Solving Function Optimization Problems with Genetic Algorithms September 26, 2001 Cho, Dong-Yeon , Tel:

© 2001 SNU CSE Biointelligence Lab 2 Function Optimization Problem Example

© 2001 SNU CSE Biointelligence Lab 3 Representation – Binary String Code length

© 2001 SNU CSE Biointelligence Lab 4 Mapping from a binary string to real number Binary NumberDecimal Number x1x x2x

© 2001 SNU CSE Biointelligence Lab 5 Framework of Simple GA Generate Initial Population Evaluate Fitness Select Parents Generate New Offspring Termination Condition? Yes No Fitness Function Crossover, Mutation Best Individual

© 2001 SNU CSE Biointelligence Lab 6 Initial Population Initial population is randomly generated.

© 2001 SNU CSE Biointelligence Lab 7 Fitness Evaluation Procedure: Evaluation  Convert the chromosome’s genotype to its phenotype.  This means converting binary string into relative real values.  Evaluate the objective function.  Convert the value of objective function into fitness.  For the maximization problem, the fitness is simply equal to the value of objective function.  For the minimization problem, the fitness is the reciprocal of the value of objective function.

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9 Selection Fitness proportional (roulette wheel) selection  The roulette wheel can be constructed as follows.  Calculate the total fitness for the population.  Calculate selection probability p k for each chromosome v k.  Calculate cumulative probability q k for each chromosome v k.

© 2001 SNU CSE Biointelligence Lab 10 Procedure: Proportional_Selection  Generate a random number r from the range [0,1].  If r  q 1, then select the first chromosome v 1 ; else, select the kth chromosome v k (2  k  pop_size) such that q k-1 < r  q k. pkpk qkqk

© 2001 SNU CSE Biointelligence Lab 11 Tournament Selection  Repeat below procedure two times  Randomly draw a tournament of size t from the population with replacement.  Choose fittest program to be parent.

© 2001 SNU CSE Biointelligence Lab 12 Genetic Operations Crossover  One point crossover  Crossover rate p c Procedure: Crossover  Select two parents.  Generate a random number r c from the range [0,1].  If r c < p c then perform undergo crossover. Mutation  Mutation alters one or more genes with a probability equal to the mutation rate p m.

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© 2001 SNU CSE Biointelligence Lab 14 Test Functions Maximization (d=5)

© 2001 SNU CSE Biointelligence Lab 15 Minimization (d=5)  Ackley’s function  Rastrigin’s function  Schwefel’s (sine root) function

© 2001 SNU CSE Biointelligence Lab 16 Experiments (Minimum Requirements) At least two test functions  Maximization Problem  One of the Minimization Problems Various experimental setup  Termination condition: maximum_generation  2 Selection Method  3 settings  10 runs  Proportional: Crossover and Mutation rate (p c, p m )  Tournament: Tournament size (t)  Elitism  The best chromosome of the previous population is just copied.

© 2001 SNU CSE Biointelligence Lab 17 Results For each test function  Result table for the best solution and your analysis  Present the optimal value and its input for the function.  Both genotype and phenotype  Draw a learning curve for the run where the best solution was found. ProportionalTournament Average  SD BestWorst Average  SD BestWorst Setting 1 Setting 2 Setting 3

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© 2001 SNU CSE Biointelligence Lab 19 References Source Codes  Simple GA coding  GA libraries (C, C++, JAVA, …)  MATLAB Tool box Web sites Books  Genetic Algorithms and Engineering Design, Mitsuo Gen and Runwei Cheng, pp. 1-15, John Wiley & Sons,  Genetic algorithms in search, optimization, and machine learning, David E. Goldberg, Addison-Wesley, 1989.

© 2001 SNU CSE Biointelligence Lab 20 Pay Attention! Due (October 17, 2001): By the end of class Submission  Source code and executable file(s)  Proper comments in the source code  Via  Report: Hardcopy!!  Running environments  Results for many experiments with various parameter settings  Analysis and explanation about the results in your own way

© 2001 SNU CSE Biointelligence Lab 21 Bonus Problems Maximization (d=30)  3-deceptive function Minimization (d=5)  Rosenbrock’s function