the p opulation and two of them are chosen as parents

to undergo uniform crossover to create one single ospring If bypurechance the ospring only

inherits alleles from parent then parent is a donor and parent is not

C Miscellaneous op erators

We b egin this survey with pap ers where the multiparent asp ect has an incidental character By an

incidental character we mean that the op erator is dened and used in a sp ecic application and has

for instance a certain xed arityoreven if the denition is general and would allow comparison

between dierentnumb er of parents this asp ect is not given attention

The recombination mechanism of Kaufman is applied for evolving mo dels for agiven

pro cess where a mo del is an arrayofa numberofblocks and mo dels may dier in the numbers of

blo cks they contain Recombination of four mo dels to create one new mo del is dened as follows

The size of the child the numb er of blo cks equals the size of eachofitsparents with probability

The ith blo ck of the child is chosen with equal probability from those parents that have

there is an exception of this latter rule of cho osing one of the at least i blo cks Let us note that

parents blo cks but that exception has a very problemsp ecic reason therefore we rather present

the general idea here

In an extensive study on bit vector function optimization sto chastic iterated genetic hill

climbing SIGH is studied and compared with other techniques such as GAs iterated hillclimbing

and simulated annealing Ackley SIGH applies a sophisticated probabilistic voting mechanism

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IOP Publishing Ltd and Oxford UniversityPress Handbook of Evolutionary Computation C

with timedep endent probability distributions co oling where m voters m b eing the size of the

p opulation determine the values of a new bitstring SIGH is shown to b e b etter than a GA with

p oint and uniform crossover on four out of the six test functions and the overall conclusion is that

it is comp etitive in sp eed with a variety of existing algorithms

In the intro ductory pap er on the parallel genetic algorithm ASPARAGOS Muhlen b ein

psexual voting recombination is applied for the quadratic assignment problem Let us remark that

the name psexual is somewhat misleading as there are no dierent genders and no restriction on

having one representative of each gender for recombination The voting recombination pro duces

one child of p parents based on a threshold value v It determines the ith allele of the child by

comparing the ith alleles of the selected parent individuals If the same allele is found more often

than the threshold v this allele is included in the child other bits are lled in randomly In the

exp eriments the values p and v are used and it worked surprisingly well but comparison

between this scheme and usual twoparent recombination was not p erformed

An interesting attempt to combine genetic algorithms with the Simplex Metho d resulted in the

ternary simplex crossover Bersini and Seront If x x x are the three parents sorted in

decreasing order of tness then the simplex crossover generates one child x by the following two

rules

i If x x then x x

i

i i i

ii if x x then x x with probability p and x x with probability p

i i

i i i i

Using the value p the simplex GA p erformed b etter than the standard GA on the DeJong

functions The authors remark that applying a mo died crossover on more than three parents is

worth to try

The problem of placing actuators on space structures is addressed byFuruya and Haftka

The authors compare dierent crossovers among others they use uniform crossover with twoaswell

as with three parents in a GA using integer representation Based on the exp erimental results they

conclude that the use of three parents did not improve the p erformance This might b e related to

another conclusion indicating that for this problem mutation is an ecient op erator and crossover

might not be imp ortant Uniform crossover with an arbitrary number of parents is also used by

Aizawa as part of a sp ecial schema sampling pro cedure in a GA but the multiparent feature

is only a sideeect and is not investigated

y Pal for a multimo dal spin A socalled triadic crossover is intro duced and tested b

lattice problem The triadic crossover is dened in terms of two parents and one extra individual

chosen randomly from the p opulation The op erator creates one child it takes the bits in p ositions

where the rst parent and the third individual have identical bits from this parent and the rest of

the bits from the other parent Clearly the result is identical to the outcome of a voting crossover

on these three individuals as parents Although the pap er is primarily concerned with dierent

selection schemes a comparison between triadic p oint and uniform crossover is made where

triadic crossover turned out to deliver the b est results

C Op erators with undened arity

In the intro duction to this section we dened the arity of a recombination op erator as the number

drawings the parents it uses In some cases this number dep ends on the outcomes of random

op erator is called without knowing in advance how manyparents would b e applied In this section

we treat three mechanisms of this kind

Global recombination in evolution strategies allows the use of more than tworecombinants Back

Schwefel In ES there are two basic typ es of recombination intermediate and discrete

recombination b oth having a standard twoparentvariant and a global variant Given a p opulation

of individuals global recombination creates one ospring x by the following mechanism

S T

i i

x or x global discrete recombination

i i

x

i

S T S

i i i

combination x x x global intermediate re

i

i i i

S T

i i

where the two parents x x S T fg are redrawn for each i anew and so is the

i i

contraction factor The ab ove denition applies to the ob ject variables as well as the strategy

i

parameter part ie for the mutation stepsizes s and the rotation angles s Observe that

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the multiparent character of global recombination is the consequence of redrawing the parents

S T

i i

x x for each co ordinate i Therefore probably more than two individuals contribute to the

ospring x but their number is not dened in advance It is clear that investigations on the

eects of dierent numb ers of parents on algorithm p erformance could not be p erformed in the

traditional ES framework The option of using multiple parents can be turned on or o that

is global recombination can be used or not but the arity of the recombination op erator is not

tunable Exp erimental studies on global versus twoparent recombination are p ossible but so

far there are almost no exp erimental results available on this sub ject In Schwefel it is

noted that appreciable acceleration is obtained by changing to bisexual from asexual scheme

ie adding recombination using two parents to the mutationonly algorithm but only slight

further increase is obtained when changing from bisexual to multisexual recombination ie using

global recombination instead of the twoparentvariant Recall the remark on the name psexual

voting The terms bisexual and multisexual are not appropriate either for the same reason

individuals have no gender or sex and recombination can be applied to any combination of

individuals

Genepool recombination GPR was intro duced byMuhlenbein and Voigt as a multi

parent recombination mechanism for discrete domains It is dened as a generalization of twoparent

recombination TPR Applying GPR is preceded by selecting a genep o ol consisting of wouldb e

parents Applying GPR the twoparent alleles of an ospring are randomly chosen for each lo cus

with replacement from the genep o ol and the ospring allele is computed using any of the standard

recombination schemes for TPR Theoretical analysis on innite p opulations shows that GPR is

mathematically more tractable than TPR If n stands for the number of variables lo ci then the

n

evolution with prop ortional selection and GPR is fully describ ed by n equations while TPR needs

equations for the genotypic frequencies In practice GPR converges ab out faster than TPR for

ONEMAX The authors conclude that GPR separates the identication and the search of promising

areas of the search space b etter b esides it searches more reasonably than do es TPR In Voigt and

Muhlenbein GPR is extended to continuous domains by combining it with uniform fuzzy

twoparent recombination UFTPR from Voigt et al The resulting uniform fuzzy gene

pool recombination UFGPR outp erforms UFTPR on the spherical function in terms of realized

heritability giving it a higher convergence sp eed The convergence of UFGPR is shown to b e ab out

higher than that of UFTPR

A very particular mechanism is the linkage evolving genetic operator LEGO as dened

in Smith and Fogarty The mechanism is designed to detect and propagate blo cks of

corresp onding genes of potentially varying length during the evolution Punctuation marks in

the chromosomes denote the b eginning and the end of eachblock and more chromosomes with the

appropriately p ositioned punctuation marks are considered as donors of a whole blo ckduringthe

creation of a child Although the multiparent feature is only a sideeect LEGO is a mechanism

where more than two parents can contribute to an ospring

C Op erators with tunable arity

Unary repro duction op erators suchasmutation are often called asexual based on the biological

analogies Sexual repro duction traditionally amounts to twoparent recombination in EC but the

op erators discussed in the previous section show that the sexual character of recombination can

be intensied in the sense that more than two parents can be recombined Nevertheless this

intensication is not graded the multiparen t option can be turned on or o but the extent of

sexuality the number of parents cannot be tuned In this section we consider recombination

op erators that make sexuality a graded rather than a Bo olean feature byhaving an arity that can

vary In other words the op erators we survey here are called with a certain numb er of parents as

input and this numb er can b e mo died by the user

An early pap er mentioning multiparent recombination is that of Bremermann et al

on solving linear equations It presents the denition of three dierentmultiparent recombination

m

mechanisms called mtuple mating Given m binary parentvectors x x themajority mating

mechanism creates one ospring vector x bycho osing

j

if half or more of the parents has x

i

x

i

otherwise

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Another mating mechanism for m binary parentvectors is called mating by crossing over Describing

it in contemp orary terms the mechanism works by selecting m crossover p oints identical in

eachparent and then comp osing one child by selecting exactly one segment from eachparent The

third op erator is called mating by averaging and it is dened for vectors of continuous variables

m

Quite naturally the child x of parents x x is dened by

m

X

j

x x

i j

i

j

P

m

is rep orted on the p erformance of these where Unfortunately only very little

j

j

op erators It is remarked that using ma jority mating and mating by crossing over the results

were somewhat inconclusive no denite b enet was obtained Using mating byaveraging however

led to sp ectacular eects within a linear programming scheme but these eects are not sp ecied

Scanning crossover hasbeenintro duced as a generalization and extension of uniform crossover

in GAs creating one child from r parents Eib en Eib en et al The name is based on

the following general pro cedure scanning parents and thus building the child from lefttoright Let

r

x x b e the selected parents of length L and let x denote the child

pro cedure scanning

begin

INITIALIZE p osition markers as i i mark st position in each parent

r

for i to i L

CHOOSE j frg

j

j

x x ith al lele of x is the i th al lele of x

i j

i

j

UPDATE p osition markers i i

r

end

The ab ove pro cedure provides a general framework for a certain style of multiparentrecombination

where the precise execution hence the exact denition of the op erator dep ends on the mechanisms

to CHOOSE and to UPDATE In the simplest case the UPDATE op eration can shift the markers

one p osition to the right ie i i j frg can be used This is appropriate for

j j

bitstrings integer and oatingp oint representation Scanning can also b e easily adapted to order

based representation where each individual is a p ermutation if the UPDATE op eration shifts to

the rst allele whichisnotinthechild yet

j

i x fx x gg j frg i minf k j k

j i j

j

k

Observe that b ecause of the term k i ab ove a marker can remain at the same p osition after an

j

UPDATE and will only b e shifted if the allele standing at that p osition is included in the child

This guarantees that each ospring will b e a p ermutation

Dep ending on the mechanism to cho ose a parent and thereby an allele there are three dierent

versions of scanning The choice can be deterministic cho osing a parent containing the allele

with the highest number of o ccurrences and breaking ties randomly occurrence based scanning

Alternatively it can b e random either unbiased following a uniform distribution thus giving each

parent an equal chance to deliver its allele uniform scanning or biased by the tness of the parents

where the chance of b eing chosen is tness prop ortional tness basedscanning Uniform scanning

for r is the same as uniform crossover although creating only one child and it also coincides

with discrete recombination in evolution strategies The o ccurrence based version is very muchlike

the voting or ma jority mating mechanism discussed b efore but without the threshold v resp ectively

with v bmc The eect of the number of parents in scanning crossover has b een studied in

several pap ers An overview of these studies is given in the next subsection

Diagonal crossover has b een intro duced as a generalization of p oint crossover in GAs Eib en

et al In its original form diagonal crossover creates r children from r parents by selecting

crossover points in the parents and comp osing the children by taking the resulting in r r

chromosome segments from the parents along the diagonals Later on a onechild version was

intro duced van Kemenade et al Figure C illustrates b oth variants It is easy to see that

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for r diagonal crossover coincides with p oint crossover and in some sense it also generalizes

traditional parent np oint crossover Tobepreciseifwe dene rssegmentation crossover as

working on r parents with s crossover p oints diagonal crossover b ecomes its rr version its

nvariant coincides with np oint crossover and p oint crossover is an instance of b oth schemes

for rs as parameters The eect of op erator arity for diagonal crossovers will be also discussed in the next subsection

parent 1

parent 2

parent 1

parent 3

parent 2

parent 3 child 1

child 2

child 3 child

Multiple children One child

Figure C Diagonal crossover left and its onechild version right for parents

A recombination mechanism with tunable arity in ES is prop osed by Schwefel and Rudolph

The ES provides the p ossibility of freely adjusting the number of parents

called ancestors by the authors The parameter stands for the number of parents and global

recombination is redened for anygiven set fx x g of parents as

j

x ary discrete recombination

i

P

x

i

k

x intermediate recombination

i

k

where j fg is uniform randomly chosen for each i indep endently Let us note that in the

original pap er the ab ove op erators are called uniform crossover resp ectively global intermediate

recombination Weintro duce the names ary discrete recombination resp ectively intermediate

recombination here for the sake of a consequent terminology A reason for using the term

intermediate recombination instead of ary intermediate recombination is given below in the

paragraph discussing the pap er Eib en and Back Observethat ary discrete recombination

ver while intermediate recombination is a sp ecial case coincides with uniform scanning crosso

of mating byaveraging At this time there are no exp erimental results available on the eect of

within this framework

Related work in evolution strategies also uses as the number of parents as an indep endent

parameter for recombination Beyer For purp oses of a theoretical analysis it is assumed

that all parents are dierent uniform randomly chosen from the p opulation of individuals Beyer

denes the intermediate recombination and ary discrete recombinations similarly to Schwefel

and Rudolph and denotes them as intermediate recombination and dominant

I D

recombination resp ectively The evolution strategy is studied on the spherical function for

the sp ecial case of By this latter assumption it is not p ossible to draw conclusions on the



eect of but the analysis shows that the optimal progress rate of the ES is a factor

higher than that of the ES for b oth recombination mechanisms Beyer hyp othesizes that

recombination has a statistical error correction eect called genetic repair and this eect can b e

improved by using more than twoparents for creating ospring Beyer

Another generalization of global intermediate recombination in evolution strategies is prop osed

by Eib en and Back The new op erator is applied after selecting parent individuals from the

T S

i i

for each i takes only these individuals and x p opulation of and the resampling of two donors x

in consideration Note that this op erator is also aryjustlikethe intermediate recombination

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IOP Publishing Ltd and Oxford UniversityPress Handbook of Evolutionary Computation C

as dened ab ove but utilizes only two donors for each allele of the ospring To express this

dierence this op erator is called intermediate recombination and the op erator of Beyer

and Schwefel and Rudolph is called intermediate recombination Observe that the

intermediate recombination is a true generalization of the original intermediate recombination the

case of coincides with lo cal intermediate recombination while for it equals global

intermediate recombination

While intermediate recombination is based on taking the arithmetical average of the realvalued

alleles of the parents the geometrical avarage is computed bythe geometrical crossover Michalewicz

et al present the denition for any k number of parents where the ospring of the

k

parents fx x g is dened as

k k

 

k k k

x hx x x x x x i

n n n

where n is the chromosome length and The exp erimental part of the pap er is

k

however based on the two parentversion hence there are no results on the eect of using more

than two parents with this op erator

The same holds for the socalled sphere crossover Scho enauer and Michalewicz the

authors give the general denition for k parents but the exp eriments are restricted to the two

k

parents version In the general case the ospring of parents fx x g is dened as

q

q

k

k

k

x h x x x x i

k k

n n

C The eects of higher op erator arities

In the last years quite a few pap ers have studied the eect of op erator arity on EA p erformance

some even in combination with varying selective pressure Here wegive a brief summary of these

results sorted by articles

The p erformance of scanning crossover for dierentnumber of parents is studied in Eib en et

al in a generational GA with prop ortional selection Bitco ded GAs for function optimization

DeJong functions F F and a function from Michalewicz as well as orderbased GAs for graph

coloring and the TSP are tested with dierentmechanisms to CHOOSE In the bitco ded case more

parents p erform b etter than two for the TSP and graph coloring parents are advisable Comparing

dierent biases in cho osing the child allele on four out of the venumerical problems tness based

scanning outp erforms the other two and o ccurrence based scanning is the worst op erator

In Eib en et al diagonal crossover is investigated compared to the classical parent

np oint crossover and uniform scanning in a steadystate GA with linear ranked biased selection

b and worsttness deletion The test suite consists of two dimensional problems F

and a function from Michalewicz and four scalable functions after AckleyGriewangk Rastrigin

and Schwefel The p erformance of diagonal crossover and np oint crossover shows a signicant

corresp ondence with r resp ectively n The b est p erformance is always obtained with high values

between where was the maximum tested Besides diagonal crossover is always b etter than

np oint crossover using the same numb er of crossover p oints r n thus representing the same

level of disruptiveness For scanning the relation b etween r and p erformance is less clear although

the b est p erformance is achieved for more than twoparents on ve out of the six test functions

The interaction between selection pressure and the parameters r for diagonal crossover

resp ectively n for np oint crossover is investigated in van Kemenade et al A steadystate

GA with tournament selection tournamentsizebetween combined with random deletion and

worsttness deletion was applied to the Griewangk and the Schwefel functions The disruptiveness

of b oth op erators increases parallely as the values for r and n are raised but the exp eriments show

that diagonal crossover consistently outp erforms np oint crossover The b est option proves to be

low selection pressure and high r in diagonal crossover combined with worsttness deletion

numerical ob jective functions the Motivated by the diculties to characterize the shap es of

eects of op erator arity are studied on tness landscap es with controllable ruggedness by Eib en

and Schipp ers The NKlandscap es of Kauman where the level of epistasis hence

the ruggedness of the landscap e can b e tuned by the parameter K are used for this purp ose The

multiplechildren and the onechild version of diagonal crossover and uniform scanning are tested

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within a steadystate GA with linear ranked biased selection b and worsttness deletion for

N and dierentvalues of K Two kinds of epistatic interactions nearest neighbor interaction

NNI and random neighbor interaction RNI are considered Similarly to earlier ndings Eib en

et al the tests show that the p erformance of uniform scanning cannot be related to the

numb er of parents The twoversions of diagonal crossover b ehaveidentically and for b oth op erators

there is a consequent improvement when increasing r However as the epistasis ruggedness of the

landscap e grows from K to K the advantage of more parents b ecomes smaller On

landscap es with signicantly high epistasis K the relationship b etween op erator arityand

algorithm p erformance seems to diminish We illustrate these observations with a gure showing

the error deviation of the b est individual from the optimum at termination for the case of NNI

in Figure C The nal conclusions of this investigation can be very well related to works of

Schaer and Eshelman Eshelman and Schaer and Hordijk and Manderick

on the usefulness of parent recombination It seems that if and when crossover is useful ie on

mildly epistatic problems then multiparent crossover can b e more useful than the parentvariants

0.18 6.6 diagonal one crossover diagonal one crossover diagonal crossover diagonal crossover 0.16 uniform scanning uniform scanning 6.4

0.14

6.2 0.12

6 0.1

0.08 5.8

0.06 5.6

0.04

5.4 0.02

0 5.2

2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16

Figure C Illustration of the eect of the numb er of parents horizontal axis on the error at

termination vertical axis on NKlandscap es with nearest neighbor interaction N K

left K right

The results of an extensive study of diagonal crossover for numerical optimization in GAs are

rep orted in Eib en and van Kemenade Diagonal crossover is compared to its one ospring

version and np oint crossover on a test suite consisting of functions monitoring the sp eed ie total

number of evaluations the accuracy ie the median of the b est ob jective function value found all

functions haveanoptimum of zero and the success rate ie the p ercentage of runs where the global

optimum is found In most of the cases an increase of p erformance can b e achieved by increasing the

disruptivity of the crossover op erator using higher values of n for np oint crossover and even more

improvementisachieved if the disruptivityofthecrossover op erator and the numb er of parents is

increased using more parents for diagonal crossver This study gives a strong indication that for

this op erator diagonal crossover an advantageous multiparent eect do es exist that is a using

with more than two parents increases GA p erformance and b this improvement is not only the

consequence of the increased numb er of crossover p oints

A recent investigation of Eib en and Back addresses the working of multiparent

recombination op erators in continuous search spaces in particular within evolution strategies This

study compares intermediate recombination ary discrete recombination which is identical

to uniform scanning crossover and diagonal crossover with one child Exp eriments are p erformed

on unimo dal landscap es sphere mo del and Schwefels double sum multimo dal functions with

regularly arranged optima and a sup erimp osed unimo dal top ology AckleyGriewangk and Rastrigin

functions and on the FletcherPowell and the Langermann functions that have an irregular random

arrangement of lo cal optima On the FletcherPowell function multiparent recombination do es not

increase EA p erformance b esides on the unimo dal double sum increasing op erator arity decreases

p erformance Other conclusions seem to dep end on the op erator in question the most consequent

improvement for raising the numb er of parents is obtained for diagonal crossover

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IOP Publishing Ltd and Oxford UniversityPress Handbook of Evolutionary Computation C

C Conclusions

The idea of applying more than twoparents for recombination in an evolutionary problem solver has

o ccurred already in the sixties Bremermann et al Several authors have designed and applied

recombination op erators with higher arities for a sp ecic task or used an existing op erator with an

arity higher than two Kaufman Muhlen b ein Bersini and SerontFuruyaand

Haftka Aizawa Pal Nevertheless investigations explicitly devoted to the eect

of op erator arity on EA p erformance are still scarce the study of the phenomenon of multiparent

recombination has just b egan What would such a study mean Similarly to the question whether

binary repro duction op erators crossover with twoparents haveadvantages over unary ones using

mutation only it can b e investigated whether or not using more than two parents is advantageous

In case of op erators with tunable arity this question can b e rened and the relationship b etween

op erator arity and algorithm p erformance can be studied It is of course questionable whether

multiparent recombination can be considered as one single phenomenon showing one b ehavioral

pattern The survey presented here discloses that there are at least three dierenttyp es of multi

parent mechanisms with tunable arity

i Op erators based on allele frequencies among the parents such as ma jority mating voting

recombination ary discrete recombination or scanning crossover

ii Op erators based on segmenting and recombining the parents such as mating by crossing over

diagonal crossover or rssegmentation crossover

iii Op erators based on numerical op erations in particular averaging of real valued alleles

such as mating byaveraging intermediate recombination intermediate recombination

geometrical and sp erical crossover

A priori it cannot be exp ected that these dierent schemes show the same resp onse to raising

op erator arities There are also exp erimental results supp orting dierentiation among various multi

parent mechanisms For instance there seems to b e no clear relationship b etween the number of

parents and the p erformance of uniform scanning crossover while the opp osite is true for diagonal

crossover Eib en and Schipp ers

Another asp ect multiparent studies havetotakeinto consideration is the exp ectedly dierent

behavior on dierent typ es of tness landscap es As no single technique would work on every

problem multiparent mechanisms will havetheir limitations to o Some studies indicate that on

irregular landscap es such as NKlandscap es with relatively high K values Eib en and Schipp ers

or the FletcherPowell function Eib en and Back they do not work On the other

hand on the same FletcherPowell function Eib en and van Kemenade observed an advantage

of increasing the number of parents for diagonal crossover in a GA framework using bitco ding

of variables although they also found indications that this can be an artifact caused simply

by the increased disruptiveness of the op erator for higher arities Investigations on multiparent

haracteristics smo othly t into the tradition of studying the eects related to tness landscap e c

disadvantages of twoparent crossovers under dierent circumstances Schaer and Eshelman

Eshelman and Schaer Sp ears Hordijk and Manderick

Let us also touch on the issue of practical diculties when using multiparent recombination

op erators Intro ducing op erator arity as a new parameter implies an obligation of setting its value

Since so far there are no reliable heuristics for setting this parameter nding good values may

require numerous tests prior to real application of the EA A solution can b e based on previous

work on adapting Davis or selfadapting Sp ears the frequency of applying dierent

op erators Alternatively a number of comp eting subp opulations could be used in the spirit of

SchlierkampVo osen and Muhlen b ein According to the latter approacheach dierent arity

is used within one subp opulation and subp opulations with greater progress ie with more p ow erful

op erators b ecome larger A rst assessment of this technique can b e found in Eib en et al

Concluding this survey we can note the following Even though there are no biological analogies

of recombination mechanisms where more than two parent genotyp es are mixed in one single

recombination act formally there is no necessity to restrict the arity of repro duction mechanisms

to one mutation or two crossover in computer simulations Studying the phenomenon of multi

parentrecombination has just b egan but there is already substantial evidence that applying more

than two parents can increase the p erformance of EAs Considering multiparent recombination

mechanisms is thus a sound design heuristics for practitioners and a challenge for theoretical analysis

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References

Ackley D An empirical study of bit vector function optimization Genetic Algorithms and Simulated

Annealing ed Davis L pages Morgan Kaufmann

Aizawa A N Evolving SSE A Sto chastic Schemata Exploiter Proceedings of the st IEEE Conference

on Evolutionary Computation pages IEEE Press

Back T Evolutionary Algorithms in Theory and Practice Oxford University Press New York

Bersini H and Seront G In searchofagoodevolutionoptimization crossover Proceedings of the nd

Conference on Paral lel Problem Solving from Nature ed Manner R and Manderick B pages

NorthHolland

Beyer HG Toward a theory of evolution strategies On the b enets of sex the theory

Evolutionary Computation

Beyer HG Dagstuhl Germany May Basic principles for a unied EAtheory Handout of the

talk given at the Evolutionary Algorithms and their Applications Workshop

Bremermann H Rogson M and Sala S Global prop erties of evolution pro cesses Natural Automata

and Useful Simulations ed Pattee H Edlsack EFein L and Callahan A pages Spartan Bo oks

Washington DC

Davis L Adapting op erator probabilities in genetic algorithms Proceedings of the rd International

Conference on Genetic Algorithms ed Schaer J pages Morgan Kaufmann

Eib en A AMethod for Designing Decision Support Systems for Operational Planning PhD Thesis Eindhoven

UniversityofTechnology

Eib en A and Back T An empirical investigation of multiparent recombination op erators in evolution

strategies Evolutionary Computation

Eib en A and Kemenade C v Diagonal crossover in genetic algorithms for numerical optimization

Journal of Control and Cybernetics

Eib en A Kemenade C v and Kok J Orgy in the computer Multiparent repro duction in genetic

algorithms Advances in Articial Life Third International Conference on Articial Life ed Moran

F Moreno A Merelo J J and Chacon Pvolume of Lecture Notes in Articial Intel ligence pages

Springer Berlin

Eib en A RauePE and Ruttkay Z Genetic algorithms with multiparentrecombination Proceedings of

the rd ConferenceonParal lel Problem Solving from Nature ed Davidor Y Schwefel HP and Manner

R numb er in Lecture Notes in Computer Science pages SpringerVerlag

Eib en A and Schipp ers C Multiparents niche nary crossovers on NKlandscap es Proceedings of the

th ConferenceonParal lel Problem Solving from Nature ed Voigt HM Eb eling W Rechenb erg I and

b er in Lecture Notes in Computer Science pages Springer Berlin Schwefel HPnum

Eib en A SprinkhuizenKuyp er I and Thijssen B Comp eting crossovers in an adaptiveGAframework

Proceedings of the th IEEE Conference on Evolutionary Computation IEEE Press to app ear

Eshelman L and Schaer J Crossovers niche Proceedings of the th International Conference on

Genetic Algorithms ed Forrest S pages Morgan Kaufmann

Furuya H and Haftka R Genetic algorithms for placing actuators on space structures Proceedings of the

th International Conference on Genetic Algorithms ed Forrest S pages Morgan Kaufmann

Hordijk W and Manderick B The usefulness of recombination Advances in Articial Life Third

International ConferenceonArticial Life ed Moran F Moreno A Merelo J J and Chacon Pvolume

of Lecture Notes in Articial Intel ligence pages Springer Berlin

Kauman S Origins of Order SelfOrganization and Selection in Evolution Oxford University Press

New York NY

Kaufman H An exp erimental investigation of pro cess identication by comp etitiveevolution IEEE

Transactions on Systems Science and Cybernetics SSC

Michalewicz Z Nazhiyath G and Michalewicz M A Note on Usefulness of Geometrical Crossover

for Numerical Optimization Problems Proceedings of the th Annual Conference on Evolutionary

Programming ed Fogel L J Angeline P J and Back Th pages MIT Press

Muhlenbein H Parallel genetic algorithms p opulation genetics and combinatorial optimization

Proceedings of the rd International Conference on Genetic Algorithms ed Schaer J pages

Morgan Kaufmann

Muhlenb ein H and Voigt HM Gene p o ol recombination in genetic algorithms MetaHeuristics Theory

and Applications ed Osman I and Kelly J pages Boston London DordrechtKluwer Academic

Publishers

Pal K Selection schemes with spatial isolation for genetic optimization Proceedings of the rd Conference

on Paral lel Problem Solving from Nature ed Davidor Y Schwefel HP and Manner R numb er in

Lecture Notes in Computer Science pages SpringerVerlag

c

IOP Publishing Ltd and Oxford UniversityPress Handbook of Evolutionary Computation C

Schaer J and Eshelman L On crossover as an evolutionary viable strategy Proceedings of the th

International Conference on Genetic Algorithms ed Belew R and Bo oker L pages Morgan

Kaufmann

SchlierkampVo osen D and Muhlen b ein H Adaptation of p opulation sizes by comp eting subp opulations

Proceedings of the rd IEEE Conference on Evolutionary Computation pages IEEE Press

Schwefel HP Evolution and Optimum Seeking WileyNewYork

Schwefel HP and Rudolph G Contemp orary evolution strategies Advances in Articial Life Third

International ConferenceonArticial Life ed Moran F Moreno A Merelo J J and Chacon Pvolume

of Lecture Notes in Articial Intel ligence pages Springer Berlin

Boundary Op erators for Constrained Parameter Optimization Scho enauer M and Michalewicz Z

Problems Proceedings of the th International Conference on Genetic Algorithms ed Back Th pages

Morgan Kaufmann

Smith J and Fogarty T Recombination strategy adaptation via evolution of gene linkage Proceedings

of the rd IEEE Conference on Evolutionary Computation pages IEEE Press

Sp ears W Crossover or mutation Foundations of Genetic Algorithms ed Whitley L pages

Morgan Kaufmann

Sp ears W Adapting crossover in evolutionary algorithms Proceedings of the th Annual Conferenceon

Evolutionary Programming ed McDonnell J Reynolds R and Fogel D pages MIT Press

van Kemenade C Kok J and Eib en A Raising GA p erformance bysimultaneous tuning of selective

pressure and recombination disruptiveness Proceedings of the nd IEEE Conference on Evolutionary

Computation pages IEEE Press

Voigt HM and Muhlen b ein H Gene p o ol recombination and utilization of covariances for the Breeder

Genetic Algorithm Proc eedings of the nd IEEE Conference on Evolutionary Computation pages

IEEE Press

Voigt HM Muhlen b ein H and Cvetkovic D Fuzzy recombination for the Breeder Genetic Algorithm

Proceedings of the th International Conference on Genetic Algorithms ed Forrest S pages

Morgan Kaufmann

c

C Handbook of Evolutionary Computation IOP Publishing Ltd and Oxford University Press