Definitions and Considerations

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Oct 23, 2013 (3 years and 10 months ago)

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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, 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