Generalized Lagrangian Duality in Set-valued Vector Optimization via Abstract Subdifferential

In this paper,we investigate dual problems for nonconvex set-valued vector optimization via abstract subdifferential.We first introduce a generalized augmented Lagrangian function induced by a coupling vector-valued function for set-valued vector optimization problem and construct related set-valued dual map and dual optimization problem on the basic of weak efficiency,which used by the concepts of supremum and infimum of a set.We then establish the weak and strong duality results under this augmented Lagrangian and present sufficient conditions for exact penalization via an abstract subdifferential of the object map.Finally,we define the sub-optimal path related to the dual problem and show that every cluster point of this sub-optimal path is a primal optimal solution of the object optimization problem.In addition,we consider a generalized vector variational inequality as an application of abstract subdifferential.