可行域为一般紧子集的集值优化问题若干有效点集连通性的研究

发布时间：2018-01-06 15:14

本文关键词：可行域为一般紧子集的集值优化问题若干有效点集连通性的研究　出处：《南昌航空大学》2016年硕士论文　论文类型：学位论文

【摘要】：集值优化问题作为当前优化理论中非常值得探讨的一门课题,在很多学术方面的应用非常之广包括金融数学、最优化问题、非线性规划、博弈论等。其中,连通的概念很有意思,它不仅具备很好的性质,而且在优化问题中起着一个很好的桥梁的作用,它主要是保证了解连续性的移动,换言之,这使得问题中的有效解到其它相关的有效解上的连续性移动得到实现,因此对于解集连通性的探讨,是十分具有现实意义的。本文主要考察的集值优化问题有效解的连通性都是在Hausdorff局部凸拓扑线性空间中进行的,并研究了在可行域是一般非空紧子集、目标映射是锥类凸及约束映射是上半连续的情况下,得到了集值优化问题中若干种类的有效点集的连通性定理。主要内容分为下面几个部分:第一章作为绪论部分,简单对有关向量优化问题起源、发展作了一些叙述,搜索了一些历史背景知识,并结合前人的一些研究成果,对国内外学术界人士的研究现状进行了概述和总结,提出了本文的创新部分。第二章介绍了部分的基础理论知识,包括拓扑线性空间、集值映射等,以及有关本文结论证明的一些引理。第三章首先介绍了有关强有效性的研究背景和相关概念,并对含约束锥类凸集值优化问题的连通性进行了详细的阐述。在此之前已有文献对含约束锥弧连通集值映射强有效点集的连通性进行了研究,而对于锥弧连通的条件要求要强于锥类凸条件要求,故本文在这个基础上,将目标映射控制在更一般的锥类凸的情况下,对集值优化问题强有效点集的连通性展开进一步的研究,并给出了其连通性的定理。这个结论是在较弱的条件下得到的,所以是对目前的强有效点集连通性的有关结论的进一步推广。第四章研究的是有关超有效点集的连通性问题。近几年来,已有许多学者在超有效性方面进行了深入研究,并取得重要成果,但至今在集值映射的超有效性上的研究,几乎都是目标映射不带含约束条件的情况下讨论的。而本章首先介绍了超有效性、锥类凸、约束集等基本概念,再在可行域是一般非空紧子集的情况下,对含约束集值优化问题超有效点集的连通性进行了讨论,并给出了相关定理的证明。第五章在前两章的基础之上,对全局真有效点集的连通性问题也进行了一般性研究,介绍了全局真有效点以及h?有效点的相关定义。我们知道,全局真有效解在集值优化问题中也是一种十分重要的有效解,故本章在可行域是一般非空紧子集、目标映射为锥类凸及约束映射为上半连续的情况下,同样证明了其全局真有效点集也是连通的。第六章对全篇的主要结果进行了总结,阐述了本文的研究意义与进步性,并相应提出了尚未解决的问题和有待今后进一步研究的问题。
[Abstract]:Set-valued optimization problem is a subject worth discussing in the current optimization theory. It is widely used in many academic fields, including financial mathematics, optimization problem, nonlinear programming, game theory and so on. The concept of connectivity is interesting. It not only has good properties, but also plays a good role as a bridge in the optimization problem. It is mainly to ensure the understanding of continuity of movement, in other words. This enables the continuous movement of the efficient solution to other related efficient solutions to be realized, so the connectivity of the solution set is discussed. In this paper, the connectedness of the efficient solution of the set-valued optimization problem is all carried out in the Hausdorff locally convex topological linear space. In the case that the feasible region is a general nonempty compact subset, the objective mapping is conical convexity and the constrained mapping is upper semicontinuous. The connectedness theorems of some kinds of effective point sets in set-valued optimization problems are obtained. The main contents are divided into the following parts: the first chapter is the introduction part. The development has made some narration, searched some historical background knowledge, and combined with some previous research results, has carried on the summary and the summary to the domestic and foreign academic circles' research present situation. The second chapter introduces the basic theoretical knowledge, including topological linear space, set-valued mapping and so on. And some Lemma about the proof of the conclusion of this paper. Chapter three introduces the research background and related concepts of strong validity. The connectedness of Cone-Convex set-valued optimization problem with constraints is discussed in detail, and the connectivity of strongly efficient point sets of Cone-Arc connected set-valued maps with constraints has been studied in previous literatures. The condition of cone-arc connectivity is stronger than that of cone-like convexity, so on this basis, the objective mapping is controlled in the case of more general conical convexity. In this paper, the connectedness of the set of strongly efficient points for set-valued optimization problems is further studied, and the theorems of its connectivity are given. This conclusion is obtained under the weaker conditions. So it is a further generalization of the related conclusions about the connectedness of strong efficient point sets. Chapter 4th deals with the connectivity of super-efficient point sets. In recent years. Many scholars have carried on the thorough research in the super efficiency aspect, and obtained the important achievement, but up to now in the set value mapping supervalidity research. Almost all of them are discussed in the case of target mapping without constraints. In this chapter, we first introduce some basic concepts, such as hyper-efficiency, conical convexity, constraint set, and so on, and then, if the feasible region is a general nonempty compact subset. In this paper, the connectedness of super-efficient set of points for set-valued optimization problems with constraints is discussed, and the proof of relevant theorems is given. Chapter 5th is based on the previous two chapters. In this paper, the connectedness of the set of globally proper efficient points is also studied, and the global proper efficient points and h? We know that the global proper efficient solution is also a very important efficient solution in the set-valued optimization problem, so this chapter is a general nonempty compact subset in the feasible region. When the object mapping is conic convex and the constrained mapping is upper semicontinuous, it is also proved that the global proper effective point set is connected. Chapter 6th summarizes the main results of the whole paper. The significance and progress of this paper are expounded, and the unsolved problems and the problems to be studied in the future are put forward accordingly.
【学位授予单位】：南昌航空大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O224

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