向量优化理论中的非线性标量化函数相关研究及应用

发布时间：2018-06-14 13:50

本文选题：Gerstewitz泛函 + 最大严格单调函数　；参考：《内蒙古大学》2016年博士论文

【摘要】：向量优化问题是指在一定的约束条件下极小化向量值函数.向量优化理论从产生、发展到逐渐成熟的过程中,与数学和经济学中的许多理论均有着密不可分的联系.目前向量优化理论和方法已形成了一个巨大体系,集中了很多不同层面和方向的研究分支以及大量丰富的研究内容和成果.鉴于标量优化理论与方法的成熟,将向量优化问题转化为标量优化问题来求解的标量化方法,被证明是一种重要和有效的方法.线性标量化方法简单易行,但同时因其对问题的凸性要求必不可少而使其应用受到了较大的限制.因此,为了处理实际中更多的非凸问题,不受到凸性限制的非线性标量化方法逐渐成为了研究的热点.这其中最为关键与核心的是非线性标量化函数的选取.本文围绕向量优化理论中的非线性标量化函数的性质分析及应用而展开,具体的工作分为以下的六个部分：第一,我们首先讨论了最大严格单调函数这一非线性标量化函数的若干基本性质并且给出该函数的对偶形式.然后提出了一般实拓扑向量空间中锥形邻域的概念和一类新的向量值映射锥半连续性的定义.此外,通过使用Gerstewitz泛函和最大严格单调函数这两个非线性标量化函数,我们得到了对向量值映射的锥半连续性完整统一的刻画.第二,利用两个非线性标量化函数,我们构造出了一种半范数并且在一种等价关系下导出了一个相关的赋范线性空间.然后基于通常的严有效性和超有效性,文中提出了锥严有效性和锥超有效性的概念并分析了新旧概念之间的关系.最后,我们得到了锥严有效性的若干标量化刻画,其中涉及到了相应标量化问题的适定性.第三,将赋范线性空间中的增广对偶锥的概念推广到了一般的局部凸空间中,在两种情形下分别给出了广义增广对偶锥的定义.然后讨论了它们的主要性质,并在合适的假设下建立了广义增广对偶锥非平凡的存在性条件.此外,在更一般的Hausdorff拓扑向量空间中,关于Gerstewitz泛函和最大严格单调函数的广义增广对偶锥的概念被提出.同时还给出了它们的一些性质及保证其非平凡性的存在性定理.第四,本文利用基泛函和增广对偶锥的概念,首次指出了范数、Gerstewitz泛函和面向距离函数等三种非线性标量化函数均具有某种和基泛函相同的特性.然后,在序锥存在有界基的假设下,通过借助增广对偶锥的结构,建立了这三种次线性函数在序锥上的等价性.然而我们证明两种超线性函数同范数之间却并没有类似的等价关系.更一般地,这三种次线性函数在负序锥外的等价性在本文中也被得到.第五,通过分别使用一种严格下水平集和最大严格单调函数,文中建立了对向量值映射的恰当锥拟凸性的水平集和标量化刻画.进一步,基于一般实拓扑向量空间中的两种常见集合偏序关系,我们先后给出了对集值映射的恰当锥拟凸性的相应刻画.我们使用的方法包括两种不同形式的水平集和最大严格单调函数.第六,在上述涉及到的一种常见的集合偏序关系下,我们提出了集值映射的标量锥拟凸概念,讨论了它与各种锥凸性的关系.同时建立了集值映射的各种锥凸性通过实值单调增加凸函数表示的标量化复合法则.最后给出了利用Gerstewitz泛函表示的对集值映射的锥拟凸性的标量化刻画.
[Abstract]:The vector optimization problem refers to the minimization of vector value functions under certain constraints. The theory of vector optimization has an inseparable connection with many theories in mathematics and economics. The theory and methods of vector optimization have formed a huge system and concentrated many different levels at present. It is proved to be an important and effective method to transform the vector optimization problem into the scalar optimization problem in view of the maturity of the scalar optimization theory and method, and the linear scalar method is simple and easy, but at the same time, the convexity of the problem is also due to its convexity. Therefore, in order to deal with more non convex problems in practice, the nonlinear scalar method, which is not restricted by convexity, has gradually become the focus of research. The key and core of this is the selection of nonlinear scalar function. This paper focuses on the nonlinearity of the vector optimization theory. The specific work of scalar function is divided into six parts: first, we first discuss some basic properties of the maximum strictly monotone function, a nonlinear scalar function, and give the dual form of the function. Then we propose a conical neighborhood of the general real topological vector space. In addition, by using the two nonlinear scalar functions of the Gerstewitz functional and the maximum strictly monotone function, we get a complete and unified description of the cone semicontinuous of the vector valued mapping. Second, we use two non linear scalar functions, we construct a kind of half. In this paper, a normed linear space is derived under an equivalent relation. Then based on the usual strict validity and superefficiency, the concept of conical validity and cone superefficiency is proposed and the relationship between the new and old concepts is analyzed. Finally, we get some scalar characterization of the conical validity. Third, the concept of the augmented dual cone in normed linear space is generalized to the general local convex space, and the generalized augmented dual cones are defined in two cases. Then the main properties of the generalized augmented dual cones are discussed, and the generalized augmented dual cone nonflat is established under the hypothetical assumption. In addition, in the more general Hausdorff topological vector space, the concept of generalized augmented dual cones for the Gerstewitz functional and the maximum strictly monotone function is proposed. Some properties of them and the existence theorems to guarantee their nontrivial properties are given. Fourth, this paper uses the basic functional and the augmented dual cone. The concept, for the first time points out the norm, the three nonlinear scalar functions, such as the Gerstewitz functional and the distance oriented function, have the same characteristics as the basic functional. Then, under the assumption that the order cone has a bounded basis, the equivalence of the three sub linear functions on the order cone is established by means of the structure of the augmented dual cone. However, we prove that the equivalence of the three sub linear functions on the order cone is established. There is no similar equivalence relation between the two superlinear functions and the norm. In general, the equivalence of the three sub linear functions outside the negative order cone is also obtained. Fifth, by using a strict lower level set and the maximum strictly monotone function, the level of the proper conical quasi convexity for the vector value mapping is established in this paper. Set and scalar characterization. Further, based on the two common set partial order relations in the general real topological vector space, we have given the corresponding characterization of the proper cone quasilateness for set valued mappings. The methods we use include two different forms of level set and the maximum strict single tone function. Sixth, a kind of ordinary one is involved in the above. Under the set partial order relation, we propose a scalar cone quasi convex concept of set valued mapping, and discuss the relation between it and various conical convexity. At the same time, we establish a scalar compound rule that the conic convexity of a set value mapping is monotonically increased by the real value. Finally, the cone quasi cones of the set value mapping using the Gerstewitz function are given. A scalar characterization of convexity.
【学位授予单位】：内蒙古大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O224

【相似文献】