210. Stationarity in nonsmooth optimization between geometrical motivation and topological relevance
Invited abstract in session TB-6: Advances in nonsmooth optimization, stream Nonsmooth and nonconvex optimization.
Tuesday, 10:30-12:30Room: B100/7013
Authors (first author is the speaker)
| 1. | Vladimir Shikhman
|
| Chemnitz University of Technology |
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 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) stationary points via Frechet normal cone include all local minimizers; (b) stationary points via Mordukhovich normal cone, which are not Frechet stationary, correspond to the singular saddle points of first order; (c) T-stationary points, which are not Mordukhovich stationary, correspond to the regular saddle points of first order; (d) stationary points via Clarke normal cone, which are not T-stationary, are irrelevant for optimization purposes. Overall, a hierarchy of stationarity notions for SNO is established.
Keywords
- Non-smooth optimization
Status: accepted
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