Second-Order Karush–Kuhn–Tucker Optimality Conditions for Set-Valued Optimization Subject to Mixed Constraints

In this paper, we propose the concept of second-order composed adjacent contingent derivatives for set-valued maps and hence discuss the included relationship to the second-order composed contingent derivatives. Under the appropriate conditions, Lipschitz properties of the first order derivatives and a chain rule for such second-order composed contingent derivatives are demonstrated. By virtue of second-order composed adjacent contingent derivatives and second-order composed contingent derivatives, we establish second-order Karush–Kuhn–Tucker sufficient and necessary optimality conditions for a set-valued optimization problem subject to mixed constraints. Applying a separation theorem for convex sets and cone-Aubin properties, we address stronger necessary optimality conditions and extend the second-order Kurcyusz–Robinson–Zowe regularity assumption for a problem with mixed constraints, in which the derivatives of the objective and the constraint functions are considered in separated ways. When the results regress to vector optimization, we also extend and improve some recent existing results.