Properties Of Pseudo Relative Interior And Applications To Set-Value Optimization Problems In Linear Spaces

In 2003, Borwein and[1]Goebel studied the concept of relative interior on the Banach spaces, and introduced a pseudo relative interior. In the dissertation[2], the concept of pseudo relative interior was applied in set-valued optimization problems in locally convex spaces. Banach spaces and locally convex spaces are topological linear spaces. There are not a lot of papers about notions and application of pseudo relative interior in which the real linear spaces are the only mathematical structure.In this dissertation, we mainly investigate pseudo relative interior in real linear spaces, and introduce a new concept of interior which is called vector pseudo relative interior. Vector pseudo relative interior is weaker than pseudo relative interior. In chapter 1, we introduce definition and properties of vector closure, and prove that it is equal to algebra closure. In chapter 2, we study the relationship between vector pseudo relative interior and pseudo relative interior, and it is concluded that vector pseudo relative interior is weaker than pseudo relative interior. In chapter 3, we mainly study properties of pseudo relative interior and vector pseudo relative interior,such as pseudo relative interior and vector pseudo relative interior of a non-empty convex set are convex. In chapter 4, we show some separation theorems on the pseudo relative interior and vector pseudo relative interior, such as alternative theorems about the pseudo relative interior. In chapter 5, we establish some optimality conditions of set-valued optimization problems by using properties and an alternative theorem of pseudo relative interior.