Derivative-Free Optimization: Genetic Algorithms

1. Derivative-Free Optimization:Genetic Algorithms Dan SimonCleveland State University

2. Outline • Biological Genetics • Genetic Algorithm: A Short History • Genetic Algorithm Example: Robot Design • Genetic Algorithm Options • Genetic Algorithm Example: Ackley Function • Continuous Genetic Algorithm • GA Code Walk-Through

3. Charles Darwin • Born in England, 1809 • “You care for nothing but shooting, dogs, and rat-catching; and you will be a disgrace to yourself and all your family.” • Medicine? • Theology? • Biology?

4. Charles Darwin • H. M. S. Beagle: 1831–1836 • The Origin of Species: 1836–… • Paper from Alfred Wallace: 1858 • Presentation of both papers by Darwin: 1858 • The Origin of Species: 1859“Only” 500 pagesThe first printing (1,250 copies) sold out the first day

5. Darwin’s Theory of Natural Selection • Survival of the fittest • Controversial • Anti-Christian? • How are traits passed to children? • Misconceptions • Traits of parents could be blended in children • Acquired traits could be passed to children

6. Gregor Mendel • Born in 1822 in Czech. • Poor farming family • Joined Augustinian monastery at age 21 • Studied botany (peas) • Discovered the idea of genes, heredity, and dominance • His publication (1865) was ignored at the time

7. Genetic Algorithms • Princeton, John von Neumann • Nils Barricelli (mathematician), 1953, artificial life • 1954: “Esempi numerici di processi di evoluzione” (Numerical models of evolutionary processes)

8. Genetic Algorithms • Alexander Fraser (biologist) – England, Hong Kong, New Zealand, Scotland, Australia – 1957: “Simulation of genetic systems by automatic digital computers” • Hans-Joachim Bremermann – University of Washington, UC Berkeley – 1958: “The evolution of intelligence”

9. Genetic Algorithms • George Box (statistician) – Imperial Chemical Industries (England) – 1957: “Evolutionary operation: A method for increasing industrial productivity”“Essentially, all models are wrong, but some are useful” (1987) • George Friedman, UCLA – 1956: “Selective Feedback Computers for Engineering Synthesis and Nervous System Analogy” (Master’s thesis)

10. GA for Robot Design 000 = 5-volt stepper 001 = 9-volt stepper 010 = 12-volt stepper 011 = 24-volt stepper 100 = 5-volt servo 101 = 9-volt serv 110 = 12-volt serv 111 = 24-volt servo 000 = 12-volt NiCd battery 001 = 24-volt NiCd battery 010 = 12-volt Li-ion battery 011 = 24-volt Li-ion battery 100 = 12-volt solar panel 101 = 24-volt solar panel 110 = 12-volt fusion reactor 111 = 24-volt fusion reactor encoding for motor spec encoding for power spec

11. GA for Robot Design Fitness = Range (hrs) + Power (W) – Weight (kg) • Experiment or simulation We are combining incompatible units Randomly create initial population: Each individual is represented with a chromosome which has two genes

12. Two Parents Two Children 0 0 1 1 1 0 1 0 1 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 crossover point GA for Robot Design Individual 1 chromosome = 010101 Individual 1’s motor genotype is 010, and its motor phenotype is “12-V stepper”

13. Individual 2 Individual 4 20 Individual 1 10 40 30 Individual 3 GA for Robot Design How do we decide which individuals to mate? Fitness proportional selection, AKA roulette-wheel selection Example: four individuals with fitness values 10, 20, 30, and 40

14. A Simple Genetic Algorithm Parents  {randomly generated population} While not (termination criterion) Calculate the fitness of each parent in the population Children =  While |Children| < |Parents| Use fitnesses to select a pair of parents for mating Mate parents to create children c1 and c2 Children  Children  { c1, c2} Loop Randomly mutate some of the children Parents  Children Next generation

15. GA Termination Criteria • Generation count • Fitness threshold • Fitness improvement threshold

16. Critical GA Design Parameters • Elitism • Encoding scheme • Fitness function and scaling • Population size • Selection method (tournament, rank, …) • Mutation rate • Crossover type • Speciation / incest

17. GA Schematic Elitism Selection Crossover Mutation Current generation Next generation

18. Encoding Binary: Neighboring phenotypes have dissimilar genotypes, and vice versa Gray: Neighboring phenotypes have similar genotypes 000 001 011 010 110 111 101 100 x = -5 : 0.1 : 2 plot(x, x.^4 + 5*x.^3 + 4*x.^2 – 4*x + 1);

19. Gray Codes Bell Labs researcher Frank Gray introduced the term reflected binary code in his 1947 patent application.

20. Ackley Function Minimization problem; global minimum = 0 (at x = y = 0) Can be generalized to any number of dimensions

21. Ackley Function • 100 Monte Carlo simulations • Population size = 50 • Mutation rate = 2% • Crossover probability = 100% • Single point crossover • Encoding: binary or gray • Elitism: 0 or 2

22. Ackley Function

23. Ackley Function Average of 100 Monte Carlo simulations

24. Ackley Function

25. Continuous Genetic Algorithms Parents crossover: [1.23, 4.76, 2.19, 7.63] [9.73, 1.09, 4.87, 8.28] Children: [1.23, 1.09, 4.87, 8.28] [9.73, 4.76, 2.19, 7.63] Usually, GAs for continuous problems are implemented as continuous GAs crossover point

26. Continuous Genetic Algorithms Blended crossover: Select a random number r [0, 1] Genotype operation: c = p1 + r(p2—p1) Parent 2 Child Parent 1

27. Continuous Genetic Algorithms Mutation: Suppose x = [9.73, 1.09, 4.87, 8.28] Problem dimension = 4 r random number  [0, 1] If r < pm then i random integer  [1, 4] r random number  [0, 1] x(i)  xmin + r(xmax – xmin) end if Aggressive Mutation

28. Continuous Genetic Algorithms Mutation: Suppose x = [9.73, 1.09, 4.87, 8.28] Problem dimension = 4 r random number  [0, 1] If r < pm then i random integer  [1, 4] r Gaussian random number  N(0, ) x(i)  x(i) + r end if Gentle Mutation

29. Rastrigin Benchmark Function Global minimum f(x) = 0 at xi = 0 for all i p dimensions Lots of local minima

30. Rastrigin Benchmark Function Population size = 50 Mutation rate = 1% Crossover prob. = 100% Single point crossover Elitism = 2 15 dimensions GA.m