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

Florentin Goyens, Armin Eftekhari, Nicolas Boumal

TL;DR

A notion of approximate first- and second-order critical point which relies on the geometric formalism of Riemannian optimization is proposed, which improves on current best theoretical bounds.

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.

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

TL;DR

A notion of approximate first- and second-order critical point which relies on the geometric formalism of Riemannian optimization is proposed, which improves on current best theoretical bounds.

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 -approximate second-order critical points of the original optimization problem in at most 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.
Paper Structure (20 sections, 21 theorems, 128 equations, 4 algorithms)

This paper contains 20 sections, 21 theorems, 128 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Assumptions
  3. Optimality conditions on layered manifolds
  4. Contributions
  5. Properties of Fletcher's augmented Lagrangian
  6. Approximate first-order criticality
  7. Approximate second-order criticality
  8. Property of the region $\mathcal{C}$
  9. Gradient-Eigenstep algorithm
  10. Algorithm
  11. First-order steps
  12. Second-order steps
  13. Worst-case global complexity
  14. Estimating the penalty parameter
  15. Conclusion and discussion
  16. ...and 5 more sections

Key Result

proposition 1

For $i=1,2,\dots,k$, consider $k$ functions $h_i\colon \mathcal{E}_i \to \mathbb{R}^{m_i}$ that satisfy assumptions assu:ROI, assu:boundedMandC and assu:lipschitzh with constants $R_i$, $\underline{\sigma}_i$ and $C_{h_i}$. Then, the function $h\colon \mathcal{E} \to \mathbb{R}^m$ with $\mathcal{E}=

Theorems & Definitions (46)

  • Example : The Stiefel manifold
  • proposition 1
  • proof
  • proposition 2: Layered manifolds
  • proof
  • proposition 3
  • Definition 1.1
  • Definition 1.2
  • Definition 2.1
  • Definition 2.2
  • ...and 36 more