4. GAs:Selected Topics

4. GAs:Selected Topics • GA theory • provides explanations why we may obtain convergence • Practical applications • 이론 따르지 않는 경우 많음 • 이유 • improper coding • limit on iteration number • limit on population size • Premature convergence 가능성 (local optimum)

4. GAs:Selected Topics • Eshelman and Schaffer [105]: premature 방지 기법 (1) a mating strategy, called incest prevention (Chap.10 TSP) (2) uniform crossover (Sec. 4.6) (3) detecting duplicate strings in the population (Sec.4.1) • 대부분 research 다음과 관련 • sampling mechanism에 의한 error (Sec.4.1) • 함수 자체의 특성 (Sec.4.2)

4.1 Sampling Mechanism • Two important issues • population diversity (exploration) • selective pressure (exploitation) • DeJong [82]: several variations of simple selection in Chap.2 • elitist model: preserving the best chromosomes • expected value model • introducing a count for each chromosome • initially f(v)/f • decreasing by 0.5 or 1 when selected for reproduction • not available for selection when the count falls below 0 • elitist expected value model • crowding factor model • a newly generated chromosome replaces an old one • and the doomed chromosome is selected from those which resemble the new one

4.1 Sampling Mechanism • Brindle[49] 1981 • further modifications • deterministic sampling • remainder stochastic sampling without replacement • stochastic tournament • remainder stochastic sampling with replacement • superiority over simple selection • remainder stochastic sampling with replacement • most successful, and adopted as standard • Baker [23] 1987 • a comprehensive theoretical study • presented a new improved version called stochastic universal sampling

4.1 Sampling Mechanism • Artificial weight 도입 • 실제 evaluation 값 대신 순위 (rank)에 따라 selection • rapid (premature) convergence는 super individual 때문이라는 근거 • 많은 방법 • 예) Baker [22] user-defined value MAX 사용 • User-defined parameter q 사용 방법 • linear function prob(rank) = q - (rank-1)r • nonlinear function prob(rank) = q(1-q)rank-1 • in case of linear function •  prob(i) = 1, so q = r(pop_size-1)/2 + 1/pop_size • if r=0, q=1/pop_size and no selective pressure • if q-(pop_size-1)r=0, then q=2/pop_size and maximum selective pressure • 즉, 1/pop_size와 2/pop_size 사이 값을 갖는 q로 selective pressure 조절 가능 • 예) pop_size=100, q=0.015

4.1 Sampling Mechanism • in case of nonlinear function • q (0..1) • larger values of q imply stronger selective pressure • 예) q=0.1, pop_size=100 • prob(1)=0.100 • prob(2)=0.1*0.9 = 0.090 • prob(3)=0.1*0.9*0.9=0.081 • ….. • prob(100)=0.000003 • some apparent drawbacks • user responsibility, chromosome의 상대적인 우열 정보 무시, 모든 경우를 uniform하게 다룸, Schema Theorem 어김 • Tournament selection [159] • k individuals 선택하고 그 중 best one selection • pop_size번 만큼 반복 • k=2 (tournament size) 많이 사용

4.1 Sampling Mechanism • Back and Hoffmeister [16]: categories of selection procedures • dynamic vs. static • static:constant probabilities between generations (e.g., ranking selection) • dynamic: no such requirement (e.g., proportional selection) • extinctive vs. preservative • preservative: non-zero selection probability for each individual • extinctive: no such requirement • pure • fitness와 무관하게 1 세대만 생존 • generational vs. on-the-fly • generational: parents fixed until all the offspring for the next generation are completely produced • on-the-fly: an offspring replaces its parent immediately

4.1 Sampling Mechanism • New two-step variation of the basic selection algorithm • Category of dynamic, preservative, generational, and elitist model • modGA in Figure 4.1 • procedure modGA • begin t0 Initialize P(t) Evaluate P(t) While (not termination-dondition) do begin t  t+1 Select-parents from P(t-1) Select-dead from P(t-1) From P(t); reproduce the parents Evaluate P(t) end • end

4.1 Sampling Mechanism • procedure modGA • Do not perform “select P(t) from P(t-1)” • But, select r chromosomes for reproduction and r chromosomes to die, based on fitness • After selection, three groups • r (not necessarily distinct) strings to reproduce (parents) • Precisely r strings to die (dead) • The remaining strings, neutral strings (at least pop_size-2r and at most pop_size-r) • modGA는 Schema theorem 만족 • modGA idea • Better utilization of the available storage space (population size) • Avoids leaving exact multiple copies of the same chromosome

4.1 Sampling Mechanism • Method of forming new generation P(t+1) • Step 1: select r parents from P(t) • Step 2: select pop-size-r distinct chromosomes from P(t) and copy them to P(t+1) • Step 3: let r parents breed to produce exactly r offspring • Step 4: insert these r new offspring into P(t+1)

4.2 Characteristics of the function • 3 basic directions • simulated annealing technique 도입 • thermodynamic operator 사용 population convergence 조절 • rank 사용 • scaling mechanism 도입하여 function 자체를 fixing • linear scaling • fi’ = a * fi + b where fi is the i-th chromosome’s fitness • sigma truncation • fi’ = fi + (f – c * ) • power law scaling • fi’ = fik ((eg., k=01.005)

4.2 Characteristics of the function • The most noticeable problem • differences in relative fitness • 예) f1(x) and f2(x)=f1(x)+const • same optimum value • but if const>>f1(x), f2 has much slow convergence (in the extreme case, totally random search) • GA 시행 결과 • 시행 때마다 다름 • caused by errors of finite sampling

4.2 Characteristics of the function • GENESIS 1.2ucsd (the best known system) • uses two parameters to control the search wrt the characteristics of functions • scaling window • sigma truncation factor • the system minimizes a function • eval(x) = F – f(x) where F > f(x) for all x • scaling window W controls how often F is updated • if W>0, set F to the greatest value of f(x) which has occurred in the last W generations • if W=0, infinite window, I.e., F=max{f(x)} over all evaluations • if W<0, use another method, I.e., sigma truncation

4.2 Characteristics of the function • Termination condition 중요 • 가장 간단한 방법: generation 수 while (t>=T) do /* in Figure 4.1 */ …… • 또다른 방법 • count the number of function evaluations • 또다른 방법 • when the chance for a significant improvement is relatively slim

4.3 Contractive mapping GA • Convergence of GA • most challenging theoretical issues • lots of researches • One possible approach for explaining GA convergence • using Banach fixpoint theorem [386] • only requirement that there should be an improvement in subsequent populations • Banach fixpoint theorem deals with contractive mappings on metric space • metric space 정의: p.69 • f is contractive iff (f(x), f(y)) <=  * (x,y) where () is a distance • Banach fixpoint Theorem • x = lim i-> fi(x0) for any x0

4.3 Contractive mapping GA • procedure CM-GA begin t0 initialize P(t) evaluate P(t) while (not termination-dondition) do begin contractive mapping f(P(t))  P(t+1) t  t+1 select P(t) from P(t-1) recombine P(t) evaluate P(t) if(Eval(P(t-1)) >= Eval(P(t)) then t  t-1 end end • P* = lim i-> fi(P(0)) for any P(0) • converge to a unique solution independent of initial population in case of infinite generations

4.4 GA with varying population size • population size • very important and may be critical in many applications • two small  converge too quickly (local optimum) • too large  waste computational resources • two important issues (Section 4.1) • population diversity • selective pressure • lots of researches

4.4 GA with varying population size • GAVaPS [12] • population size varies over time • introducing concept of ‘age’ of a chromosome • number of generations the chromosome stays alive • replace the concept of selection • during the ‘recombine P(t)’ step, a new auxiliary population created • population of offspring • AuxPopSize(t) =  PopSize(t) *   •  is a reproduction ratio • 각 chromosome은 fitness와 무관하게 동일 확률로 선택 • 대신 ‘lifetime’ parameter 사용 • Popsize(t+1) = PopSize(t) + AuxPopSize(t) – D(t) • D(t): number of die-off chromosome

4.4 GA with varying population size • procedure GAVaPS begin t0 initialize P(t) evaluate P(t) while (not termination-dondition) do begin t  t+1 increase the age of each individual by 1 recombine P(t) evaluate P(t) remove from P(t) all individuals with age greater than their lifetime end end

4.4 GA with varying population size • Assignment of lifetime value • 방법 1: constant value for all the individuals  bad performance • 방법 2: fitness와 state of search 고려 • state of search • AvgFit, MaxFit, MinFit: 현재 population의 ave, max, min fitness • AbsFitMax, AbsFitMin: max, min fitness found so far • MaxLT and MinLT: max and min allowable lifetime •  = (MaxLT-MinLT)/2

4.4 GA with varying population size • Assignment of lifetime value • 방법 2: fitness와 state of search 고려 (1) proportional allocation min(MinLT+ (fitness[i]/AvgFit), MaxLT) • roulette-wheel selection • a serious drawback: ‘objective goodness’ 사용 안함 (2) linear allocation MinLT+2((fitness[i]-AbsFitMin)/(AbsFitMax-AbsFitMin)) • population 급히 증가할 위험 (3) bi-linear allocation MinLT+((fitness[i]-MinFit)/(AvgFit-MinFit)) if AvgFit>=fitness[i] (MinLT+MaxLT)/2+((fitness[i]-AvgFit)/(MaxFit-AvgFit)) otherwise • (1)과 (2)의 타협점

4.4 GA with varying population size • Testing of GAVaPS • 네개함수를 testbed로 사용 • covering the wide spectrum of possible function types • p.75 G1 ~ G4 • SGA(Simple GA)와 비교 실험 • chromosome length=20 • initial pop-size = 20 • =0.4, pm=0.015, pc=0.65 • MinLT=1, MaxLT=7 • two parameters (성능 측정 위한) • cost: evalnum (average number of function evaluations over all runs) • performance: avgmax (average of maximal values found over all runs) • Figure 4.4: generation number에따른 fitness와 pop-size • Figure 4.5&4.6: 에 따른 변화 • Figure 4.7&4.8: initial pop-size에 따른 변화 • Table4.1: 성능 비교

4.5 GA, Constraints, and the Knapsack Problem • Constraint handling techniques • solution은 주어진 constraint (조건)을 만족해야하는 문제 • feasible한 해와 infeasible한 해 • 예) TSP, Knapsack problem, ….. • GA 기법 세 가지: penalty function, repair, decoder (1) Penalty function • constraint violate하는 solution은 function evaluation에서 “goodness”를 줄임으로써 penalty 부여 penalty function • 즉, constraint problem ----------------------> unconstraint problem • penalty 값 • constant 부여, 또는 • violation 정도에 따라 부여, 또는 • violating solution을 population에서 제거 (즉, death penalty)

4.5 GA, Constraints, and the Knapsack Problem (2) Repair • infeasible solution을 feasible하게 repairing • repairing 계산량 많고, 경우에 따라 repairing이 매우 어려움 (3) Decoder • use of special representation mappings (decoders) which guarantee the generation of feasible solutions, or • use of problem-specific operators which preserve feasibility of the solutions • 모든 constraints 구현 가능한 건 아니고, 특정 문제에 맞게 tailoring 필요

4.5.1 The 0/1 Knapsack problem and the test data • 0/1 knapsack problem • n개 물건과 한 개 knapsack • W[i]: weight, P[i]: profit, C: knapsack capacity • find a binary vector x = <x[1], x[2], …., x[n]> such that • i=1,n x[i]W[i] <= C and • i=1,n x[i]P[i] is maximum

4.5.1 The 0/1 Knapsack problem and the test data • Three sets of test data • uncorrelated • W[i]: (uniformly) random ([1..v]), and • P[i]: (uniformly) random ([1..v]) • weakly correlated • W[i]: (uniformly) random ([1..v]), and • P[i]: W[i] + (uniformly) random ([-r..r]) • strongly correlated • W[i]: (uniformly) random ([1..v]), and • P[i]: W[i] + r (* W와 P간의 correlation정도가 문제 난이도와 관련, 즉, higher correlation --> higher difficulty)

4.5.1 The 0/1 Knapsack problem and the test data • Data 생성 • v=10 and r=5 • n=100, 250, 500 • 두 knapsack type • restricted • C1 = 2v • 적은 수의 item만 넣을 수 있음 • average • C2 = 0.5 i=1,n W[i] • 대략 전체의 반 정도 items를 넣을 수 있음

4.5.2 Description of Algorithms • Three types of algorithms • algorithms based on penalty function: Ap[i] • algorithms based on repair method: Ar[i] • algorithms based on decoders: Ad[i] • Algorithm Ap[i] • eval(x) = i=1,n x[i]P[i] - Pen(x) • Pen(x) = 0 for all feasible solutions > 0 otherwise • logarithmic, linear, and quadratic penalty functions • Ap[1]: Pen(x) = log2(1+ (i=1,n x[i]W[i] - C)) • Ap[2]: Pen(x) =  (i=1,n x[i]W[i] - C) • Ap[3]: Pen(x) = ( (i=1,n x[i]W[i] - C))2 • ( = max{P[i]/W[i]})

4.5.2 Description of Algorithms • Algorithm Ar[i] • eval(x) = i=1,n x’[i]P[i] • x = repaired version of the original vector x • replacement rate • 5% rule[301]: 5% is better than any other rate • 두 가지 select 방법 • Ar[1]: random repair • Ar[2]: greedy repair (based on P[i]/W[i])

4.5.2 Description of Algorithms Procedure repair(x) begin knapsack_overfilled = false x’ = x if i=1,n x[i]W[i] > C then knapsack_overfilled = true while (knapsack_overfilled) do begin i = select an item from the knapsack remove the selected item from the knapsack (i.e., x’[i]=0) if i=1,n x’[i]W[i] <= C then knapsack_overfilled = false end end

4.5.3 Experiments and Results • Experiments • pop_size=100 • pm=0.05, pc=0.65 • Table 4.2 • 5% repair rule • no influence for 0/1 knapsack problem: Table 4.3 • (5% rule [301]: network design problem, graph coloring problem에서 유효성 확인) • main conclusions • no feasible solution for C1 by Ap[] • quite intuitive [332] • Ap[1] is the best for C2 • Ar[2] is the best for C1

4.6 Other Ideas • Multi-point crossover • single-point crossover • S1=(001********01) • S2=(****11*******) 에서 • v1=(0010001101001)과 v2=(1110110001000)으로 • S3=(001*11*****01) 생성 불가능 • two-point crossover • v1’=(001|01100|01001) • v2’=(111|00011|01000) • v1’ is matched by S3 • uniform crossover • bitwise 교환

4. GAs:Selected Topics