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We provide systematic studies of the variational properties of decomposablefunctions which are compositions of an outer support function and an innersmooth mapping under certain constraint qualifications. We put a particularfocus on the strict twice epi-differentiability and the associated strictsecond subderivative of such functions. Calculus rules for the (strict) secondsubderivative and twice epi-differentiability of decomposable functions arederived which allow us to link the (strict) second subderivative ofdecomposable mappings to the simpler outer support function. Applying thevariational properties of the support function, we establish the equivalencebetween strict twice epi-differentiability of decomposable functions,continuous differentiability of its proximity operator, and the strictcomplementarity condition. This allows us to fully characterize the strictsaddle point property of decomposable functions. In addition, we give a formulafor the strict second subderivative for decomposable functions whose outersupport set is sufficiently regular. This provides an alternativecharacterization of the strong metric regularity of the subdifferential ofdecomposable functions. Finally, we verify that these introduced regularityconditions are satisfied by many practical functions.

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