An “attractor” for a numerical minimization method was defined in (Levy, in Springer Briefs in Optimization, Springer Nature Switzerland AG, Cham, 2018) in terms of “iteration mappings” that assign the output of an iteration of the method to its input, and the related concept of “basin of attraction” was defined as the collection of initial inputs from which the method clusters toward a given attractor. This idea was broadened in (Levy, in Set-Valued Var. Anal. 29:1–28, 2021) to $\epsilon $ -basins of attraction that collect the inputs from which repeated application of the iteration mapping eventually generates outputs within $\epsilon >0$ of the attractor. In the present paper, we introduce a third type of basin of attraction and refine and unify the understanding of all three types of basins by analyzing basin intersections and boundaries. We introduce several new notions of continuity for iteration mappings including a type of Lipschitz continuity whose absence identifies basin boundaries. We also study inverse-iteration and provide a characterization of $\epsilon $ -basins as a union of inverse-images. We use illustrative examples and simulations, and we demonstrate how our results allow us to address limitations of simulation.
