## Simple Genetic Algorithms With Linear Fitness

Michael D. Vose
Computer Science Dept.
107 Ayres Hall
The University of Tennessee
Knoxville, TN 37996-1301
vose@cs.utk.edu
(615) 974-5067
And
Alden H. Wright
*(This work was done while visiting the computer science department
of the University of Tennessee.)*
Computer Science Dept.
The University of Montana
Missoula, MT. 59812
wright@cs.umt.edu
(406) 243-2883
### ABSTRACT:

A general form of stochastic search is described random heuristic
search and some of its general properties are proved. This provides a
framework in which the simple genetic algorithm (SGA) is a special case.
The framework is used to illuminate relationships between seemingly
different probabilistic perspectives on SGA behavior. Next the SGA
is formalized as an instance of random heuristic search. The
formalization is then used to show expected population fitness is
a Lyapunov function in the infinite population model when mutation is
zero and fitness is linear. In particular, the infinite population
algorithm must converge, and average population fitness increases from
one generation to the next. The consequence for a finite population
SGA is that the expected population fitness increases from one
generation to the next. Moreover, the only stable fixed point of the
expected next population operator corresponds to the population consisting
entirely of the optimal string. This result is extended by way of
a perturbation argument to allow nonzero mutation.