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We consider the minimization of a sum of a smooth function with a nonsmoothcomposite function, where the composition is applied on a random linearmapping. This random composite model encompasses many problems, and canespecially capture realistic scenarios in which the data is sampled during theoptimization process. We propose and analyze a method that combines theclassical Augmented Lagrangian framework with a sampling mechanism and adaptiveupdate of the penalty parameter. We show that every accumulation point of thesequence produced by our algorithm is almost surely a critical point.

An Augmented Lagrangian Approach for Problems With Random Matrix Composite Structure