Fractal Techniques to Measure the Numerical Instability of Optimization Methods

Abstract

All process that can be defined in the form of an iterative algorithm of the form x=g(x), may be considered as a dynamical system. The complexity of the system depends on how complex is the function y=g(x). Even for the simplest cases, the behavior of such dynamical systems may be chaotic. In such processes it may be obtaines a map coloring the initial points with different colors, depending on the fixed point toward they converge. These maps are fractals if the system is chaotic and its fractal dimension may represent a measure of the chaotic quality (or instability) of the system. In this context, some optimization methods such as Newton-Raphson, Secant
Method, Cuasi-Newton, and Second Order Methods are analized.