Convergence of Augmented Lagrangian Methods for Composite Optimization Problems

TL;DR

This paper proves local convergence of the augmented Lagrangian method for composite optimization problems, introducing a new second-order property and demonstrating Q-linear convergence under certain conditions.

Contribution

It introduces the semi-stability of second subderivatives and establishes local convergence results for ALM in composite optimization.

Findings

Semi-stability of second subderivatives is widely satisfied.

Q-linear convergence of ALM is established under second-order conditions.

Convergence holds even with nonunique Lagrange multipliers.

Abstract

Local convergence analysis of the augmented Lagrangian method (ALM) is established for a large class of composite optimization problems with nonunique Lagrange multipliers under a second-order sufficient condition. We present a new second-order variational property, called the semi-stability of second subderivatives, and demonstrate that it is widely satisfied for numerous classes of functions, important for applications in constrained and composite optimization problems. Using the latter condition and a certain second-order sufficient condition, we are able to establish Q-linear convergence of the primal-dual sequence for an inexact version of the ALM for composite programs.