Finding Deepest Attraction Basins for Multimodal Global Minimization Algorithms with Schoenberg-Steinhaus Space-Filling Curves

This paper is about the problem of finding good starting points for global optimization of numerical functions, defined on subsets of Rn , and a new approach for doing that is proposed by means of Schoenberg-Steinhaus space-lling curves - the HQF ASA method is used in the examples, after explaining the new paradigm itself. Although typical global optimization approaches use purely random initialization, it is shown in this work that it is possible to obtain substantially better results by using deterministic initialization with space-lling curves. By launching (possibly in a fully parallel way) several independent threads of optimization departing from points obtained in the preliminary phase, the final set of results produced by this procedure happens to be very effective, representing an attractive tool to boost any existing approach, with the outstanding feature that the framework corresponding to simultaneous execution of optimization threads is easy to implement, not demanding a large programming effort in terms of the synchronization apparatus. Space-filling curves are unique, considering that their paths pass through each point of their images in compact subsets of Rn and other important sets - even the points with irrational coordinates are included. In this fashion, by establishing the correct sampling granularity, it is possible to obtain seeds already inside attraction basins of global minimizers.