Variational Properties of Decomposable Functions. Part I: Strict Epi-calculus and Applications

Source abstract

Abstract. This work provides a systematic study of the variational properties of decomposable functions which are compositions of an outer support function and an inner smooth mapping under certain constraint qualifications. A particular focus is put on the strict twice epi-differentiability and the associated strict second subderivative of such functions. A lower bound for the strict second subderivative of decomposable functions is derived which allows linking the strict second subderivative of decomposable mappings to the simpler outer support function. Leveraging the variational properties of the support function, we establish that continuous differentiability of the proximity operator implies strict twice epi-differentiability, which further necessitates the strict complementarity condition to hold. Under a strict constraint qualification, all of these conditions are shown to be equivalent. As an application, this allows us to fully characterize the strict saddle point property of decomposable functions. In addition, an explicit formula for the strict second subderivative of decomposable functions is derived if the outer support set is [Formula: see text]-cone reducible. This yields an alternative characterization of the strong metric regularity of the subdifferential of decomposable functions at a local minimizer. Finally, we verify that the introduced regularity conditions are satisfied by many practical functions and applications.