Year: 2026
Type: preprint
Abstract: This paper addresses the challenge of obtaining strong optimality guarantees in constrained nonsmooth nonconvex optimization under mild regularity conditions, namely local Lipschitz continuity and existence of directional derivatives. While standard methods typically ensure weak stationarity notions, achieving directional (d-)stationarity remains nontrivial. We show that a random direction exploration step is sufficient to attain d-stationarity. The proposed approach augments any feasible base method with one additional randomized exploration step per iteration, which samples a direction and step size and accepts the candidate based on a function value comparison. The resulting scheme guarantees that all accumulation points are d-stationary almost surely, independently of the behavior of the underlying method. Moreover, under conditions on the base method, it preserves convergence rates of the base method, as established for the difference-of-convex algorithm (DCA), prox-linear-type schemes, and interpolated normalized gradient descent (INGD). The theoretical results are complemented by numerical experiments illustrating the effect and guarantees of the exploration step.
Source: arXiv (Cornell University)
Authors: Greenstein, Dan, Hallak, Nadav
Institution: Technion – Israel Institute of Technology
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7162125447