SONG Zhihui, XU Yihongy, LIU Yueqing
Duality theory is an important branch of vector optimization theory, which plays an important role in establishing the optimality conditions and solving vector optimization problems, and is widely used in fields such as game theory and economic equilibrium problems. In this paper, the conjugate duality and Lagrangian duality for generalized vector optimization are investigated. Firstly, under the order relationship induced by convex cones, a new conjugate mapping is introduced by the weak supremal of sets, and an example is provided to illustrate its economic significance. And the conjugate duality for generalized vector optimization is defined by using a perturbation mapping. The weak duality, strong duality and inverse duality theorems are obtained, and an example is provided to illustrate the strong duality theorem. Secondly, a new Lagrangian mapping is introduced, with which a Lagrangian duality for generalized vector optimization is introduced. The objective value of the original problem is characterized by a Lagrangian mapping, and the Lagrangian duality theory is established. Finally, a kind of saddle point is defined, and the saddle point theorem is obtained. The corresponding results in the references are generalized.
