New higher-order weak lower inner epiderivatives and application to Karush-Kuhn-Tucker necessary optimality conditions in set-valued optimization

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.