Monte Carlo Methods and the Genetic Algorithm Definitions and Considerations John E. Nawn MAT 5900 March 17 th, 2011

What is the Genetic Algorithm? Heuristic search method employing randomness in order to determine the optimal solution to a wide range of problems Applications include: ◦ Economics ◦ Number Theory ◦ Rankings ◦ Path Length Determination (TSP, etc.) Based in Neo-Darwinian theory

History of Genetic Algorithms Operational Research (1940s and 1950s) – birth of heuristics Evolutionsstrategie – Rechenberg and Schwefel (1960s) Adaptation in Natural and Artificial Systems – John Holland (1975) Increased computational complexity (1990s – 2000s)

Evolution: A Survey On the Origin of Species – Charles Darwin (1859) Proposed natural selection – environment creates selection pressure for individuals in a species Selected advantages may be heritable: provides method for determining fitness of offspring What Darwin (and biologists) didn’t know…

Genetics: A Survey Gregor Mendel (1863) Individuals within a species carry directions for their promulgation Segregation (First Law) Independent Assortment (Second Law) Increasing technology and the discovery of mutations and crossovers Genotype and phenotype

Terminology Population ◦ Set of possible solutions in any given generation Chromosomes ◦ Basic units that undergo reproduction in the algorithm ◦ Two types: binary and non-binary ◦ Minimum size requirements ◦ Genes and alleles Reproduction

Terminology Mutation ◦ Process of changing allele values in a chromosome ◦ Inversions ◦ How often? ◦ What type? Crossover ◦ Process of combining parental chromosomes to yield new chromosomes ◦ What type?

Terminology Selection ◦ Criterion ◦ Fitness functions ◦ Reeves and Rowe:  Tournament selection  Ranking Termination ◦ Diversity thresholds ◦ Generation limits ◦ Computational limits

Minimum String Length Requirements Reeves, Colin R.; p. 28

Mutations Simplicity of method Binary ◦ Reversal of alleles Non-binary ◦ Stochastic selection of new alleles ◦ Differing mutation rates ◦ Selecting complete mutations and error repair

Crossovers (X) Binary ◦ NX – N-point crossovers ◦ UX – Uniform crossover, or linear operator “masks” Non-Binary ◦ Difficulty in applying n-point crossovers ◦ PMX – Partially matched crossover ◦ UX – “in/out” order crossovers Further possibilities – Fox/ McMahon and Poon/ Carter

Fitness Functions Method comparing gene success Roulette wheel model of selection Selection pressure = individual fitness/ total fitness Benefit of larger selection pressure Niches

Critiques of the Genetic Algorithm: Biological and Philosophical Arguments What is natural selection selecting for? Evolution as a theory or fact: Lisa Gatlin Individual genes and group interactions Lamarckian or Darwinian evolution?

Critiques of the Genetic Algorithm: Mathematical Arguments Lack of theory in heuristic applications Newton’s Method problem Best possible solution or best solution? Pseudo-randomness Similarities to Markov chains and processes (a.k.a. t – 1 dependency)

What to Expect Next Crossover possibilities Holland’s method - schemata approaches Three applications: ◦ General Path Problems or the Traveling Salesman Problem (TSP) ◦ Ranking Styles ◦ Stock Selection

Selected Bibliography Craig, Nancy L. et. al. Molecular Biology: Principles of Genome Function. New York: Oxford University Press, Print. Krzanowski, Roman and Jonathan Raper. Spatial Evolutionary Modeling. New York: Oxford University, Inc., Print. Reeves, Colin R. and Johathan E. Rowe. Genetic Algorithms: Principles and Perspectives: A Guide to GA Theory. Boston: Kluwer Academic Publishers, Print. Russell, Peter J. iGenetics: A Mendelian Approach. San Francisco: Pearson Education, Inc., Print