向量优化中集合的一些相对代数性质和相对拓扑性质

Some relative algebraical properties and relative topological properties of sets in vector optimization

基于Flores-Bazn等人的思想,提出了假设B1和假设B2,证明了集合和的相对代数内部等于相对代数内部的和;集合代数闭包与相对代数内部的和等于和的相对代数内部;集合和的相对拓扑内部等于相对拓扑内部的和;集合拓扑闭包与相对拓扑内部的和等于和的相对拓扑内部,建立了集合代数闭包相等与代数内部相等,拓扑闭包相等与拓扑内部相等之间的一些等价关系.

In this paper, the Assumption B1 and B2 are proposed basing on the idea of Flores-Baz′an et al. The relative algebraic interior of the sum for two sets is equal to the sum of the relative algebraic interior for these sets, the sum of the algebraic closure of a set and the relative algebraic interior of a set is equal to the sum of the relative algebraic interior for the two sets, the relative topological interior of the sum for two sets is equal to the sum of the relative topological interior for these sets, the sum of topological closure of set and the relative topological interior of set is equal to the sum of the relative topological interior for the two sets are proved. Furthermore, the equivalent relations between equality of the algebraic closure and the equality of algebraic interior are established. We also obtain the similar equivalent relations for the topological closure and the relative topological interior.