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.

References

In more concrete terms, the question of whether the worst-case complexity guarantees established for the unconstrained optimization setting carry over to the equality constrained one, remains open.

— A Sequential Cubic Programming Method with Second-Order Complexity Guarantees for Equality Constrained Optimization  (2604.02747 - Dimou et al., 3 Apr 2026) in Section 1.1 (Contributions)

We also note that the present analysis relies on the full row rank of the constraint Jacobian $A_k$ at every iterate; extending the framework to rank-deficient constraints, for instance through weaker constraint qualifications or regularized normal steps, remains an open question for future work.

— An Adaptive Cubic Regularisation Algorithm Based on Interior-Point Methods for Optimization with General Inequality Constraints  (2610.21673 - Pei et al., 15 Oct 2026) in Section Conclusion