Monte Carlo Methods and the Genetic Algorithm

Definitions and Considerations

John E. NawnMAT 5900March 17th, 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 AlgorithmsOperational Research (1940s and

1950s) – birth of heuristicsEvolutionsstrategie – Rechenberg

and Schwefel (1960s)Adaptation in Natural and

Artificial Systems – John Holland (1975)

Increased computational complexity (1990s – 2000s)

Evolution: A SurveyOn 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 SurveyGregor Mendel (1863)Individuals within a species carry

directions for their promulgationSegregation (First Law)Independent Assortment (Second

Law)Increasing technology and the

discovery of mutations and crossovers

Genotype and phenotype

TerminologyPopulation

◦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?

TerminologySelection

◦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

MutationsSimplicity of methodBinary

◦Reversal of allelesNon-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 FunctionsMethod comparing gene successRoulette wheel model of selectionSelection pressure =

individual fitness/ total fitnessBenefit of larger selection

pressureNiches

Critiques of the Genetic Algorithm:Biological and Philosophical ArgumentsWhat is natural selection

selecting for?Evolution as a theory or fact: Lisa

GatlinIndividual genes and group

interactions Lamarckian or Darwinian

evolution?

Critiques of the Genetic Algorithm:Mathematical ArgumentsLack of theory in heuristic

applicationsNewton’s Method problemBest possible solution or best

solution?Pseudo-randomnessSimilarities to Markov chains and

processes (a.k.a. t – 1 dependency)

What to Expect NextCrossover possibilitiesHolland’s method - schemata

approachesThree applications:

◦General Path Problems or the Traveling Salesman Problem (TSP)

◦Ranking Styles◦Stock Selection

Selected BibliographyCraig, Nancy L. et. al. Molecular Biology:

Principles of Genome Function. New York: Oxford University Press, 2010. Print.

Krzanowski, Roman and Jonathan Raper. Spatial Evolutionary Modeling. New York: Oxford University, Inc., 2001. Print.

Reeves, Colin R. and Johathan E. Rowe. Genetic Algorithms: Principles and Perspectives: A

Guide to GA Theory. Boston: Kluwer Academic Publishers, 2003. Print.

Russell, Peter J. iGenetics: A Mendelian Approach. San Francisco: Pearson Education, Inc., 2005. Print