Conjugate Duality In Vector Optimization

In topological vector spaces, based on weak efficiency, we study the conjugate dual problems for constrained vector optimization problems under different perturbations and some inclusion relations among their dual objective maps. The detailed contents are listed below:Firstly, we recall respectively the definitions of weak supremum and infimum of a set which is introduced by Tanino. By considering the perturbation of constraints, we introduce a conjugate map and a Lagrangian dual problem for a constrained vector optimization problem. We investigate its weak duality theorem, strong duality theorem and stability criteria. Then, by considering the perturbations of constraints and objective mapping respectively, we introduce a Fenchel-Lagrange dual problem for the constrained vector optimization problem and also obtain its weak duality theorem, strong duality theorem and stability criteria. Simultaneously,we show some inclusion relations between the dual objective maps of the Lagrangian dual problem and the Fenchel-Lagrange dual problem. We also define a Lagrangian map and a saddle point for the vector optimization problem and discuss their properties. Finally, by using the two conjugate dual problems, we obtain two set-valued gap functions for a vector equilibrium problem and some inclusion relations between the two set-valued gap functions, respectively.