Attractors are cluster points of sequences of iterate-multisets, and a basin of attraction is the set of initial iterate-multisets whose resulting sequences of iterate-multisets are attracted to an attractor. We define notions of stability and asymptotic stability for attractors, where the second notion relies on the additional requirement of a positive radius of attraction (signaling that the attractor attracts every viable initial iterate-multiset whose elements are close to the attractor). We apply our generalized Banach fixed-point theorems to deduce conditions on the iteration mapping that ensure a positive radius of attraction. We also give conditions under which a positive radius of attraction implies that the attractor is a local minimizer, and we provide a companion result involving a weaker notion of radius of restricted attraction. In addition, we prove that the reverse implication of the companion result holds when the attractor is stable. These results rely on important properties of local dense viability (viable initial iterate-multisets are sufficiently abundant near an attractor) and minvalue-monotonicity (the minimum value of the objective function does not increase with iteration).
- 16.Levy, A.B.: Stationarity and Convergence in Reduce-or-Retreat Minimization. Springer Briefs in Optimization. Springer, New York (2012)Google Scholar