The inexact power augmented Lagrangian method for constrained nonconvex optimization

TL;DR

This paper proposes a novel inexact augmented Lagrangian method with a power-based augmenting term for constrained nonconvex optimization, providing a comprehensive complexity analysis and demonstrating improved primal convergence with lower powers.

Contribution

It introduces an unconventional augmenting term in the augmented Lagrangian method and analyzes its complexity, showing benefits of lower powers for faster constraint satisfaction.

Findings

Lower powers lead to faster constraint satisfaction.

The method achieves improved primal complexity with lower powers.

Numerical experiments confirm practical performance benefits.

Abstract

This work introduces an unconventional inexact augmented Lagrangian method where the augmenting term is a Euclidean norm raised to a power between one and two. The proposed algorithm is applicable to a broad class of constrained nonconvex minimization problems that involve nonlinear equality constraints. In a first part of this work, we conduct a full complexity analysis of the method under a mild regularity condition, leveraging an accelerated first-order algorithm for solving the H\"older-smooth subproblems. Interestingly, this worst-case result indicates that using lower powers for the augmenting term leads to faster constraint satisfaction, albeit with a slower decrease of the dual residual. Notably, our analysis does not assume boundedness of the iterates. Thereafter, we present an inexact proximal point method for solving the weakly-convex and H\"older-smooth subproblems, and demonstrate that the combined scheme attains an improved rate that reduces to the best-known convergence rate whenever the augmenting term is a classical squared Euclidean norm. Different augmenting terms, involving a lower power, further improve the primal complexity at the cost of the dual complexity. Finally, numerical experiments validate the practical performance of unconventional augmenting terms.