Conjugate Duality In Set-valued Vector Optimization

Conjugate duality is a class of important issues in vector optimization. It is characteristic of using the conjugate function to construct the dual problem, and using the properties of conjugate function to prove duality theorems. Based on the definition of the partially ordering, we consider conjugate duality theorems of set-valued vector optimization in finite and infinite dimensional spaces. The main results can be summarized as follows:1. In Chapter 1 and 2, we firstly introduce the development of the vector opti-mization theory, then the main progress in conjugate duality, and based on the research of specialists and scholars, we design this paper.2. In Chapter 3, we consider conjugate duality theorem of set-valued vector opti-mization in finite dimensional spaces. New perturbation function is presented for a class of set-valued vector optimization, (?): R~n×R~n(?)R~p∪{∞},in order to obtain corresponding conjugate duality optimization and duality theorems. It should be mentioned that in the general conjugate duality theory the weak duality assertion is fulfilled without any assumptions. But in our weak duality theorem it requires the externally stability in order to simplify the result. We show under a stability criteria that the form of weak and strong duality becomes simple, general and convenient in applications.3. In Chapter 4, we consider the conjugate duality problem of set-valued vec-tor optimization in infinite dimensional spaces. New perturbation function (?)(x, z) = F(x-z) is presented for a class of set-valued vector optimization, obtained corresponding conjugate duality optimization, and duality theorems. Furthermore we get two lemmas, i.e., (1).If objective function is S-convex, then perturbation function is also S-convex; (2).If objective function is sub-addition, then perturbation function is also sub-addition. Moreover under sub-addition of objective function hypotheses, we provides that the dual gap is zero between the primal problem and the dual problem. Finally, the sufficient and necessary condition of dual gap is proved.4. In Chapter 5, we summarize the paper.