Duality Of Vector Equilibrium And Set-valued Optimization Problems

In topological vector spaces, based on weak efficiency, we study the Fenchel dual for a mixed vector equilibrium problem and some saddle-point theorems. Moreover, based onε? weak efficiency, we study theε? conjugate dual for the sum of two set-valued optimization problems. The detailed contents are listed below:Part one: Firstly, we recall respectively the definitions of weak supremum and infimum of a set which is introduced by Tanino. Then, we introduce a mixed vector equilibrium problem (MVEP) and an equivalent vector optimization problem. We investigate the dual mixed vector equilibrium problem (DMVEP) by using the method of the Fenchel conjugate function. Under a stabilization condition, we discuss the relationships between the solutions of a mixed vector equilibrium problem and a dual mixed vector equilibrium problem. Moreover, under the same condition, we also define a Lagrangian map and a saddle point for the mixed vector equilibrium problem and prove the solutions of (MVEP) and (DMVEP) are relating to the saddle points of an associated Lagrangian mapping. Finally, as applications, we study dualities of vector convex optimization problems and vector variational inequality problems, respectively.Part two: Firstly, we recall respectively the definitions ofε? weak minimal point,ε-subdifferentials of set-valued maps,ε?conjugate map of set-valued maps and some related propositions. Then, we establish theε? conjugate dual for the sum of set-valued optimization problems. Finally, we discuss the relationships between the solutions of the sum of set-valued optimization problems and theε? conjugate dual of the sum of two set-valued optimization problems.