Further Study On The Connectness Of Some Set-valued Optimization Problems About Efficient Solution Set

An important problem of vector optimization is to study architecture of theefficient solution set. In the topological properties of these sets, connectedness is veryinteresting, it provides a possibility of continuously moving from one efficient solutionto any other along efficient alternatives only. The generalized convex of set-valuedmapping can replace the convexity requirements in set-valued optimization theory andits associated similar problems. Super efficient points can be scalarized and strongefficiency is a generation of it, strong efficiency promote super efficiency. This paperstudies the connectedness of strong efficient solution set and super efficient solution sefor constrained set-valued optimization problem with arc-wise connectedness feasibleregion. in the hausdorff locally convex topological linear spaces. The main contents areas follows:In the first chapter, we introduce the background of vector optimization problemsand the effectiveness research, analyse and summarize the current research of scholars,describe innovation of this paper.In the second chapter, we introduce some necessary basics knowledge in studyingthe connectedness of set-valued optimization problem solution set, includingtopological space, set-valued mapping, continuous, connectedness and so on.In the third chapter, we discusse the connectedness of strong efficient solution setfor set-valued optimization problems with constraints. At present the set-valuedmapping for the connectedness research of solution set for set-valued optimizationproblems is unconstrained, or the feasible region is not arc-wised. In this chapter, wefirst introduce some prior knowledge, such as strong efficiency, cone-arcwise connectedmapping and so on, then we present the theorem of the connectedness of strong efficientpoint set and strong efficient solution set for set-valued optimization problems withconstraints under the condition that the domain is an arcwise connectedness set, and itextends some results of the connectedness of strong efficient solution set for set-valuedoptimization problems.In the fourth chapter, we discusse the connectedness of super efficient solution set forset-valued optimization problems with constraints. First of all, we introduce someconcepts such as super efficiency, constraint set, cone-arcwise connected mapping in the hausdorff locally convex topological linear spaces. As we all know, the cone-arcwiseconnected mapping must be cone-convexlikeness, then we can say that some propertiesunder the cone-convexlikeness set-valued maps, are possessed under the cone-arcwiseconnected set-valued mapping. At last we present the theorem of the connectedness ofsuper efficient point set and super efficient solution set for set-valued optimizationproblems with constraints under the condition that the domain is an arcwiseconnectedness set.In the fifth chapter, we study the connectedness of strong efficiency forcone-convexlikeness set-valued mapping. The cone-convexlikeness mapping is moregeneral than the cone-arcwise connected, in this chapter we present the theorem of theconnectedness of strong efficient point set under the condition that the domain is ageneral set based on the research for connectedness of strong efficient solution set forset-valued optimization problems with constraints.In the sixth chapter, we summarize the main results of the full text and propase somequestions for further research in the future.