Characterizations of Pointwise Additivity of Subdifferential

We prove that the additivity of subdifferential in a given point of a locally convex space X is equivalent to other important optimality properties of an associated family of optimization problems. As a consequence, the subdifferential additivity is characterized by a dual closedness condition in $X^* \times \mathbb{R}$, where $\mathbb{R}$ are the reals, endowed with the weak-star topology. Also, some special cases in which this closedness condition can be given in $X^*$ are presented.