User:MatthewKean2020/sandbox

== Chaos Game Optimization (CGO) (Talatahari and Azizi 2020) == Chaos theory is a branch of mathematics concentrating on the specific characteristics of dynamical systems which are extremely sensitive to initial conditions. Considering the randomness of these dynamical systems, chaos theory denotes on the existence of some primary patterns such as similar loops, repeated templates, fractals and multiple sub-systems in the behavior of these systems which represent them as self-similar and self-organized dynamical systems. In mathematics, the chaos game is the methodology of creating fractals utilizing an initial polygon shape and a randomly selected initial point. The main purpose is to create a sequence of points in an iterative manner in order to achieve a sketch which has similar shape in different scales. Sierpinski triangle is one of these fractals.

Chaos Game Optimization (CGO) is a novel meta-heuristic which is conceptualized by general configuration of a Sierpinski triangle. As a simple example, the step by step creation of a Sierpinski triangle by the methodology of chaos game is presented which his utilized in the CGO methodology. At first, three vertices are selected in order to create the main shape of the fractal which results in a triangle shape in this case. Each of the selected vertices is marked by one of the red, blue and green colors. A dice is taken which has two red faces, two blue faces and two green faces. An initial random point is selected as the starting point of the fractal which is considered as a seed in this example. As the dice is rolled, based on which color comes up, the seed in the initial point is moved toward the related vertex half the distance between the seed and the vertex. The new position of the seed is utilized in the next iteration as the starting point in which the dice is rolled again and the seed is moved to the intended vertex accordingly. By rolling the dice many times, the Sierpinski triangle is achieved as the final shape.

As previously described, the principles of chaos theory concern the existence of some primary patterns in the behavior of dynamical systems which represent them as the self-similar and self-organized systems. The created initial seeds (eligible seeds), represent the primary patterns of the dynamical systems based on the chaos theory. The eligibility of these seeds to be as the primary patterns (the self-similarity) can be modeled with solution candidates for an optimization problem. The solution candidates with the highest and lowest eligibility levels are equivalent to the best and worst fitness values, respectively.

Fig. 2. The pseudo-code of the CGO algorithm.