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arXiv · 2309.01980

Local properties and augmented Lagrangians in fully nonconvex composite optimization

Alberto De Marchi·Patrick Mehlitz
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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.

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Alberto De Marchi, Patrick Mehlitz. 2023-09-05. Local properties and augmented Lagrangians in fully nonconvex composite optimization. https://doi.org/10.46298/jnsao-2024-12235

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