The purpose of the paper is to establish higher-order Karush–Kuhn–Tucker higher-order necessary optimality conditions for set-valued optimization where the derivatives of objective and constraint functions are separated. We first introduce concepts of higher-order weak lower inner epiderivatives for set-valued maps and discuss some useful properties about new epiderivatives, for instance, convexity, subadditivity and chain rule. With the help of the new concept and its properties, we establish higher-order Karush–Kuhn–Tucker necessary optimality conditions which is the classical type Karush–Kuhn–Tucker optimality conditions and improve and enhance some recent existing results in the literatures. Several examples are provided to illustrate our results. Finally, we provide weak and strong duality theorems in set-valued optimization.

中文翻译：

---

新的高阶弱低内表导数及其在设定值优化中 Karush-Kuhn-Tucker 必要最优条件的应用

本文的目的是建立高阶 Karush-Kuhn-Tucker 高阶必要最优性条件，用于目标函数和约束函数的导数分离的集值优化。我们首先介绍集合值映射的高阶弱低内表导数的概念，并讨论有关新表导数的一些有用属性，例如凸性、次可加性和链式法则。借助新概念及其性质，我们建立了高阶 Karush-Kuhn-Tucker 必要最优条件，即经典类型的 Karush-Kuhn-Tucker 最优条件，并改进和增强了一些最近已有的文献结果。提供了几个例子来说明我们的结果。最后，我们在集值优化中提供了弱对偶和强对偶定理。