Do unconstrained optimal iteration-complexity bounds carry over to equality-constrained optimization?

Determine whether the worst-case iteration complexity guarantees established for unconstrained nonconvex optimization extend to smooth equality-constrained optimization without deterioration, specifically whether the optimal bounds for achieving approximate second-order KKT points carry over unchanged when smooth nonconvex equality constraints are present.

Background

The paper highlights a gap between single-phase and two-phase methods for equality-constrained optimization: while two-phase methods attain optimal complexity with respect to the Lagrangian gradient, state-of-the-art single-phase SQP-type methods have better feasibility complexity bounds but lack guarantees for higher-order stationarity. This motivates the question of whether the optimal unconstrained iteration bounds also hold under equality constraints.

The authors frame this as an explicit open question in the Contributions section and then develop a sequential cubic programming method that aims to resolve it by achieving worst-case bounds that match the unconstrained setting for the Lagrangian gradient and Hessian while providing improved feasibility complexity.