Evolutionary Computation : A Unified Approach

5.2. EA-BASED OPTIMIZATION 97
5.2.2.7 Non-homogeneous Parameter Optimization
Perhaps one of the biggest OPT-EA parameter optimization payoff areas is the situation
frequently encountered in practice, in which the parameters of an optimization problem
have non-homogeneous data types. For example, suppose I am interested in tuning the
parameters of a complex engineering design or a military wargame simulation. In such
cases it is most likely that the parameters will be a mixture of real numbers (e.g., beam
thickness), integers (e.g., number of floors), and nominal parameters (e.g., facade type).
Generally, mathematical optimization techniques require one to “embed” such problems
into a real-valued space by converting everything to real numbers, which can lead to messy
issues relating to continuity and derivative calculations involving the pseudo-real-valued
parameters.
By contrast, it is relatively easy to modify an OPT-EA to handle non-homogeneous
parameters. In the case of genotypic representations, nothing new is required except an
indication of the data type of each parameter, since the mapping between internal (binary)
strings and external parameter values is now dependent on the data type of each parameter.
Once converted, all of the standard reproduction operators can be used.
If the internal representation is phenotypic, the standard crossover operations can still be
used as is. However, mutation must now be generalized to handle different parameter data
types. But, as we have seen in the previous sections, there are simple mutation operators
for each of the three data types. So, for non-homogeneous problems, the parameter data
type determines which mutation operator to use.
5.2.3 Constrained Optimization
So far, there has been a tacit assumption that, while solving a parameter optimization
problem, the values of individual parameters can be changed independently of one another.
In practice, however, there are frequently complex, nonlinear interdependencies between
parameters that constrain the search space and must be dealt with in one form or another
by any parameter optimizer, EA-based or otherwise.
One standard approach to handling constraints is to embed the constrained space in an
unconstrained space and augment the objective function with a “penalty function” for points
outside the constrained space. As a simple example, suppose that the solution space to be
searched is a two-dimensional unit circle centered at the origin. This means that in order
to stay within the boundary of the unit circle, both parameters must simultaneously satisfy
x 2 1 + x 2 2