An Introduction to Genetic Algorithms - Boente

Chapter 4: Theoretical Foundations of Genetic Algorithms
still conjectures; there is as yet no detailed understanding of the nature of the GA surface when F and are
combined. Understanding this surface is a worthwhile (and still open) endeavor.
4.4 STATISTICAL−MECHANICS APPROACHES
I believe that a more useful approach to understanding and predicting GA behavior will be analogous to that
of statistical mechanics in physics: rather than keep track of the huge number of individual components in the
system (e.g., the exact genetic composition of each population), such approaches will aim at laws of GA
behavior described by more macroscopic statistics, such as "mean fitness in the population" or "mean degree
of symmetry in the chromosomes." This is in analogy with statistical mechanics' traditional goal of describing
the laws of physical systems in terms of macroscopic quantities such as pressure and temperature rather than
in terms of the microscopic particles (e.g., molecules) making up the system.
One approach that explicitly makes the analogy with statistical mechanics and uses techniques from that field
is that of the physicists Adam Prügel−Bennett and Jonathan Shapiro. Their work is quite technical, and to
understand it in full requires some background in statistical mechanics. Here, rather than go into full
mathematical detail, I will sketch their work so as to convey an idea of what this kind of approach is all about.
Prügel−Bennett and Shapiro use methods from statistical mechanics to predict macroscopic features of a GA's
behavior over the course of a run and to predict what parameters and representations will be the most
beneficial. In their preliminary work (Prügel−Bennett and Shapiro 1994), they illustrate their methods using a
simple optimization problem: finding minimal energy states in a one−dimensional "spin glass." A spin glass is
a particular simple model of magnetic material. The one−dimensional version used by Prügel−Bennett and
Shapiro consists of a vector of adjacent "spins," ( where each Si is either ‘ or
+1. Each pair of neighboring spins (i,i + 1) is "coupled" by a real−valued weight Ji. The total energy of
the spin configuration ( is
Setting up spin−glass models (typically, more complicated ones) and finding a spin configuration that
minimizes their energy is of interest to physicists because this can help them understand the behavior of
magnetic systems in nature (which are expected to be in a minimal−energy state at low temperature).
The GA is set with the problem of finding an ( that minimizes the energy of a one−dimensional spin glass
with given Ji's (the Ji values were selected ahead of time at random in [ ‘, +1]). A chromosome is simply a
string of N +1 spins (‘ or +1). The fitness of a chromosome is the negative of its energy. The initial
population is generated by choosing such strings at random. At each generation a new population is formed by
selection of parents that engage in single−point crossover to form offspring. For simplicity, mutation was not
used. However, they did use an interesting form of selection. The probability p ± that an individual ± would be
selected to be a parent was
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