The Leverage Space Model 317
Although you can use any of the aforementioned multidimensional op-
timization algorithms, I have opted for the genetic algorithm because it is
perhaps the single most robust mathematical optimization technique, aside
from the very crude technique of attempting every variable combination.
It is ageneraloptimization and search method that has been applied
to many problems. Often it is used in neural networks, since it has the
characteristic of scaling well to noisy or large nonlinear problems. Since the
technique does not require gradient information, it can also be applied to
discontinuous functions, as well as empirical functions, just as it is applied
to analytic functions.
The algorithm, although frequently used in neural networks, is not lim-
ited solely to them. Here, we can use it as a technique for finding the optimal
point in then+1 dimensional landscape.
The Genetic Algorithm
In a nutshell, the algorithm works by examining many possible candidate
solutions and ranking them on how well their value output, by whatever ob-
jective function, is used. Then, like the theory of natural selection, the most
fit survive and reproduce a new generation of candidate solutions, which
inherit characteristics of bothparentsolutions of the earlier generation.
The average fitness of the population will increase over many generations
and approach an optimum.
The main drawback to the algorithm is the large amount of processing
overhead required to evaluate and maintain the candidate solutions. How-
ever, due to its robust nature and effective implementation to the gamut
of optimization problems, however large, nonlinear, or noisy, it is this au-
thor’s contention that it will become the de facto optimization technique
of choice in the future (excepting the emergence of a better algorithm
which possesses these desirable characteristics). As computers become
ever more powerful and inexpensive, the processing overhead required of
the genetic algorithm becomes less of a concern. Truly, if processing speed
were zero, if speed were not a factor, the genetic algorithm would be the
optimization method of choice for nearly all mathematical optimization
problems.
The basic steps involved in the algorithm are as follows:
1.Gene length. You must determine the length of agene. A gene is
the binary representation of one member of the population of candidate
solutions, and each member of this population carries a value for each
variable (i.e., anfvalue for each scenario spectrum). Thus, if we allow a
gene length of 12 times the number of scenario spectrums, we have 12 bits