The Functions

The functions are randomised self-similar (or fractal) landscapes. This means that, like real landscapes, the detail doesn't diminish as the resolution increases. This avoids some of the biases inherent in other traditional benchmark functions. More discussion on the motivations for the functions can be found in the references.

How the Functions are Constructed

Each function is composed of a large number of base functions or unit functions, so called because their non-zero values lie within a unit square. In order to preserve the self-similarity property, the average number of base functions increases with the inverse square (for the 2-d case used here) of their size. 

For each unit function, a range of fractal functions can be generated, depending on the depth to which the unit functions are applied (fractal depth), the average number of unit functions per order of magnitude square (density), and the sequence number (index) which is used to seed the random generator. 

The functions proposed for use in this competition are as follows:

Base function	Example Fractal Function	Properties
		Original sphere-based function, similar to that used on 2006 Huygens Probe competition Self-similar Continuous Not continuously differentiable   Fractal depth: 48 Density: 3
		A smooth function (tho it may not look it to the naked eye) Self-similar Continuous Continuously differentiable   Fractal depth: 48 Density: 3
		A discrete step-wise function Self-similar Not continuous Not continuously differentiable   Fractal depth: 48 Density: 16

Benchmarking Series

We call the functions resulting from a fixed fractal depth and density a series. The problem instances in the series can thus be obtained simply by setting the index. For example, the next three instances in the Fractal Sphere series, with depth 48 and density 3, are:

In this competition the three series are fixed. The problem instances within those series are chosen randomly (but the same for each contenstant).