On the hardness of deterministic second-order optimization of functions with Lipschitz gradients

Jiewen Guan; Anthony Man-Cho So

On the hardness of deterministic second-order optimization of functions with Lipschitz gradients

Optimization and Control 2026-07-27 v1

Authors: Jiewen Guan , Anthony Man-Cho So

Abstract

We show that no deterministic zero-respecting algorithm (resp., (general) deterministic algorithm) can compute Goldstein approximate second-order stationary points of functions with Lipschitz continuous gradients within a finite number of (resp., no more than $n-3$ with $n$ being the input dimension) second-order oracle calls. This, among other consequences, shows that deterministic second-order weakly convex optimization is intractable.

Keywords

second-order optimization Goldstein stationary points oracle complexity Lipschitz gradients weakly convex optimization intractability

Cite

@article{arxiv.2607.24120,
  title  = {On the hardness of deterministic second-order optimization of functions with Lipschitz gradients},
  author = {Jiewen Guan and Anthony Man-Cho So},
  journal= {arXiv preprint arXiv:2607.24120},
  year   = {2026}
}