Tightness or improvement of the constraint-violation complexity bound in equality-constrained optimization

Establish whether the worst-case iteration complexity bound O(epsilon_c^{-1}) for reducing the constraint violation in smooth nonconvex equality-constrained optimization is optimal by deriving matching lower bounds; alternatively, construct an algorithm that achieves strictly faster convergence in the constraint violation while preserving the O(epsilon_g^{-3/2}) and O(epsilon_H^{-3}) worst-case bounds on the Lagrangian gradient norm and second-order stationarity, respectively.

Background

The paper proves an O(epsilon_c^{-1}) worst-case iteration bound for reducing constraint violation, alongside optimal O(epsilon_g^{-3/2}) and O(epsilon_H^{-3}) bounds for the Lagrangian gradient and Hessian. While the latter are known to be tight from the unconstrained literature, the authors state that it is not currently known whether the O(epsilon_c^{-1}) bound is tight for feasibility.

They explicitly pose an open question: either prove matching lower bounds for the O(epsilon_c^{-1}) dependence or design methods that improve the feasibility rate without degrading the optimal gradient and Hessian complexities.