Space-Filling Curves and Fuzzy ASA

Abstract

This chapter introduces a multi-start global optimization algorithm that uses dimensional reduction techniques based upon approximations of space-filling curves and fuzzy adaptive simulated annealing, aiming at finding global minima of real-valued (possibly multimodal) functions that are not necessarily well behaved, that is, are not required to be differentiable, continuous, or even satisfying Lipschitz conditions. The overall idea is as follows: given a real-valued function with a multidimensional and compact domain, the method builds an equivalent, onedimensional problem by composing it with a space-filling curve, searches for a small group of candidates and returns to the original higher-dimensional domain, this time with a small set of promising starting points. In this fashion, it is possible to overcome difficulties related to capture in inconvenient attraction basins and, simultaneously, to bypass the complexity associated to finding the global minimum of the auxiliary one dimensional problem, whose graph is typically fractal-like, as we shall see along the chapter. New space-filling curves are built with basis on the well-known Sierpiński SFC, a subtle modification of a theorem by Hugo Steinhaus and several results of Ergodic Theory.