Computing Second-Order Points Under Equality Constraints: Revisiting Fletcher’s Augmented Lagrangian

Abstract

We address the problem of minimizing a smooth function under smooth equality constraints. Under regularity assumptions on these constraints, we propose a notion of approximate first- and second-order critical point which relies on the geometric formalism of Riemannian optimization. Using a smooth exact penalty function known as Fletcher’s augmented Lagrangian, we propose an algorithm to minimize the penalized cost function which reaches \(\varepsilon \)-approximate second-order critical points of the original optimization problem in at most \({\mathcal {O}}(\varepsilon ^{-3})\) iterations. This improves on current best theoretical bounds. Along the way, we show new properties of Fletcher’s augmented Lagrangian, which may be of independent interest.