The Approximate Weakly Efficient Solutions And The Approximate Global Properly Efficient Solutions For Set Optimization When The Topological Interior Is Empty

When the algebraic interior of an ordering cone is not empty,we establish a new order relation.And the approximate weakly efficient solutions and the approximate efficient solutions for set optimization are introduced by using above order relation.Then,we introduce the convergence,compactness and continuity in the sense of algebra,which are used to establish sufficient and necessary optimality conditions for LPi well-posedness and we get the optimality conditions for the approximate weakly efficient solutions and the approximate efficient solutions of set optimization by applying Minkowski nonlinear functionals without topological structure,respectively.By applying the Cantor intersection theorem in the Banach space and some properties of above order relation,we get the existence theorem of the approximate weakly efficient solutions.Finally,we introduce the global properly efficient solutions and the approximate global properly efficient solutions,and we get the inclusion relation among above solutions.