Local Convergence Analysis of Augmented Lagrangian Methods for Piecewise Linear-Quadratic Composite Optimization Problems

TL;DR

This paper proves that second-order sufficient conditions guarantee linear convergence of the augmented Lagrangian method for piecewise linear-quadratic composite optimization, even with non-unique multipliers, and establishes their equivalence to quadratic growth conditions.

Contribution

It shows second-order conditions alone ensure convergence of the augmented Lagrangian method in a class of composite problems, expanding understanding of convergence criteria.

Findings

Second-order sufficient conditions imply linear convergence.

Convergence holds even with non-unique Lagrange multipliers.

Second-order conditions are equivalent to quadratic growth conditions.

Abstract

Second-order sufficient conditions for local optimality have been playing an important role in local convergence analysis of optimization algorithms. In this paper, we demonstrate that this condition alone suffices to justify the linear convergence of the primal-dual sequence, generated by the augmented Lagrangian method for piecewise linear-quadratic composite optimization problems, even when the Lagrange multiplier in this class of problems is not unique. Furthermore, we establish the equivalence between the second-order sufficient condition and the quadratic growth condition of the augmented Lagrangian problem for this class of composite optimization problems.