The notions of upper and lower global directional derivatives are introduced for dealing with nonconvex and nonsmooth optimization problems. We provide calculus rules and monotonicity properties for these notions. As a consequence, new formulas for the Dini directional derivatives, radial epiderivatives and generalized asymptotic functions are given in terms of the upper and lower global directional derivatives. Furthermore, a mean value theorem, which extend the well-known Diewert’s mean value theorem for radially upper and lower semicontinuous functions, is established. We also provide necessary and sufficient optimality conditions for a point to be a local and/or global solution for the nonconvex minimization problem. Finally, applications for nonconvex and nonsmooth mathematical programming problems are also presented.

中文翻译：

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通过全局导数的非凸非光滑优化的最优条件

引入上下全局方向导数的概念来处理非凸和非光滑优化问题。我们为这些概念提供了微积分规则和单调性。因此，根据上下全局方向导数给出了 Dini 方向导数、径向表导数和广义渐近函数的新公式。此外，还建立了一个平均值定理，它扩展了著名的 Diewert 的径向上半连续函数和下半连续函数的平均值定理。我们还提供了必要和充分的最优条件，使一个点成为非凸最小化问题的局部和/或全局解。最后，还介绍了非凸和非光滑数学规划问题的应用。