Evolutionary Computation : A Unified Approach

116 CHAPTER 6. EVOLUTIONARY COMPUTATION THEORY
and assumptions are often made that are motivated by the goal of improving analytical
tractability rather than by properties of the underlying object of study. Classic examples of
this are assumptions of linearity or independence of variables. Obviously, the wider the gap,
the more concern there is as to whether the analytical results obtained from such models
carry over to the “real” thing.
These issues confront us immediately in our attempt to develop useful abstract models
of evolutionary computation methods. In order to be useful, these models must be capable
of capturing the important algorithmic details of particular EAs. However, the inclusion
of such details often leads to analytic intractability, since the important EA components
interact with each other in complex, nonlinear ways.
Yet, if theory is to have any impact on practice, these gaps must be addressed. This can
be done in a number of ways. First, we can require a particular theory to make predictions
about the behavior of “real” EAs, and evaluate a theory’s usefulness in terms of the accuracy
of its predictions. However, for some theories, the gap is so wide that making practical
predictions is not even reasonable or possible. These gaps can often be narrowed by incrementally
removing some of the simplifying assumptions, and studying the effects they have
on the earlier analyses. Another gap reduction technique is to perform experimental studies
on the models that are mathematically intractable, not unlike the computational techniques
used to “solve” nonlinear systems of equations. And, finally, one can perform experimental
analyses of the EAs themselves in ways that can be generalized to other situations.
Consequently, as we survey the important theoretical results, we will perform some “gap
analysis” as well. In the process we will obtain a better understanding of the current gaps
that exist between EC theory and practice, and how they are being addressed.
There are a wide range of analytical tools one might use to develop a theoretical framework
for EC. The ones that have proved most useful historically are:
• dynamical systems tools for characterizing EA population dynamics,
• Markov process tools for analyzing the stochastic properties of EAs,
• statistical mechanics tools for studying aggregate EA properties,
• analysis of algorithm tools for assessing the computational complexity of EAs, and
• mathematical tools for characterizing the optimization properties of EAs.
The fact that there is more than one set of tools and techniques for EC analysis is a
reflection of the fact that most theories are designed to capture specific properties and answer
specific questions. For systems of any complexity it is seldom the case that there is
a single “unified” theory capable of addressing all interesting issues at all levels of abstraction.
Rather, our overall understanding of complex systems emerges from a collection of
theoretical models that complement one another.
In the case of EC theory, this collection of models falls into two broad categories:
1) application-independent theories that focus on understanding EA dynamics, and 2)
application-dependent theories that focus on understanding EA problem-solving properties.
The remainder of this chapter explores each of these categories in more detail.