Basin Sensitivity via Continuity of Multiset-Valued Mappings

Basins of attraction consist of the initial iterates from which a minimization method applied to an objective function clusters at the same attractor, and we aim to analyze the sensitivity of basins of attraction to perturbations in the objective function. We do this by developing various continuity properties for "multiset-valued mappings" that generalize traditional notions of continuity for mappings. We apply those continuity properties to the particular case of "basin mappings" that map a parameter to the basins of attraction resulting from that perturbation in the objective. We explore our results through simulation and we develop new "grid continuity" properties where sampling is intrinsic. Finally, we connect grid continuity to our generalized versions of more traditional continuity properties.