Deterministic Oracle Complexity for Goldstein Stationarity in Nonsmooth Nonconvex Optimization with Constraints

Abstract

We consider the deterministic oracle complexity of finding a $(δ,ε)$-Goldstein stationary point of a Lipschitz function under constraints. The problem is posed in $\mathbb{R}^d$, where $d$ denotes the dimension; constraints are specified either by a closed feasible set or by a single explicit inequality. For closed feasible sets that are star-shaped with respect to an interior ball, including all convex bodies, we use a resisting-oracle construction to show that every deterministic algorithm needs $Ω(ε^{-2})$ objective-oracle calls in the worst case when $d\gtrsimε^{-2}$ and $δ\lesssimε$. For explicit inequalities satisfying uniform Goldstein constraint qualification and strict feasibility, finding Goldstein KKT stationary points requires exponentially many oracle calls in $d$ in the worst case, even at fixed stationarity accuracy. This exponential dependence holds under dimension-independent constraint regularity and strict-feasibility bounds, with a sufficiently small neighborhood radius and complementary slackness error. The result covers common constraints such as ball and box constraints and also allows nonconvex feasible regions. For convex coercive constraints with a Slater point, the resisting-oracle argument also gives an $Ω(ε^{-2})$ oracle-call bound in the same dimension and radius regime while allowing a larger complementary slackness error. These results hold for all deterministic algorithms using a local oracle, whether or not the returned point was queried. Each constructed objective has a strictly feasible global minimizer; the lower bounds therefore concern the information required to find stationary points, not their existence.