This paper proposes three enlargements of the conventional Moreau–Rockafellar subdifferential: the sup-, sup^⋆- and symmetric subdifferentials. They satisfy the most fundamental properties of the conventional subdifferential: convexity, weak^∗-closedness and, if the function is bounded, weak^∗-compactness. Moreover, they are nonempty for the Rainwater function, possess certain calculus rules, and provide necessary and sufficient optimality conditions for not necessarily convex functions.

中文翻译：

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Moreau-Rockafellar次微分的放大

本文提出了常规的Moreau–Rockafellar次微分的三个扩展：sup-，^sup⋆-和对称次微分。它们满足了常规次微分的最基本性质：凸度，弱^＊-闭合度，如果函数是有界的，则弱^＊-紧致度。而且，它们对于雨水函数是非空的，具有一定的演算规则，并为不一定是凸函数提供了必要和充分的最优性条件。