Local properties and augmented Lagrangians in fully nonconvex composite optimization

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.