Augmented Lagrangian methods for fully convex composite optimization

TL;DR

This paper advances augmented Lagrangian methods for fully convex composite optimization by extending classical relationships, analyzing convergence in different cases, and proposing an elastic safeguarding scheme to improve dual convergence.

Contribution

It introduces an elastic safeguarding scheme that enhances convergence guarantees for convex problems, combining benefits of existing methods and addressing their limitations.

Findings

Global primal-dual convergence established

Elastic safeguarding preserves primal convergence

Dual sequence convergence improved in regular cases

Abstract

This paper is concerned with augmented Lagrangian methods for the treatment of fully convex composite optimization problems. We extend the classical relationship between augmented Lagrangian methods and the proximal point algorithm to the inexact and safeguarded scheme in order to state global primal-dual convergence results. Our analysis distinguishes the regular case, where a stationary minimizer exists, and the irregular case, where all minimizers are nonstationary. Furthermore, we suggest an elastic modification of the standard safeguarding scheme which preserves primal convergence properties while guaranteeing convergence of the dual sequence to a multiplier in the regular situation. Although important for nonconvex problems, the standard safeguarding mechanism leads to weaker convergence guarantees for convex problems than the classical augmented Lagrangian method. Our elastic safeguarding scheme combines the advantages of both while avoiding their shortcomings.