A discrete representation for real optimization with unique search properties

Walking triangle representations for real optimization are linear representations drawn from the group that acts on simplices of Euclidean space. The representation encodes a series of modifications to an initial simplex, evaluating the quality of the point at its center of mass for the function being optimized. Different operations available in the representation permits easy tailoring of the degree of exploration and exploitation implemented and also permit control over the order in which they happen. Some operations perform search with linear differences in the position of the search point while others exponentially increase or decrease the distance between adjacent points. The current work focuses on developing theory and experimentally testing a walking triangle representation based on the walk, center, and uncenter moves representing changes in the position of the modeled point that are linear, exponentially decreasing, and exponentially increasing, respectively. The experimental results are compared with one another and with a standard evolutionary algorithm. A new test function called the eight hill function, specifically intended to test the ability of an algorithm to explore, is presented.