Study On The Connectness Of Some Kinds Of Efficient Solution Sets For Set-Valued Optimization Problem Under The Condition That The Domain Is A Nonempty Compact Set

Set-valued optimization problems as a topic of the current optimization theory is very worthy of discussion and have very wide applications in many academic aspects,including financial mathematics, optimization problems,nonlinear programming,game theory, etc. One of them, the conception of connectedness is very interesting, it not only has good properties, but also plays a good bridge role in the optimization problems. it is mainly to ensure the continuous movement of efficient solution, in other words, it makes the effective solutions continuously move to other efficient solutions. So the discussion of connectedness of solution set possesses very realistic significance. This paper studies the connectedness of effective solution for set-valued optimization problems all in hausdorff locally convex topological linear spaces, and under the condition that the domain is a nonempty compact set and the objective mapping is a cone-convexlike and the constraint mapping is upper semi-continuous set-valued,proves the theorems of connectedness of several kinds of efficient point sets for set-valued optimization problems. The main contents are as follows:The first chapter as the introduction part, narrate simply about the origin,development of some vector optimization problems, and search for some historical background, and combining with some previous research results, reviews and summarizes the research status of the domestic and overseas academics.At last it describes some innovations of this paper.In the second chapter, we introduce some necessary basics knowledge in studying the connectedness for set-valued optimization problem solution set, including topological linear space, set- valued mapping, etc, and some lemmas about the conclusion proof of this article.In the third chapter, we firstly introduce the research background and related conceptions of the strong effectiveness, and elaborate the connectedness for set-valued optimization with constrains in detail. Previously, there were some literatures which had studied the connectedness of strong efficient solution set for set-valued optimization with constraints, but the condition of con-arcwise is stronger than con-convexlike. On this basis, we let the objective mapping be more general condition, and study furtherly the connectedness of strong efficient solution set for set-valued optimization, and haveproved the theorems. This conclusions are acquired under the more weak condition, so,it is a further extension about the current connectedness of strong efficient solution.In the fourth chapter, we discuss about the connectedness of super efficient solution. In recent years, there are many scholars who have studied super effectiveness,and achieved important results, but so far, scholars hardly studied the super effectiveness for set-valued mapping and the objective mapping with constraints. This paper firstly introduce the basic conception of super effectiveness, con-convexlike,constraint set, etc, then discusses the connectedness of super efficient solution set for set-valued optimization with constraints under the condition that the domain is a nonempty compact set, and proves some related theorems.In the fifth chapter, on the basic of previous chapters, we study the connectedness of the global proper efficient set, and introduce the related definitions of the global proper efficient point and h ? efficient point. We all know that the global proper efficient point is very important efficient point, so, this paper has proved the connectedness theorem of the global proper efficient set on the same condition.In the sixth chapter, we summarize the main results of the full text and elaborate the research significance and advantages of this article, proposing some unsolved problems and remains to be further research problems in the future.