Computing second-order points under equality constraints: revisiting Fletcher's augmented Lagrangian

TL;DR

This paper introduces an improved algorithm for constrained smooth optimization using Fletcher's augmented Lagrangian, achieving better theoretical bounds for reaching approximate second-order critical points under equality constraints.

Contribution

It proposes a novel algorithm leveraging Riemannian optimization and Fletcher's augmented Lagrangian, improving convergence bounds for second-order critical points in constrained problems.

Findings

Achieves $ ext{O}( ext{ε}^{-3})$ iteration complexity for approximate second-order points.

Provides new properties of Fletcher's augmented Lagrangian.

Enhances theoretical understanding of constrained optimization methods.

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.