# A Fractal Representation for Real Optimization

### Daniel Ashlock and Justin Schonfeld

Accepted to the 2007 Congress on Evolutionary Computation

The chaos game, in which a moving point is repeatedly averaged
toward randomly selected vertices of a triangle, is one method of
generating the fractal called the Sierpinski triangle. The sequence of
vertices, called generators, used to reach a given point of the
Sierpinski triangle yields a map from strings over a three-character
alphabet to points in the plane. This study generalizes that
representation to give a character-string representation for points in
Rn . This is a novel representation for evolutionary
optimization. With the correct generating points the method is proven
to search its entire target domain at an easily controlled
resolution. The representation can be used to achieve the same goals
as niche specialization at a far lower computational cost because the
optima located are specified by strings which can be stored and
searched in standard string dictionaries. An implementation of the
algorithm called the multiple optima Sierpinski searcher(MOSS) is
found to be substantially faster at locating diverse collections of
optima than a standard optimizer. The Sierpinski representation has a
number of natural mathematical properties that are described in the
paper. These include the ability to adapt both it search domain and
its resolution on the fly during optimization.