Stationarity in nonsmooth optimization between geometrical motivation and topological relevance

TL;DR

This paper compares different notions of stationarity in structured nonsmooth optimization, revealing a hierarchy and clarifying their roles in identifying local minimizers and saddle points.

Contribution

It introduces and analyzes a hierarchy of geometrically motivated and topologically relevant stationarity notions in structured nonsmooth optimization.

Findings

$\, ext{ extasciitilde}N$-stationary points include all local minimizers.

$N$-stationary points are singular saddle points.

T-stationary points are regular saddle points.

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 $\widehat N$-, $N$-, and $\overline{N}$-stationarity in terms of Fr\'echet, 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) $\widehat N$-stationary points include all local minimizers; (b) $N$-stationary points, which are not $\widehat N$-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) $\overline{N}$-stationary points, which are not T-stationary, are irrelevant for optimization purposes. Overall, a hierarchy of stationarity notions for SNO is established.