Synthesis of Space-Filling Curves Through Measure- Preserving Transformations and Their Application to Global Optimization

This paper proposes a new multi-start, stochastic global optimization algorithm that uses dimensional reduction techniques based upon approximations of space-filling curves and simulated annealing, aiming to find 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, one-dimensional problem by composing it with a space-filling curve (SFC), 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 manner, it is possible to overcome difficulties related to capture in inconvenient attraction basins and, simultaneously, bypassing the complexity associated to finding the global minimum of the auxiliary one dimensional problem, whose graph is typically fractal-like, as we shall see in the sequel. New SFCs 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.