Towards Supremum Sum Subdifferential Calculus Free of Qualification Conditions
We give a formula for the subdifferential of the sum of two convex functions where one of them is the supremum of an arbitrary family of convex functions. This is carried out under a weak assumption expressing a natural relationship between the lower semicontinuous envelopes of the data functions in the domain of the sum function. We also provide a new rule for the subdifferential of the sum of two convex functions, which uses a strategy of augmenting the involved functions. The main feature of our analysis is that no continuity type condition is required. Our approach allows us to unify, recover, and extend different results in the recent literature.
SIAM J. Optim.