Chapter 4 GAs: Selected Topics

Premature Convergence

Why GA cannot to find the optimal solutions in the practical applications?

Coding problem Limit number of iterations Limit population size Super individuals Multiple copies

Combating Premature Convergence

A. Sampling Mechanism

Characteristics of Function

A. Termination Condition

B. Population size

C. Penalty

Exploitation and Exploration

Exploreing the search space (population diversity) Too much random search no convergence

Exploiting the best individuals (selective pressure) Too much Hillclimbing local search

Sampling Mechanism

1. Stochastic sampling Proportional selection Stochastic universal sampling

2. Deterministic sampling Truncation selection (T%) Block selection (s)

)( ke

)( kSP

Stochastic Universal Sampling

procedure Stochastic Universal Samplingbegin

sum 0;ptr rand();for k 1 to pop_size do

sum sum ek; ‘expected value: while (sum > ptr) do

select chromosome k;ptr ptr + 1;

endend

end

sizepop

jjkk

kk

ffp

psizepope_

1

_

3. Mixed sampling Tournament selection (tournament size = 2) Boltzmann tournament

Brindle’s remainder stochastic sampling

4. Others Elitist model Expected value model ( ) Elitist expected value model Crowding factor model (CF)

Tjfif

e)()(

1

1

0)/)( cfvf

Ranking

Linear ranking

*

*

Nonlinear ranking

1_

)1()(

0

sizepop

qqr

rrankqrankprob

]1

1min)(max[min)( 1

n

rankrankprob n

1)1()( rankqqrankprob

Classification of Selection Strategies

Dynamic vs. Static: The selection probabilities vary across

generations.

Dynamic (yes) proportionate selection Static (no) ranking

Extinctive vs. Preservative

Guarantee each individual a non-zero selection probability.

Preservative (yes) proportional selection Extinctive (no) truncation selection

Pure? Parents are allowed to reproduce in one

generation only.

Pure (yes) SGA, Not pure(no) SGA (Elite)

ES),(

Generational? The set of parents is fixed until all offspring for the

next generation are completely produced.

Generational (yes) SGA; Not generational (no) tournament

ES),(

Modified Genetic Algorithm (modGA)

modGA_1, modGA_2 a two-step selection algorithm: “select-parents” and

“select-dead” three groups of strings: parents, dead, neutral strings Stochastic universal sampling Avoid exact multiple copies Compare with SGA, Crowding Factor Mode, and

Breeder GA (BGA) selection: dynamic, preservative, generational, elitist.

modGA_1 Algorithm

Procedure modGA_1begin

t 0initialize P(t)evaluate P(t)while (not termination-condition) dobegin

t t+1select-parents from P(t-1)select-dead from P(t-1)form P(t): reproduce the parentsevaluate P(t)

endend

modGA_2 AlgorithmProcedure modGA_2begin

t 0initialize P(t)evaluate P(t)while (not termination-condition) dobegin

t t+1select r parents from P(t-1)

select pop_size - r distinct chromosomes from P(t-1) and copy them to P(t)

form P(t): r parents breed r offspringevaluate P(t)

endend

Characteristics of the function

Scaling Linear scaling

Sigma truncation

Power law scaling

bfaf ii

))((

,0

0 ,)(

cffTf

otherwise

xifxxT

ii

ki i

ff

Window Scaling For maximum problem

W : all individuals in the last W generations ( W >= 0 )

For minimum problem

)(min)()(' yfxfxfWy

)()(max)(' xfyfxfWy

Termination Condition

Some number of evolution cycles Pre-defined value of fitness Some variation of individuals between

different generations Some percentage of the converged

alleles

Banach Fixpoint Theorem

Let

1. be a complete metric space

2. be a contractive mapping.

Then there has a unique fixpoint such that for any ,

where and .

,S

SSf :

f Sx

000 )( xxf

)(lim 0xfx i

i

Sx 0

))(()( 001 xffxf ii

A set together with is a metric space,

, if, for any elements ,

If is a metric space and , is contractive iff there is a constant such that for all

S RSS :

Syx ,,S

yxyxyx iff 0),( and ;0),( ),(),( xyyx

).,()(),( zxzyyx

),())(),(( yxyfxf

,S SSf :

)1,0[Syx ,

f

The sequence of metric space is a Cauchy sequence iff for any there is such that for all , .

A metric space is complete if any Cauchy sequence has a limit .

,, 10 pp

0knm , ),( nm pp

,S

k

nn

pp

lim,, 10 pp

Contractive Mapping GA(CM-GA)

Procedure CM-GABegin

t 0Initialize P(t)Evaluate P(t)While (not termination-condition) doBegin (contractive mapping f ( P(t) ) P(t+1))

t t + 1 Select P(t) from P(t-1) Alter P(t)

Evaluate P(t) if Eval( P(t-1)) >= Eval( P(t) ) ‘then t = t - 1

EndEnd

Px

in

i

xevalPEval )()( 1

CM-GA satisfies the assumptions of Banach fixpoint theorem: The distance

The space of populations is a metric space.

The metric space is complete.

The iteration is contractive.

CM-GA converges to population , which is a unique fixpoint in the space of all populations.

,S

212121 iff 0),( and ;0),( PPPPPP ),(),( 1221 PPPP

),(),(),( 313221 PPPPPP

)))1(())(((

))(),(()))(()),((( 2121

tPEvaltPEval

tPtPtPftPf

)1()(: tPtPf

))0((lim* PfP i

i

*P

,S

ii

PP

lim

otherwise

PEvalM

PEvalMPPif

PP

)(1

)(10

),(

2

1

21

21

Population Size

Too small converge too quickly Too large waste computational resources

Varying population Size (GAVaPS)

Varying Population Size(GAVaPS)

Procedure GAVaPS

begin

t = 0

initialize P(t)

evaluate P(t)

while (not termination-condition) do

begin

t = t+1

increase the age to each individual by 1

recombine P(t) ‘

evaluate P(t) ‘ p : reproduction ration

remove from P(t) all individuals

with age greater than their lifetime

end ‘

end

ptPopSizetAuxPopSize *)()(

PopSize t PopSize t AuxPopSize t Dead t( ) ( ) ( ) ( ) 1

Lifetime value: Proportional allocation

Linear allocation

Bi-linear allocation

)( where

),][

min(

21 MinLTMaxLT

MaxLTAvgFit

ifitnessMinLT

AbsFitMinAbsFitMax

AbsFitMinifitnessMinLT

][

2

MinFitAvgFit

MinFitifitnessMinLT

][

][ if ifitnessAvgFit AvgFitMaxFit

AvgFigifitnessMaxLTMinLT

][

)(2

1

][ if ifitnessAvgFit

Example

Initial_size = 20 Reproduction ratio = 0.4 Mutation ratio =0.015 Crossover ratio = 0.65 Length of chromosome = 20 MaxLT =7 MinLT = 1 Termination = no progress for consecutive 20 generations Independent run = 20 times

GAVaPS(1): proportional

GAVaPS(1): linear

GAVaPS(1): bi-linear

Handling Constraints

d

b

c

a

Solution Space

Xx

mmixh

mixg

xf

i

i

,,1 ,0)(

,,2,1 ,0)( tosubject

)(max

1

1

Penalty Strategy

Rejecting/Death Strategy Repairing Strategy Modifying GA Strategy Penalty Strategy Others:

Dynamic penalty functions Self-adaptive penalty function Combine penalty methods with repair algorithms

0-1 Knapsack Problem(example)

A thief robbing a store finds n items; the ith item is worth Pi dollars and weighs Wi pounds, where Pi and Wi are integers. He wants to take as valuable a load as possible, but he can carry at most C pounds in his knapsack. What items should he take? (This is called the 0-1 knapsack problem because each item must either be taken or left behind; the thief cannot take a fractional amount of an item or take an item more than once.)

ProblemMaximize

Subject to

n

i

iPixxf1

][][)(

CiWixn

i

1

][][

Penalty Strategy

)()()( xPenxfxeval

(max/min)otherwise 0

feasible is if

/

0)(

xxPen

Penalty Function:

Penalty Strategies

Penalty Functions: Logarithmic:

Linear:

Quadratic:

)()()( xPenxfxeval

n

i

CiWixxPen1

2 ))][][(1(log)(

n

i

CiWixxPen1

)][][()(

n

i

CiWixxPen1

2))][][(()(

) ][/][max (where1

iWiPni

Repair Strategy

Two different repair algorithms: Random repair Greedy repair (profit / weight)

n

i

iPixxeval1

][][)(

Algorithm:Procedure Repair (x)begin

knapsack-overfilled := false

if then knapsack-overfilled := truewhile (knapsack-overfilled) dobegin

i := select an item from the knapsackremove the selected item from the

knapsack:i.e.

if then knapsack-overfilled := false

end end

xx :

n

i

CiWix1

][][

0:][ ix

n

i

CiWix1

][][

Decoder Strategy

Ordinal representation L = ( 1 2 3 4 5 6 ) v = ( 4 3 4 1 1 1 )

Two different decoding algorithms: Random decoding Greedy decoding (profit / weight)

Algorithm:Procedure Decode (x)begin build a list L of itemsWeightSum := 0ProfitSum := 0

i := 1while i <= n dobeginj := x[i]remove the j-th item from the list Lif WeightSum + W[j] <= C thenbeginWeigthSum := WeightSum + W[j]ProfitSum := ProfitSum + P[j]endi := i+1endend

 

0.10.0 ))(001.01(

5.0sin0.5 :G4

0.10.0 )sgn( :G3

0.10.0 8/)8(int :G2

0.10.2 1)10sin( :G1

222

222

xyx

yx

xxx

xxeger

xxx

Crossover

One-point crossover Two-point crossover Multi-point crossover Segmented crossover Uniform crossover Gene pool recombination Simplex crossover

Messy Genetic Algorithm(mGA)

Representation tag, length, redundant and contradictory

v1 = ((7,1)(1,0)) v2 = ((3,1)(9,0)(3,1)(3,1)(3,1)) v3 = ((2,1)(2,0)(4,1)(5,0)(6,0)(7,1)(8,1))

overspecification – tie-breaking (first come, first serve) underspecification – probability / competitive templates

Selection tournament

Crossover splice & cut

Mutation bit inversion