On the complexity of finding first-order critical points in constrained nonlinear optimization

Abstract

The complexity of finding \(\epsilon \)-approximate first-order critical points for the general smooth constrained optimization problem is shown to be no worse that \(O(\epsilon ^{-2})\) in terms of function and constraints evaluations. This result is obtained by analyzing the worst-case behaviour of a first-order short-step homotopy algorithm consisting of a feasibility phase followed by an optimization phase, and requires minimal assumptions on the objective function. Since a bound of the same order is known to be valid for the unconstrained case, this leads to the conclusion that the presence of possibly nonlinear/nonconvex inequality/equality constraints is irrelevant for this bound to apply.