Pathological subgradient dynamics

TL;DR

This paper constructs examples of Lipschitz functions with unusual subgradient dynamics where trajectories do not reveal critical points, highlighting complexities in nonsmooth optimization.

Contribution

It introduces pathological examples of subgradient dynamics in both continuous and discrete settings, showing bounded trajectories that miss critical points.

Findings

Bounded trajectories do not necessarily indicate critical points.

Pathological examples exist in both continuous and discrete subgradient methods.

Challenges in detecting critical points in nonsmooth optimization.

Abstract

We construct examples of Lipschitz continuous functions, with pathological subgradient dynamics both in continuous and discrete time. In both settings, the iterates generate bounded trajectories, and yet fail to detect any (generalized) critical points of the function.