An Introduction to Genetic Algorithms - Boente

Chapter 4: Theoretical Foundations of Genetic Algorithms
and the expected proportion of string y in the next population is
Since pi(y) is equivalent to the expected proportion of string y in the next population, we can finally write
down a finished expression for Qi,j:
The matrix QI,j gives an exact model of the simple GA acting on finite populations.
Nix and Vose used the theory of Markov chains to prove a number of results about this model. They showed,
for example, that as n ’ , the trajectories of the Markov chain converge to the iterates of G (or G p) with
probability arbitrarily close to 1. This means that for very large n the infinite−population model comes close
to mimicking the behavior of the finite−population GA. They also showed that, if Gp has a single fixed point,
as n ’ the GA asymptotically spends all its time at that fixed point. If G p has more than one fixed point, then
as n ’ , the time the GA spends away from the fixed points asymptotically goes to 0. For details of the proofs
of these assertions, see Nix and Vose 1991.
Vose (1993) extended both the infinite−population model and the finite−population model. He gave a
geometric interpretation to these models by defining the "GA surface" on which population trajectories occur.
I will not give the details of his extended model here, but the main result was a conjecture that, as n ’ , the
fraction of the time the GA spends close to nonstable fixed points asymptotically goes to 0 and the time the
GA spends close to stable fixed points asymptotically goes to 1. In dynamical systems terms, the GA is
asymptotically most likely to be at the fixed points having the largest basins of attraction. As n ’ , the
probability that the GA will be anywhere else goes to 0. Vose's conjecture implies that the short−term
behavior of the GA is determined by the initial population—this determines which fixed point the GA initially
approaches—but the long−term behavior is determined only by the structure of the GA surface, which
determines which fixed points have the largest basins of attraction.
What are these types of formal models good for? Since they are the most detailed possible models of the
simple GA, in principle they could be used to predict every aspect of the GA's behavior. However, in practice
such models cannot be used to predict the GA's detailed behavior for the very reason that they are so
detailed—the required matrices are intractably large. For example, even for a very modest GA with, say,l = 8
and n = 8, Nix and Vose's Markov transition matrix Q would have more than 10 29 entries; this number grows
very fast with l and n. The calculations for making detailed predictions simply cannot be done with matrices
of this size.
This does not mean that such models are useless. As we have seen, there are some less detailed properties that
can be derived from these models, such as properties of the fixed−point structure of the "GA surface" and
properties of the asymptotic behavior of the GA with respect to these fixed points. Such properties give us
some limited insight into the GA's behavior. Many of the properties discussed by Vose and his colleagues are
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