| In this paper,the set-valued optimization problem is generalized to set optimization problem by using the Minkowski difference of set operations.The concepts of Benson properly efficient solution,approximate Benson properly efficient solution,Henig properly efficient solution and approximate Henig properly efficient solution of set optimization problems are introduced in topological space,and their relations are discussed.It is proved that approximate Benson properly efficient solution set is closed under some assumptions.The Painlevé-Kuratowski convergence of approximate Henig properly efficient solution sets under some conditions is proved.The necessary conditions for obtaining approximate Benson properly efficient solution for unconstrained and constrained set optimization problems are established by using a nonlinear functional.The concepts of Benson properly efficient solution,approximate Benson properly efficient solution,Henig properly efficient solution and approximate Henig properly efficient solution of set optimization problems are introduced in the space without topology,the relations among them are discussed.A functional is introduced to characterize approximate Henig properly efficient solution of unconstrained and constrained optimization problems. |
