Stationarity in nonsmooth optimization between geometrical motivation and topological relevance
Year: 2025
Type: article
Abstract: The goal of this paper is to compare alternative stationarity notions in structured nonsmooth optimization (SNO). Here, nonsmoothness is caused by complementarity, vanishing, orthogonality type, switching, or disjunctive constraints. On one side, we consider geometrically motivated notions of $$\widehat{N}$$ N ^ -, N-, and $$\overline{N}$$ N ¯ -stationarity in terms of Fréchet, Mordukhovich, and Clarke normal cones to the feasible set, respectively. On the other side, we advocate the notion of topologically relevant T-stationarity, which adequately captures the global structure of SNO. Our main findings say that (a) $$\widehat{N}$$ N ^ -stationary points include all local minimizers; (b) N-stationary points, which are not $$\widehat{N}$$ N ^ -stationary, correspond to the singular saddle points of first order; (c) T-stationary points, which are not N-stationary, correspond to the regular saddle points of first order; (d) $$\overline{N}$$ N ¯ -stationary points, which are not T-stationary, are irrelevant for optimization purposes, at least from the topological point of view. Overall, a hierarchy of stationarity notions for SNO is established.
Source: Journal of Global Optimization
Author: Vladimir Shikhman
Institution: Chemnitz University of Technology
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
Funder: Technische Universität Chemnitz
OpenAlex ID: W4414619949