Local properties and augmented Lagrangians in fully nonconvex composite optimization

TL;DR

This paper explores the properties and convergence of augmented Lagrangian methods for fully nonconvex composite optimization problems, providing new insights into optimality conditions and local convergence rates without convexity assumptions.

Contribution

It introduces a comprehensive analysis of local properties and augmented Lagrangians in nonconvex composite optimization, including optimality conditions and convergence rates.

Findings

Established first- and second-order optimality conditions for nonconvex composite problems

Analyzed the challenges of Lagrangian frameworks without convexity

Derived local convergence rates for augmented Lagrangian methods in nonconvex settings

Abstract

A broad class of optimization problems can be cast in composite form, that is, considering the minimization of the composition of a lower semicontinuous function with a differentiable mapping. This paper investigates the versatile template of composite optimization without any convexity assumptions. First- and second-order optimality conditions are discussed. We highlight the difficulties that stem from the lack of convexity when dealing with necessary conditions in a Lagrangian framework and when considering error bounds. Building upon these characterizations, a local convergence analysis is delineated for a recently developed augmented Lagrangian method, deriving rates of convergence in the fully nonconvex setting.