### GA Demo: Multiple global optima of the Vincent function

The Vincent function is defined as follows: $f(\vec{X}) = - \sum_{i=1}^n sin(10.log(x_i))$, with $\vec{X}$ varying in [0.25:10].  This is a plot of the function in 1 dimension:

Vincent Function- 1 Dimension

As you can see there, are 6 global minima. This function is such that in N dimension, the function with have 6^D global minima. For example, in 2D it has 36 minima:

Vincent Function - 2D

The job of a multi-modal optimizer is to find all the global optima in a multi-modal function.  See this demo:

See the interesting bit? As the population of the GA evolves, it settles into the 36 minima, forming beautiful clusters around them.

The algorithm used here is based on our work  Deb, K., Saha, A., Finding Multiple Solutions for Multimodal Optimization Problems Using a Multi-Objective Evolutionary Approach (Accepted to be presented as full paper in GECCO-2010). (Let me know if you want to take a look at the paper).

You may also be interested in my past two demos: