The first part of the paper provides new characterizations of the normal cone to the effective domain of the supremum of an arbitrary family of convex functions. These results are applied in the second part to give new formulas for the subdifferential of the supremum function, which use both the active and nonactive functions at the reference point. Only the data functions are involved in these characterizations, the active ones from one side, together with the nonactive functions multiplied by some appropriate parameters. In contrast with previous works in the literature, the main feature of our subdifferential characterization is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of this domain) does not appear. A new type of optimality conditions for convex optimization is established at the end of the paper.

本文的第一部分为法线锥到任意凸函数族的有效域的有效域提供了新的特征。这些结果将在第二部分中应用，以给出至高函数次微分的新公式，该公式在参考点上同时使用有效函数和非有效函数。这些表征中仅涉及数据功能，一侧是活动功能，而非活动功能则乘以一些适当的参数。与文献中的先前工作相比，我们的亚微分表征的主要特征是，没有显示出至上有效区域（或该区域的有限维截面）的法线锥。