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The first part of the paper provides new characterizations of the normal coneto the effective domain of the supremum of an arbitrary family of convexfunctions. These results are applied in the second part to give new formulasfor the subdifferential of the supremum function, which use both the active andnonactive functions at the reference point. Only the data functions areinvolved in these characterizations, the active ones from one side, togetherwith the nonactive functions multiplied by some appropriate parameters. Incontrast with previous works in the literature, the main feature of oursubdifferential characterization is that the normal cone to the effectivedomain of the supremum (or to finite-dimensional sections of this domain) doesnot appear. A new type of optimality conditions for convex optimization isestablished at the end of the paper.

Alternative representations of the normal cone to the domain of supremum functions and subdifferential calculus