Journal of Convex Analysis 32 (2025), No. 2, 487--510
Copyright Heldermann Verlag 2025
Basin Sensitivity via Continuity of Multiset-Valued Mappings
Adam B. Levy
Department of Mathematics, Bowdoin College, College Station, Brunswick, Maine, U.S.A.
alevy@bowdoin.edu
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.
Keywords: Basin of attraction, numerical minimization, sensitivity analysis, multiset-valued mappings, continuity.
MSC: 49J53, 49K40, 65K99, 90C31.
[ Fulltext-pdf (15772 KB)] for subscribers only.