F-stationarity of proximal-gradient cluster points under continuous differentiability

Determine whether every cluster point of a sequence generated by the proximal gradient method for minimizing the sum F+φ is a Fréchet stationary point of F+φ when F is continuously differentiable, without assuming that the gradient of F is locally Lipschitz continuous.

Background

The paper studies proximal gradient methods for minimizing F+φ, where F may be nonsmooth or nonconvex and φ is proper, lower semicontinuous, and prox-bounded. Under the paper’s local Lipschitz smoothness assumption on F, the authors prove that every cluster point of a proximal-gradient sequence satisfying the standard intermediate convergence conditions is a proximal stationary point, which is stronger than Fréchet stationarity.

The unresolved question concerns weakening the regularity of F from locally Lipschitz-continuous differentiability to mere continuous differentiability. The authors explicitly identify whether cluster points retain Fréchet stationarity under this weaker assumption as an open question and do not resolve it in the paper.