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

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 the equivalence between the strict twice epi-differentiability of decomposable functions, continuous differentiability of the proximity operator, and the strict complementarity condition. 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 sufficiently regular. 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.