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This work provides calculus to the subdifferential of the pointwise supremum given by an arbitrary family of lower semicontinuous functions. We start our study using basic properties of the subdifferential, which are satisfied by many subdifferential available in the literature, for example: The Dini-Hadammard subdifferential, the Fru0027echet subdifferential and the viscosity subdifferentials. Posteriorly, we study in finite- and infinite-dimensional setting the Limiting/Mordukhovich subdifferential and The G-subdifferential of Ioffe of the supremum function. Finally, we apply our result to the study of the convex subdifferential, here we recover general formulae for the subdifferential of an arbitrary family of convex functions.

Subdifferential Formulae for the Supremum of an Arbitrary Family of Functions