2012 09 claio2012 brkga tutorial both days

7/13/2019 2012 09 CLAIO2012 Brkga Tutorial Both Days

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CLAIO/SBPO 2012September 2012 BRKGA tutorial

T

utorial given at CLAIO/SBPO 2012

Rio e !aneiro" Bra#il September 2012

Bia$e ranom%&e' geneti(

algorit)m$* A tutorial

+auri(io G, C, Re$eneAT-T Lab$ Re$ear()

.lor)am Par&" e !er$e'

mg(rre$ear(),att,(om


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AT-T S)annon Laborator'.lor)am Par&" e !er$e'


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Summar'* a' 1

3 Ba$i( (on(ept$ o4 (ombinatorial an (ontinuou$ global optimi#ation3 Ba$i( (on(ept$ o4 geneti( algorit)m$

3 Ranom%&e' geneti( algorit)m o4 Bean 516678

3 Bia$e ranom%&e' geneti( algorit)m$ 5BRKGA8

9 :n(oing / e(oing

9 Initial population

9 :volutionar' me()ani$m$

9 Problem inepenent / problem epenent (omponent$

9 +ulti%$tart $trateg'

9 Re$tart $trateg'

9 +ulti%population $trateg'

9 Spe(i4'ing a BRKGA

3 Appli(ation programming inter4a(e 5API8 4or BRKGA


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Summar'* a' 2

3 Appli(ation$ o4 BRKGA9 Set (overing

9 Pa(&ing re(tangle$

9 Pa(&et routing on t)e Internet

9 ;anover minimi#ation in mobilit' netor&$

9 Continuou$ global optimi#ation

3 Overvie o4 literature - (on(luing remar&$


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Combinatorial an

Continuou$ GlobalOptimi#ation


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Combinatorial Optimi#ation

Combinatorial optimi#ation*pro(e$$ o4 4ining t)e

be$t" or optimal" $olution 4or problem$ it) a i$(rete $eto4 4ea$ible $olution$,

Appli(ation$*routing" $()euling" pa(&ing" inventor' anprou(tion management" lo(ation" logi(" an a$$ignment o4

re$our(e$" among man' ot)er$,

:(onomi( impa(t*tran$portation 5airline$" tru(&ing" rail"an $)ipping8" 4ore$tr'" manu4a(turing" logi$ti($" aero$pa(e"energ' 5ele(tri(al poer" petroleum" an natural ga$8"

agri(ulture" biote()nolog'" 4inan(ial $ervi(e$" antele(ommuni(ation$" among man' ot)er$,


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Given*

i$(rete $et o4 4ea$ible $olution$ 8* > < * 45>8 ? 45'8" ' a(tl'an

reuire goo $olution met)o$,


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Appro>imation algorit)m$are guarantee to4in in pol'nomial%time a $olution it)in a

given 4a(tor o4 t)e optimal,


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Appro>imation algorit)m$are guarantee to4in in pol'nomial%time a $olution it)in a

given 4a(tor o4 t)e optimal,

Sometime$ t)e 4a(tor i$ too big" i,e, guarantee$olution$ ma' be 4ar 4rom optimal

Some optimi#ation problem$ 5e,g, ma> (liue"

(overing b' pair$8 (annot )ave appro>imation$()eme$unle$$ PP


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Aim o4 )euri$ti( met)o$4or (ombinatorialoptimi#ation i$ to ui(&l'prou(e goo%

ualit' $olution$" it)outne(e$$aril'

proviing an' guaranteeo4 $olution ualit',


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Continuou$ Global Optimi#ation

Given*

(ontinuou$ $et o4 4ea$ible $olution$ 8* > < * 45>8 ? 45'8" ' < * 45>8 ? 45'8" ' 8 (an be ell%be)ave or

not" e,g, it (an be

non%(onve>" i$(ontinuou$"non%i44erentiable" a bla(&%bo>"

et(,

Continuou$ Bo>%Con$traine Global


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Continuou$ Bo>%Con$traine Global

Optimi#ation

;ere" t)e (ontinuou$ $et o4 $olution$

< l1"u1D l2"u2D ln"unDi$ a )'per%re(tangle" i,e, variable$ )ave loer an

upper boun$,


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+eta)euri$ti($

+eta)euri$ti($are )euri$ti($ to evi$e )euri$ti($,


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+eta)euri$ti($

+eta)euri$ti($are )ig) level pro(eure$ t)at (oorinate$imple )euri$ti($" $u() a$ lo(al $ear()" to 4in $olution$ t)at

are o4 better ualit' t)an t)o$e 4oun b' t)e $imple )euri$ti($

alone,


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+eta)euri$ti($

+eta)euri$ti($are )ig) level pro(eure$ t)at (oorinate$imple )euri$ti($" $u() a$ lo(al $ear()" to 4in $olution$ t)at

are o4 better ualit' t)an t)o$e 4oun b' t)e $imple )euri$ti($

alone,

:>ample$*GRASP an C%GRASP" $imulate annealing"geneti( algorit)m$" tabu $ear()" $(atter $ear()" ant (olon'

optimi#ation" variable neig)bor)oo $ear()" an bia$e

ranom%&e' geneti( algorit)m$ 5BRKGA8,


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Geneti( algorit)m$


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Geneti( algorit)m$

Iniviual* $olution

Aaptive met)o$ t)at are u$e to $olve $ear()

an optimi#ation problem$,

;ollan 516EF8

G i l i )


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Geneti( algorit)m$

Iniviual* $olution 5()romo$ome $tring o4 gene$8

Population* $et o4 4i>e number o4 iniviual$

G ti l it)


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Geneti( algorit)m$ evolve population appl'ingarin$ prin(iple o4 $urvival o4 t)e 4itte$t,

Geneti( algorit)m$

G ti l it)


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A $erie$ o4 generation$ are prou(e b'

t)e algorit)m, T)e mo$t 4it iniviual o4 t)e

la$t generation i$ t)e $olution,

Geneti( algorit)m$

G ti l it)


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A $erie$ o4 generation$ are prou(e b'

t)e algorit)m, T)e mo$t 4it iniviual o4 t)e

la$t generation i$ t)e $olution,

Iniviual$ 4rom one generation are (ombine

to prou(e o44$pring t)at ma&e up ne>t

generation,

Geneti( algorit)m$

Geneti( algorit)m$


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Probabilit' o4 $ele(ting iniviual to mate

i$ proportional to it$ 4itne$$* $urvival o4 t)e

4itte$t,

a

b

Geneti( algorit)m$

Geneti( algorit)m$


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a

b

Parent$ ran 4rom

generation K

(

C)il in

generation KH1

Probabilit' o4 $ele(ting iniviual to mate

i$ proportional to it$ 4itne$$* $urvival o4 t)e

4itte$t,

Geneti( algorit)m$

a

b

Combine

parent$(

mutation

C t ti


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Cro$$over an mutation

a

b

Combine

parent$(

mutation

C v t ti


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a

b

Combine

parent$(

mutation

Cro$$over* Combine$ parent$ @ pa$$ing along to o44$pring

()ara(teri$ti($ o4 ea() parent @

Inten$i4i(ation o4 $ear()



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a

b

Combine

parent$(

mutation

+utation* Ranoml' ()ange$ ()romo$ome o4 o44$pring ,,,

river o4 evolutionar' pro(e$$ ,,,

iver$i4i(ation o4 $ear()

:volution o4 $olution$


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:v



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:n(oing $olution$

it) ranom &e'$

:n(oing it) ranom &e'$


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3 A ranom &e' i$ a real ranom number in t)e(ontinuou$ interval 0"18,



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g '


3 A ve(tor


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g '


3 A ve(tor


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g '


3 A ve(tor


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:n(oing

1" 2" " 7" F D

< 0,066" 0,21J" 0,02" 0,J" 0,JF D

:n(oing it) ranom &e'$* Seuen(ing


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:n(oing

1" 2" " 7" F D

< 0,066" 0,21J" 0,02" 0,J" 0,JF D

e(oe b' $orting ve(tor o4 ranom &e'$ 1" 2" 7" F" D

< 0,066" 0,21J" 0,J" 0,JF" 0,02 D

:n(oing it) ranom &e'$* Seuen(ing


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:n(oing

1" 2" " 7" F D

< 0,066" 0,21J" 0,02" 0,J" 0,JF D

e(oe b' $orting 1" 2" 7" F" D

< 0,066" 0,21J" 0,J" 0,JF" 0,02 D

T)ere4ore" t)e ve(tor o4 ranom &e'$*

< 0,066" 0,21J" 0,02" 0,J" 0,JF D

en(oe$ t)e $euen(e*19 2 9 7 9 F 9

:n(oing it) ranom &e'$* Sub$et

$ele(tion 5$ele(t o4 F element$8


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:n(oing 1" 2" " 7" F D

< 0,066" 0,21J" 0,02" 0,J" 0,JF D

e(oe b' $orting ve(tor o4 ranom &e'$

:n(oing it) ranom &e'$* Sub$et$ele(tion 5$ele(t o4 F element$8


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:n(oing 1" 2" " 7" F D

< 0,066" 0,21J" 0,02" 0,J" 0,JF D


< 0,066" 0,21J" 0,J" 0,JF" 0,02 D:n(oing

e(oe b' $orting ve(tor o4 ranom &e'$

:n(oing it) ranom &e'$* Sub$et$ele(tion 5$ele(t o4 F element$8


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:n(oing 1" 2" " 7" F D

< 0,066" 0,21J" 0,02" 0,J" 0,JF D


< 0,066" 0,21J" 0,J" 0,JF" 0,02 D:n(oing

T)ere4ore" t)e ve(tor o4 ranom &e'$*

< 0,066" 0,21J" 0,02" 0,J" 0,JF D

en(oe$ t)e $ub$et*1" 2" 7 M

:n(oing it) ranom &e'$* A$$igning integer

eig)t$ 0"10D to a $ub$et o4 o4 F element$


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:n(oing 1" 2" " 7" F N 1" 2" " 7" F D

< 0,066" 0,21J" 0,02" 0,J" 0,JF N0,7J7" 0,FJ11" 0,2EF2" 0,7E7" 0,07 D

:n(oing it) ranom &e'$* A$$igning integer



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g "1 D

:n(oing 1" 2" " 7" F N 1" 2" " 7" F D

< 0,066" 0,21J" 0,02" 0,J" 0,JF N0,7J7" 0,FJ11" 0,2EF2" 0,7E7" 0,07 D

e(oe b' $orting t)e 4ir$t F &e'$an a$$ign a$ t)e eig)t t)e value

i floor 10 e(oer$" e,g, t)o$e )i() )ave a lo(al

$ear() moule,

originalp)a$e 1

lo(al

$ear()4ea$ible l l

C)romo$omea=u$tment


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original

ranom &e' ve(toro4

e(oer

$ear()

o4

e(oer

4ea$ible

$olution

e(oer

a=u$te

ranom &e' ve(tor

p)a$e 1

o4e(oer

lo(al

$ear()

o4

e(oer

lo(al

optimum lo(aloptimum

e(oer

a=u$te

ranom &e' ve(tor

lo(al

optimum

lo(al

optimum

()romo$ome a=u$tment

original

ranom &e' ve(tor

Flow 1 2 3 Dist 1 2 3

uarati( a$$ignment problem C)romo$omea=u$tment


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CLAIO/SBPO 2012September 2012BRKGA tutorial

1 20 30

2 20 40

3 30 40

1 10 1

2 10 5

3 1 5

original

ranom &e' ve(tor*

5 ," ,1" ,E2 8

p)a$e 1o4

e(oer

12 21

(o$t 451"28 5"28 H 451"8 5"18 H

452"8 52"18 100 H 0 H 700 F0

12 21

lo(al$ear()

o4

e(oer

12 1 2

lo(al $ear() $appe lo(ation$ o4

4a(ilitie$ 1 an 2" re$ulting in(o$t 200

a=u$te

ranom &e' ve(tor*

5 ,1" ," ,E28

Flow 1 2 3 Dist 1 2 3



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1 20 30

2 20 40

3 30 40

1 10 1

2 10 5

3 1 5

a=u$te

ranom &e' ve(tor*

5 ,1" ," ,E2 8

p)a$e 1o4

e(oer

12 1 2

(o$t 451"28 52"8 H 451"8 52"18 H

452"8 5"18 100 H J0 H 70 200

Flow 1 2 3 Dist 1 2 3



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1 20 30

2 20 40

3 30 40

1 10 1

2 10 5

3 1 5

a=u$te

ranom &e' ve(tor*

5 ,1" ," ,E2 8

p)a$e 1o4

e(oer

12 1 2

(o$t 451"28 52"8 H 451"8 52"18 H

452"8 5"18 100 H J0 H 70 200

ot onl' i$ e>pen$ive lo(al $ear() avoie @C)ara(teri$ti($ o4 lo(al optimum are pa$$e on to 4uture

generation$ @, T)e' ill be repre$ente in t)e population b'

a=u$te ranom &e' ve(tor,


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Geneti( algorit)m$

an ranom &e'$

GA$ an ranom &e'$


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3 Introu(e b' Bean 516678

4or $euen(ing problem$,

GA$ an ranom &e'$


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CLAIO/SBPO 2012

September 2012 BRKGA tutorial



3 Iniviual$ are $tring$ o4

real%value number$5ranom &e'$8 in t)e

interval 0"1D,

S 5 0,2F" 0,16" 0,JE" 0,0F" 0,6 8

$518 $528 $58 $578 $5F8

GA$ an ranom &e'$


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3 Iniviual$ are $tring$ o4

real%value number$5ranom &e'$8 in t)e

interval 0"18,

3 Sorting ranom &e'$ re$ult$in a $euen(ing orer,

S 5 0,2F" 0,16" 0,JE" 0,0F" 0,6 8

$518 $528 $58 $578 $5F8

S 5 0,0F" 0,16" 0,2F" 0,JE" 0,6 8

$578 $528 $518 $58 $5F8

Seuen(e* 7 9 2 9 1 9 9 F

GA$ an ranom &e'$


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3 +atingi$ one u$ing

parametri#e uni4orm

(ro$$over 5Spear$ - e!ong " 16608

a 5 0,2F" 0,16" 0,JE" 0,0F" 0,6 8b 5 0,J" 0,60" 0,EJ" 0,6" 0,0 8

GA$ an ranom &e'$


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a 5 0,2F" 0,16" 0,JE" 0,0F" 0,6 8b 5 0,J" 0,60" 0,EJ" 0,6" 0,0 8

3 +ating i$ one u$ing

parametri#e uni4orm


3 .or ea() gene" 4lip a bia$e

(oin to ()oo$e )i()

parent pa$$e$ t)e allele

5&e'" or value o4 gene8 tot)e ()il,

GA$ an ranom &e'$


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a 5 0,2F" 0,16" 0,JE" 0,0F" 0,6 8b 5 0,J" 0,60" 0,EJ" 0,6" 0,0 8

( 5 8


parametri#e uni4orm



(oin to ()oo$e )i()



GA$ an ranom &e'$


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a 5 0,2F" 0,16" 0,JE" 0,0F" 0,6 8b 5 0,J" 0,60" 0,EJ" 0,6" 0,0 8

( 5 0,2F 8


parametri#e uni4orm



(oin to ()oo$e )i()



GA$ an ranom &e'$


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a 5 0,2F" 0,16" 0,JE" 0,0F" 0,6 8b 5 0,J" 0,60" 0,EJ" 0,6" 0,0 8

( 5 0,2F" 0,60 8


parametri#e uni4orm


3 .or ea() gene" 4lip a bia$e(oin to ()oo$e )i()



GA$ an ranom &e'$


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a 5 0,2F" 0,16" 0,JE" 0,0F" 0,6 8b 5 0,J" 0,60" 0,EJ" 0,6" 0,0 8

( 5 0,2F" 0,60" 0,EJ 8


parametri#e uni4orm





GA$ an ranom &e'$


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a 5 0,2F" 0,16" 0,JE" 0,0F" 0,6 8b 5 0,J" 0,60" 0,EJ" 0,6" 0,0 8

( 5 0,2F" 0,60" 0,EJ" 0,0F 8


parametri#e uni4orm





GA$ an ranom &e'$


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a 5 0,2F" 0,16" 0,JE" 0,0F" 0,68b 5 0,J" 0,60" 0,EJ" 0,6" 0,0 8

( 5 0,2F" 0,60" 0,EJ" 0,0F" 0,6 8


parametri#e uni4orm





GA$ an ranom &e'$


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a 5 0,2F" 0,16" 0,JE" 0,0F" 0,68b 5 0,J" 0,60" 0,EJ" 0,6" 0,0 8

( 5 0,2F" 0,60" 0,EJ" 0,0F" 0,6 8

I4 ever' ranom%&e' arra' (orre$pon$

to a 4ea$ible $olution* +ating ala'$prou(e$ 4ea$ible o44$pring,


parametri#e uni4orm





GA$ an ranom &e'$

Initial population i$ mae up o4 P


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Initial populationi$ mae up o4 P

ranom%&e' ve(tor$" ea() it)

&e'$" ea() )aving a value

generate uni4orml' at ranom in

t)e interval 0"18,

GA$ an ranom &e'$

At t)e K%t) generation"Population K


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(ompute t)e (o$t o4 ea()$olution ,,, :lite $olution$

on%elite

$olution$

GA$ an ranom &e'$



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(ompute t)e (o$t o4 ea()$olution an partition t)e

$olution$ into to $et$*

:lite $olution$

on%elite

$olution$

GA$ an ranom &e'$



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elite $olution$an non%elite$olution$,

:lite $olution$

on%elite

$olution$

GA$ an ranom &e'$



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elite $olution$an non%elite$olution$, :lite $et $)oul

be $maller o4 t)e to $et$

an (ontain be$t $olution$,

:lite $olution$

on%elite

$olution$

GA$ an ranom &e'$

:volutionar' 'nami($Population K Population KH1


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:lite $olution$

on%elite

$olution$

GA$ an ranom &e'$

:volutionar' 'nami($Population KH1Population K


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9 Cop' elite $olution$ 4rom populationK to population KH1

:lite $olution$:lite $olution$

on%elite

$olution$

GA$ an ranom &e'$

:volutionar' 'nami($

Population KH1Population K


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9 A R ranom $olution$ 5mutant$8

to population KH1

:lite $olution$

+utant

$olution$

:lite $olution$

on%elite

$olution$

GA$ an ranom &e'$

:volutionar' 'nami($

Population KH1Population K


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9 A R ranom $olution$ 5mutant$8

to population KH1

9 )ile KH1%t) population Q P3 RAO+%K: GA*$e an' to

$olution$ in population Kto prou(e

()il in population KH1, +ate$ are

()o$en at ranom,

:lite $olution$

+utant

$olution$

t population

Generate mutant$ inne>t population

Combine elite an

non%elite $olution$an a ()ilren to

ne>t population

no

$top

'e$

.rameor& 4or bia$e ranom%&e' geneti( algorit)m$

Generate P ve(tor$

o4 ranom &e'$

e(oe ea() ve(tor

o4 ranom &e'$

Problem inepenent


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Stopping rule

$ati$4ieU

Sort $olution$ b'

t)eir (o$t$

Cla$$i4' $olution$ a$

elite or non%elite

Cop' elite $olution$to ne>t population


Combine elite an


ne>t population

$top

Problem inepenent

no 'e$

.rameor& 4or bia$e ranom%&e' geneti( algorit)m$

Generate P ve(tor$

o4 ranom &e'$

e(oe ea() ve(tor

o4 ranom &e'$

Problem inepenent

Problem epenent


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Stopping rule

$ati$4ieU

Sort $olution$ b'

t)eir (o$t$


elite or non%elite



Combine elite an


ne>t population

$top

Problem inepenent

no 'e$

e(oing o4 ranom &e' ve(tor$ (an be one in parallel

Generate P ve(tor$

o4 ranom &e'$

e(oe ea() ve(tor

o4 ranom &e'$


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Stopping rule

$ati$4ieU

Sort $olution$ b'

t)eir (o$t$


elite or non%elite



Combine elite an


ne>t population

$top

no 'e$

BRKGA in multi%$tart $trateg'

t i

output

in(umbent$top

'e$


97/520


Generate P ve(tor$o4 ranom &e'$

e(oe ea() ve(tor

o4 ranom &e'$

Re$tart rule

$ati$4ieU

Sort $olution$ b'

t)eir (o$t$


elite or non%elite

Cop' elite $olution$

to ne>t population

Generate mutant$ in

ne>t population

Combine elite an

non%elite $olution$

an a ()ilren to

ne>t population

no

'e$

begin

i4 in(umbent

improve" upate

in(umbent

$topping

rule

$ati$4ie U

no


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Ranomi#e )euri$ti( iteration

(ount i$tribution* (on$tru(teb' inepenentl' running t)e

algorit)m a number o4 time$" ea()

time $topping )en t)e algorit)m

4in$ a $olution at lea$t a$ goo a$ a

given target,


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In mo$t o4 t)e inepenent run$" t)e algorit)m 4in$ t)e target $olution in

relativel' 4e iteration$*


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relativel' 4e iteration$* 2FV o4 t)e run$ ta&e 4eer t)an 101 iteration$


101/520



relativel' 4e iteration$* F0V o4 t)e run$ ta&e 4eer t)an 162 iteration$


102/520



relativel' 4e iteration$* EFV o4 t)e run$ ta&e 4eer t)an 7F iteration$


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;oever" $ome run$ ta&e mu() longer* 10V o4 t)e run$ ta&e over 1000

iteration$


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;oever" $ome run$ ta&e mu() longer* FV o4 t)e run$ ta&e over 2000

iteration$


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;oever" $ome run$ ta&e mu() longer* 2V o4 t)e run$ ta&e over 6E1F

iteration$


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;oever" $ome run$ ta&e mu() longer* t)e longe$t run too& 11J0E

iteration$

Probabilit' t)at algorit)m ill ta&e

over 7F iteration$* 2FV 1/7


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Probabilit' t)at algorit)m ill ta&e

over 7F iteration$* 2FV 1/7

B' re$tarting algorit)m a4ter 7F

iteration$" probabilit' t)at ne runill ta&e over J60 iteration$* 2FV


108/520


1/7

Probabilit' t)at algorit)m it)

re$tart ill ta&e over J60iteration$*probabilit' o4 ta&ing over 7F ample" probabilit' t)atalgorit)m it) re$tart ill $till be


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algorit)m it) re$tartill $till be

running a4ter 1E2Fiteration$ 5F

perio$ o4 7Fiteration$8* 1/7F 0,06EEV

T)i$ i$ mu() le$$ t)an t)eFV

probabilit' t)at t)e algorit)m

it)out re$tartill ta&e over2000

iteration$,

Re$tart $trategie$

3 .ir$t propo$e b' Lub' et al,51668

3 T)e' e4ine a re$tart $trateg' a$ a 4inite $euen(e


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' g'

o4 time interval$ S 1"

2"

3" @ M)i() e4ine

epo()$ 1" 1H2" 1H2H3" @)en t)ealgorit)m i$ re$tarte 4rom $(rat(),

3 Lub' et al, 51668 prove t)at t)e optimal re$tart

$trateg' u$e$ 1 2 3

*")ere *i$ a(on$tant,

Re$tart $trategie$

3 Lub' et al, 51668

3 Kaut# et al, 520028

3 Palube(&i$ 520078


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3 Palube(&i$ 520078

3 Sergien&o et al, 520078

3 oi(&i - Smutni(&i 5200F8

3 WApu##o et al, 5200J8

3 S)'lo et al, 52011a8

3 S)'lo et al, 52011b8

3 Re$ene - Ribeiro 520118

Re$tart $trateg' 4or BRKGA

3 Re(all t)e re$tart $trateg' o4 Lub' et al, )ere eual time

interval$ 1 2 3 * pa$$ beteen re$tart$,


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3 Strateg' reuire$ * a$ input,

3 Sin(e e )ave no prior in4ormation a$ to t)e runtime

i$tribution o4 t)e )euri$ti(" e run t)e ri$& o4*9 ()oo$ing * too $mall* re$tart variant ma' ta&e long to

(onverge

9 ()oo$ing * too big* re$tart variant ma' be(ome li&e no%re$tart variant

Re$tart $trateg' 4or BRKGA

3 e (on=e(ture t)at number o4 iteration$ beteen

improvement o4 t)e in(umbent5be$t $o 4ar8 $olutionvarie$ le$$ r t )euri$ti(/ in$tan(e/ target t)an run


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varie$ le$$ ,r,t, )euri$ti(/ in$tan(e/ target t)an run

time$,

3 e propo$e t)e 4olloing re$tart $trateg'*Keep tra(& o4t)e la$t generation )en t)e in(umbent improve an

re$tart BRKGA i4 Kgeneration$ )ave gone b' it)out

improvement,

3 e (all t)i$ $trateg' re$tart5K8

:>ample o4 re$tart $trateg' 4or BRKGA* Loa balan(ing

Given an orere $euen(e o4 1027 integer$ p0D" p1D" @" p102D


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Pla(e (on$e(utive number$ in 2 bu(&et$ b0D" b1D" @" b1D



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b0D b1D b2D bD b7D b26D b0D b1D

A t)e number$ in ea() bu(&et b0D" b1D" @" b1D



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p p p p p p p p

Pla(e t)e bu(&et$ in 1J bin$ B0D" B1D" @" B1FD



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p pp

pp

p

pp

B0D B1D B2D B1FD

A up t)e number$ in ea() bin B0D" B1D" @" B1FD



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p pp

pp

p

pp

B0D B1D B2D B1FD

T0D=(p) T1D=(p) T2D=(p) T1FD=(p)

OB!:CTIX:* +inimi#e +a>imum 5T0D" T1D" @" T1FD8 M



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p pp

pp

p

pp

B0D B1D B2D B1FD

T0D=(p) T1D=(p) T2D=(p) T1FD=(p)


:n(oing

< >1D" >2D" @" >2D N >2H1D" >2H2D" @" >2H1JD D


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e(oing

>1D" >2D" @" >2D are u$e to e4ine brea& point$ 4or bu(&et$

>2H1D" >2H2D" @" >2H1JD are u$e to etermine to

)i() bin$ t)e bu(&et$ are a$$igne


:n(oing

< >1D" >2D" @" >2D N >2H1D" >2H2D" @" >2H1JD D


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e(oing

>1D" >2D" @" >2D are u$e to e4ine brea& point$ 4or bu(&et$Si#e o4 bu(&et i 4loor 51027 >iD/5>1DH>2DHH>2D88" i1",,,"1F

Si#e o4 bu(&et 1J 1027 9 $um o4 $i#e$ o4 4ir$t 1F bu(&et$


:n(oing

< >1D" >2D" @" >2D N >2H1D" >2H2D" @" >2H1JD D


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e(oing

>1D" >2D" @" >2D are u$e to e4ine brea& point$ 4or bu(&et$

>2H1D" >2H2D" @" >2H1JD are u$e to etermine to

)i() bin$ t)e bu(&et$ are a$$igne

Bin t)at bu(&et ii$ a$$igne to 4loor 51J >2HiD8 H 1


e(oing 5Lo(al $ear() p)a$e8

9 while5t)ere e>i$t$ a bu(&et in t)e mo$t loae bin t)at

(an be move to anot)er bin an not in(rea$e t)e


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ma>imum loa8 then

3 move t)at bu(&et to t)at bin9 end while

+a&e ne(e$$ar' ()romo$ome a=u$tment$ to la$t 1J

ranom &e'$ o4 ve(tor o4 ranom &e'$ to re4le(t ()ange$

mae in lo(al $ear() p)a$e* A or $ubtra(t an integervalue 4rom ()romo$ome o4 bu(&et t)at move to ne bin,


re$tart $trateg'*re$tart520008 no re$tart


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BRKGA it) p parallel population$

evolve evolve evolve evolve

HHgen


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evolve

pop 1

evolve

pop 2

evolve

pop

evolve

pop p

gen mo > 0

top v iniviual$

emigrate 4rom

ea() pop to

ot)er pop$

'e$

no

BRKGA it) p parallel population$

evolve evolve evolve evolve

HHgen


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evolve

pop 1

evolve

pop 2

evolve

pop

evolve

pop p

gen mo > 0

top v iniviual$

emigrate 4rom

ea() pop to

ot)er pop$

'e$

no

:ver' >generation$

t)e vmo$t 4it iniviual$

4rom ea() population are

(opie to t)e ot)er p 9 1population$ i4 not t)ere

alrea',

Initiali#e population it) $ome non%ranom iniviual$

It i$ o4ten u$e4ul to initiali#e t)e 4ir$t population it)

$ome iniviual$ not generate totall' at ranom,

9 Generate $ome iniviual$ u$ing $imple )euri$ti($"


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Generate $ome iniviual$ u$ing $imple )euri$ti($"

e,g, Buriol" +,G,C,R," Ribeiro" - T)orup 5200F8

9 .ormulate 0%1 integer program an $olve linearprogramming 5LP8 rela>ation an u$e LP $olution a$

iniviual" e,g, Anrae" +i'a#aa" +,G,C,R," - To$o

520128

3 :n(oing i$ ala'$ one t)e $ame a'" i,e, it) a ve(tor o4

ranom%&e'$ 5parameter mu$t be $pe(i4ie8

3 e(oert)at ta&e$ a$ input a ve(tor o4 ranom%&e'$ an

output$ t)e (orre$poning $olution o4 t)e (ombinatorial

Spe(i4'ing a bia$e ranom%&e' GA


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optimi#ation problem an it$ (o$t 5t)i$ i$ u$uall' a )euri$ti(8

3 Parameter$


Parameter$*

9 Si#e o4 population

9 Parallel population parameter$


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9 Si#e o4 elite partition

9 Si#e o4 mutant $et

9 C)il in)eritan(e probabilit'

9 Re$tart $trateg' parameter

9 Stopping (riterion


Parameter$*

9 Si#e o4 population* a 4un(tion o4 " $a' or 2

9 Parallel population parameter$


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Parameter$*


9 Parallel population parameter$* $a'"p " v 2" an > 200


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Parameter$*




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9 Si#e o4 elite partition* 1F%2FV o4 population






Parameter$*



Si 4 li i i 1F 2FV 4 l i


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9 Si#e o4 mutant $et* F%1FV o4 population





Parameter$*



Si 4 lit titi 1F 2FV 4 l ti


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9 C)il in)eritan(e probabilit'* 0,F" $a' 0,E




Parameter$*



Si 4 lit titi 1F 2FV 4 l ti


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9 Re$tart $trateg' parameter* a 4un(tion o4 " $a' 2 or 10



Parameter$*



Si#e o4 elite partition* 1F 2FV o4 population


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9 Re$tart $trateg' parameter* a 4un(tion o4 " $a' 2 or 10

9 Stopping (riterion* e,g,time" Y generation$" $olution ualit'"

Y generation$ it)out improvement

br&gaAPI* A CHH API 4or BRKGA

3 :44i(ient an ea$'%to%u$e ob=e(t oriente appli(ation

programming inter4a(e 5API8 4or t)e algorit)mi( 4rameor&o4 BRKGA,


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3 Cro$$%plat4orm librar' )anle$ large portion o4 problem


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p ' g p p

inepenent moule$ t)at ma&e up t)e 4rameor&" e,g,

9 population management

9 evolutionar' 'nami($






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p ' g p p




3 Implemente in CHH an ma' bene4it 4rom $)are%memor'

paralleli$m i4 available,






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3 Implemente in CHH an ma' bene4it 4rom $)are%memor'

paralleli$m i4 available,3 $er onl' nee$ to implement problem%epenent e(oer,


Paper*Rorigo ., To$o an +,G,C,R," ZA CHH

Appli(ation Programming Inter4a(e 4or

Bia$e Ranom%Ke' Geneti( Algorit)m$"[


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Bia Ran om K ' G n i Algori )m "AT-T Lab$ Te()ni(al Report" .lor)am Par&" Augu$t 2011,

So4tare*)ttp*//,re$ear(),att,(om/\mg(r/$r(/br&gaAPI

Re4eren(e

!,., Gon]alve$ an +,G,C,R," ZBia$e ranom%&e'

geneti( algorit)m$ 4or (ombinatorial optimi#ation"[!, o4 ;euri$ti($" vol,1E" pp, 7E%F2F" 2011,
http://www.research.att.com/http://www.research.att.com/


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Te() report ver$ion*

)ttp*//,re$ear(),att,(om/\mg(r/o(/$r&ga,p4

Re4eren(e

+,G,C,R," ZBia$e ranom%&e' geneti( algorit)m$it) appli(ation$ in tele(ommuni(ation$"[ TOP"

vol, 20" pp, 120%1F" 2012,


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Te() report ver$ion*

)ttp*//,re$ear(),att,(om/\mg(r/o(/br&ga%tele(om,p4

T)an&$^


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T)e$e $lie$ an all o4 t)e paper$ (ite in t)i$tutorial (an be onloae 4rom m' )omepage*

)ttp*//,re$ear(),att,(om/\mg(r

T)an&$^


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T)e$e $lie$ an all o4 t)e paper$ (ite in t)i$tutorial (an be onloae 4rom m' )omepage*

)ttp*//,re$ear(),att,(om/\mg(rSee you tomorrow for some applications of BRKGA.

Biased random-key genetic

algorithms: A tutorial


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Tutorial given at CLAIO/SBPO 2012

Rio de Janeiro, Brazil September 2012

Mauricio G. C. ResendeAT&T Labs Research

Florham Park, New Jersey

[email protected]


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AT&T Shannon Laboratory

Florham Park, New Jersey


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Part 2 of tutorial

Summary: Day 1

Basic concepts of combinatorial and continuous global optimization

Basic concepts of genetic algorithms

Random-key genetic algorithm of Bean (1994)


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Biased random-key genetic algorithms (BRKGA) Encoding / Decoding

Initial population

Evolutionary mechanisms

Problem independent / problem dependent components

Multi-start strategy

Restart strategy

Multi-population strategy

Specifying a BRKGA

Application programming interface (API) for BRKGA

Summary: Day 2

Applications of BRKGA Set covering

Packing rectangles


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Packing rectangles

Packet routing on the Internet

Handover minimization in mobility networks

Continuous global optimization Overview of literature & concluding remarks

Encoding is always done the same way, i.e. with a vector of

N random-keys (parameter Nmust be specified)

Decoderthat takes as input a vector of Nrandom-keys and

Specifying a biased random-key GA


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outputs the corresponding solution of the combinatorialoptimization problem and its cost (this is usually a heuristic)

Parameters


Parameters: Size of population


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Parallel population parameters Size of elite partition

Size of mutant set

Child inheritance probability Restart strategy parameter

Stopping criterion


Parameters: Size of population: a function of N, say N or 2N


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Parallel population parameters Size of elite partition

Size of mutant set


Stopping criterion




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Parallel population parameters: say,p = 3 , v = 2, and x = 200 Size of elite partition

Size of mutant set


Stopping criterion




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Parallel population parameters: say,p = 3 , v = 2, and x = 200 Size of elite partition: 15-25% of population

Size of mutant set


Stopping criterion




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Size of mutant set: 5-15% of population


Stopping criterion




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Child inheritance probability: > 0.5, say 0.7 Restart strategy parameter

Stopping criterion



Parallel population parameters: say p = 3 v = 2 and x = 200


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Parallel population parameters: say,p = 3 , v = 2, and x = 200

Size of elite partition: 15-25% of population


Child inheritance probability: > 0.5, say 0.7 Restart strategy parameter: a function of N, say 2N or 10N

Stopping criterion



Parallel population parameters: say p = 3 v = 2 and x = 200


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Parallel population parameters: say,p = 3 , v = 2, and x = 200

Size of elite partition: 15-25% of population


Child inheritance probability: > 0.5, say 0.7 Restart strategy parameter: a function of N, say 2N or 10N

Stopping criterion: e.g.time, # generations, solution quality,

# generations without improvement


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Some applicationsof BRKGA


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Steiner triple covering

M.G.C.R., R.F. Toso, J.F. Gonalves, and R.M.A.

Silva, A biased random-key genetic


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y g

algorithm for the Steiner triple covering

problem, Optimization Letters, vol. 6, pp. 605-619, 2012.

tech report: http://www.research.att.com/~mgcr/doc/brkga-stn.pdf


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Steiner triple

covering problem

Kirkman school girl problem [Kirkman, 1850]

Fifteen young ladies in a school walk out three


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abreast for seven days in succession:

It is required to arrange them daily, so that no two

shall walk twice abreast.

Kirkman school girl problem [Kirkman, 1850]

Sunday Monday Tuesday Wednesday Thursday Friday Saturday

If girls are numbered 01, 02, ..., 15, a solution is:


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01, 06, 11 01, 02, 05 02, 03, 06 05, 06, 09 03, 05, 11 05, 07, 13 04, 11, 13

02, 07, 12 03, 04, 07 04, 05, 08 07, 08, 11 04, 06, 12 06,08, 14 05, 12, 14

03, 08, 13 08, 09, 12 09, 10, 13 01, 12, 13 07, 09, 15 02, 09, 11 02, 08, 15

04, 09, 14 10, 11, 14 11, 12, 15 03, 14, 15 01, 08, 10 03, 10, 12 01, 03, 09

05, 10, 15 06, 13, 15 01, 07, 14 02, 04, 10 02, 13, 14 01, 04, 15 06, 07, 10

Ball, Rouse, and Coxeter (1974)

Steiner triple system

A Steiner triple system on a set Xof nelements is a


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collection Bof 3-sets (triples)such that, for any twoelements xandyin X, the pair {x, y}appears in

exactly one triple in B.




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collection Bof 3-sets (triples)such that, for any twoelements xandyin X, the pair {x, y}appears in


First studied by Kirkmanin 1847. Then by Steinerin 1853and

hence the name.



collection Bof 3-sets (triples)such that, for any two


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elements xandyin X, the pair {x, y}appears in


The school girl problem has the additional constraint that the

collection of|B|= 75 = 35triples be divided into seven sets offive triples, one for each day, such that each girl appears exactly once

in the set of five triples for that day.





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A Steiner triple system exists for a set X if and only if either

|X|= 6k+1 or |X|=6k+3 for some k > 0 [ Kirkman, 1847 ]





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One non-isomorphic Steiner triple system exists for |X| = 7 and 9.

This number grows quickly after that. For |X| = 19, there are over

1010non-isomorphic Steiner triple systems.





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A Steiner triple system can be represented by a binary matrix Awith

one column for each element in Xand a row for each triple in B. In

this matrix A(i,j) = 1if and only if elementjis in triple i.

Each row iof Ahas exactly 3 entrieswith A(i,j) = 1.

1-width of a binary matrix

The 1-width of a binary matrix Ais the minimum

number of columns that can be chosen from A such


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that every row has at least one 1in the selected

columns.

The 1-width of a binary matrixAis the solution of the set

covering problem: minj xj subject to Ax 1m , xj{ 0, 1 }

Recursive procedure to generate Steinertriple systems

Let A3be the 1 3matrix of all ones. A recursive

d d ib d b H ll (1967) t


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procedure described by Hall (1967) can generateSteiner triple systems for which n3korn15 3k-1, for k1, 2, ...

Recursive procedure to generate Steinertriple systems

Let A3be the 1 3matrix of all ones. A recursive

d d ib d b H ll (1967) t


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procedure described by Hall (1967) can generateSteiner triple systems for which n3korn15 3k-1, for k1, 2, ...

Starting from A3, the procedure can generate A

9, A

27, A

81, A

243, A

729,

Starting from A15

[Fulkerson et al., 1974], the procedure can generate

A45

, A135

, A405

, ...

Solving Steiner triple covering

Fulkerson, Nemhauser, and Trotter (1974) were first to

point out that the Steiner triple covering problem was a


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computationally challenging set covering problem.

Solving Steiner triple covering

Fulkerson, Nemhauser, and Trotter (1974) were first to

point out that the Steiner triple covering problem was a


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computationally challenging set covering problem.

They solved stn9(A9), stn15(A

15), and stn27(A

27) to optimality, but not

stn45(A45

), which was solved in 1979 by Ratliff.

Mannino and Sassano (1995) solved stn81and recently Ostrowski et al.(2009; 2010) solved stn135in 126 days of CPU and stn243in 51 hours.

Independently, Ostergard and Vaskelainen (2010) also solved stn135.

Heuristics for Steiner triple covering (stn81and stn135)

Feo and R. (1989) proposed a GRASP, finding a cover ofsize 61for stn81, later shown to be optimal by Mannino

and Sassano (1995).


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and Sassano (1995).

Karmarkar Ramakrishnan and R (1991) found a cover


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Karmarkar, Ramakrishnan, and R. (1991) found a coverof size 105for stn135with an interior point algorithm.

In the same paper, they used a GRASP to find a better

cover of size 104. Mannino and Sassano (1995) also

found a cover of this size.



and Sassano (1995).

Karmarkar Ramakrishnan and R (1991) found a cover


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Karmarkar, Ramakrishnan, and R. (1991) found a coverof size 105for stn135with an interior point algorithm.

In the same paper, they used a GRASP to find a better

cover of size 104. Mannino and Sassano (1995) also

found a cover of this size.

Odijk and van Maaren (1998) found a cover of size 103,which was shown to be optimal by Ostrowski et al. and

Ostergard and Vaskelainen in 2010.

Heuristics for Steiner triple covering (stn243)

The GRASP in Feo and R. (1989) as well as the interior

point method in Karmarkar, Ramakrishnan, and R.

(1991) produced covers of size 204for stn243.


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Karmarkar Ramakrishnan and R (1991) used theGRASP F R


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Karmarkar, Ramakrishnan, and R. (1991) used theGRASP of Feo and R. (1989) to improve the best known

cover to 203.





Karmarkar Ramakrishnan and R (1991) used theGRASP f F d R (1989) i h b k


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cover to 203.

Mannino and Sassano (1995) improved it further to202.





Karmarkar Ramakrishnan and R. (1991) used theGRASP f F d R (1989) t i th b t k


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cover to 203.

Mannino and Sassano (1995) improved it further to202.

Odijk and van Maaren (1998) found a cover of size 198,

which was shown to be optimal by Ostrowski et al.

(2009; 2010).

Heuristics for Steiner triple covering (stn405 and stn729)

No results have been previously presented for stn405.


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Ostrowski et al. (2010) report that the best solution

found by CPLEX 9 on stn729 after two weeks of CPU


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time was 653.



Ostrowski et al. (2010) report that the best solution



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time was 653.

Using their enumerate-and-fix heuristic, they were ableto find a better cover of size 619.

Best known solutions to date

instance n m BKS opt? reference

stn9 9 12 5 yes Fulkerson et al. (1974)



stn45 45 330 30 yes Ratliff (1979)


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stn45 45 330 30 yes Ratliff (1979)

stn81 81 1080 61 yes Mannino and Sassano(1995)

stn135 135 3015 103 yes Ostrowski et al. (2009; 2010) andOstergard and Vaskelainen (2010)

stn243 243 9801 198 yes Ostrowski et al. (2009;2010)

stn405 405 27270 335 ? M.G.C.R. et al. (2012)

stn729 729 88452 617 ? M.G.C.R. et al. (2012)

BRKGA for Steiner

triple covering


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triple covering

Encoding a solution to a vector of random keys

A solution is encoded as an n-vector X= (X1, X

2, ..., X

n)

of random keys where nis the number of columnsof matrix A


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yof matrix A.



2, ..., X

n)

of random keys where nis the number of columnsof matrix A


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yof matrix A.

Each key is a randomly generated number in the

real interval [0,1).



2, ..., X

n)

of random keys where nis the number of columns

of matrix A


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of matrix A.

Each key is a randomly generated number in the

real interval [0,1).

Thej-th component of Xcorresponds to thej-th

column of A.

Decoding a solution from a vector of random keys

Decoder takes as input ann-vectorX= (X1, X

2, ..., X

n)of

random keys and returns a coverJ*

{1, 2, ..., n }

corresponding to the indices of the columns of A


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corresponding to the indices of the columns ofAselected to cover the rows ofA.


Decoder takes as input ann-vectorX= (X1, X

2, ..., X

n)of

random keys and returns a coverJ*

{1, 2, ..., n }

corresponding to the indices of the columns of A


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corresponding to the indices of the columns ofA

selected to cover the rows ofA.

LetY= (Y1, Y

2, ..., Y

n)be a binary vector whereY

j= 1 if

and only ifj J*.


Decoder has three phases:

Phase I: For j = 1 2 n set Y 1 if X set Y 0


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Phase I:Forj = 1, 2, ..., n, set Yj= 1 if X

j, set Y

j= 0

otherwise.



Phase I: For j = 1 2 n set Y 1 if X set Y 0


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Phase I:Forj = 1, 2, ..., n, set Yj= 1 if X

j, set Y

j= 0

otherwise.

The indices implied by the binary vector can correspond to either a

feasible or infeasible cover.

If cover is feasible, Phase IIis skipped.



Phase II: If J* is not a valid cover build a cover with


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Phase II:IfJ is not a valid cover, build a cover with

a greedy algorithm for set covering(Johnson, 1974)

starting from the partial coverJ*.



Phase II: If J* is not a valid cover build a cover with


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Phase II:IfJ is not a valid cover, build a cover with

a greedy algorithm for set covering(Johnson, 1974)

starting from the partial coverJ*.

Greedy algorithm:WhileJ*is not a valid cover, select to add inJ*thesmallest indexj {1,2,...,n} \ J*for which the inclusion ofjinJ*covers the maximum number of yet-uncovered rows.



Phase III: Local search attempts to remove


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Phase III:Local search attempts to remove

superfluous columns from coverJ*.



Phase III: Local search attempts to remove


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Phase III:Local search attempts to remove

superfluous columns from coverJ*.

Local search:While there is some elementj J*such thatJ* \ { j }is

still a valid cover, then such element having the smallest index isremoved fromJ*.

Implementation


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Implementationissues

Implementation issues

BRKGA framework (R. and Toso, 2010), a C++

framework for biased random-key genetic

algorithms.


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Object oriented

Multi-threaded: parallel decoding using OpenMP General-purpose framework: implements all problem

independent components and provides a simple hook for

chromosome decoding Chromosome adjustment


Chromosome adjustment:decoder not only returns

the coverJ*but also modifies the vector Xof random

keys such that it decodes directly intoJ*with the


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application of only the first phase of the decoder:


Chromosome correcting:decoder not only returns

the coverJ*but also modifies the vector Xof random

keys such that it decodes directly intoJ*with the


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application of only the first phase of the decoder:

Xjis unchanged if X

j andj J*or if X

j andj J*

Xjchanges to1Xjif Xj andj J*or if Xj andj J*

Experimental


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CLAIO/SBPO 2012


Experimentalresults

Experiments: objectives

Investigate effectivenessof BRKGA to find

optimal covers for instances with known

optimum.


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CLAIO/SBPO 2012





optimum.


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For the two instances (stn405and stn729) for

which optimal solutions are not known, attemptto produce better coversthan previously found.




optimum.


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For the two instances (stn405and stn729) for

which optimal solutions are not known, attemptto produce better coversthan previously found.

Investigate effectiveness of parallelimplementation.

Experiments: instances

Set of instances: stn9, stn15, stn27, stn45, stn81,

stn135, stn243, stn405, stn729


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Instances can be downloaded from:

http://www2.research.att.com/~mgcr/data/steiner-triple-covering.tar.gz

Experiments: computing environment

Computer: server with four 2.4 GHz Quad-core Intel XeonE7330 processors with 128 Gb of memory, running CentOS 5

Linux. Total of 16 processors.


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Compiler:g++ version 4.1.2 20080704 with flags -O3 -fopenmp





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Compiler:g++ version 4.1.2 20080704 with flags -O3 -fopenmp

Random number generator: Mersenne Twister (Matsumoto

& Nishimura, 1998)

Experiments: multi-population GA

We evolve 3 populationssimultaneously (but sequentially).


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Every 100generations the best two solutions from each

population replace the worst solutionsof the other two


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p p p

populations if not already present there.



Every 100generations the best two solutions from each

population replace the worst solutionsof the other two


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populations if not already present there.

Parallel processingis only done when calling the decoder. Up to

16 chromosomes are decoded in parallel.

Experiments: other parametersPopulation size: 10n, where nis the number of columns of A


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Population partition: 1.5n elite solutions;11.5n=8.5n non-elite solutions


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Mutants: 5.5n are created at each generation


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Probability child inherits gene of elite/non-elite parent: biased coin

60% : 40%


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Probability child inherits gene of elite/non-elite parent: biased coin

60% : 40%


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Stopping rule: we use different stopping rules for each of the threetypes of experiments

Experiments on instances with known optimal covers

For each instance: ran GA independently 100 times, stopping

when an optimal cover was found.


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On all 100 runsfor each instance, the algorithm found an optimal

cover.


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cover.

On the smallest instances (stn9, stn15, stn27) an optimal cover was

always found in the initial population.


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cover.

On the smallest instances (stn9, stn15, stn27) an optimal cover was

always found in the initial population.


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On stn81an optimal cover was found in the initial population in 99

of the 100 runs. In the remaining run, an optimal cover was found in

the second iteration.


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Instance stn45


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Optimal cover found in initial population in 54/100runs


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Largest number of iterations in 100runs was 12


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Time per 1000generations: 4.70s(real), 70.55s (user),2.73s (sys)


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Instancestn135


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Most difficultinstance of those with known optimal cover


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9 of the 100 runsfound an optimal cover in less than 1000 iterations


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39 of the 100 runsrequired over 10,000 iterations


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No run required fewer than 23 iterations


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Longest run took 75,741 iterations


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Instance stn243


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Appears to be much easier than stn135


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39/100 runs required fewer than100 generations


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95/100 runs required fewer than200 generations


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The longest of the 100 runs took 341 generations


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Simulation of random multi-start on stn243

To show that success of BRKGA to consistently find covers of

size 198 on stn243 was not due to the decoder alone ...


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Simulation of random multi-start on stn243

To show that success of BRKGA to consistently find covers of

size 198 on stn243 was not due to the decoder alone ...

We conducted 100 independent runs

2012 09 claio2012 brkga tutorial both days

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