The Computational Complexity of Finding Second-Order Stationary Points

Non-convex minimization problems are universally considered hard, and even guaranteeing that a computed solution is locally minimizing is known to be NP-hard. In this general context, our paper focuses on the problem of finding stationary points that satisfy an approximate second-order optimality condition, which serves to exclude strict saddles and other non-minimizing stationary points. Our main result is that the problem of finding approximate second-order stationary points (SOSPs) is PLS-complete, i.e., of the same complexity as the problem of finding first-order stationary points (FOSPs), thus resolving an open question in the field. In particular, our results imply that, under the widely believed complexity conjecture that PLS $\neq$ FNP, finding approximate SOSPs in unconstrained domains is easier than in constrained domains, which is known to be NP-hard. This comes in stark contrast with earlier results which implied that, unless PLS = CLS, finding approximate FOSPs in unconstrained domains is harder than in constrained domains.