本文選題：Gerstewitz泛函 + 最大嚴格單調(diào)函數(shù)��； 參考：《內(nèi)蒙古大學》2016年博士論文

【摘要】：向量優(yōu)化問題是指在一定的約束條件下極小化向量值函數(shù).向量優(yōu)化理論從產(chǎn)生、發(fā)展到逐漸成熟的過程中,與數(shù)學和經(jīng)濟學中的許多理論均有著密不可分的聯(lián)系.目前向量優(yōu)化理論和方法已形成了一個巨大體系,集中了很多不同層面和方向的研究分支以及大量豐富的研究內(nèi)容和成果.鑒于標量優(yōu)化理論與方法的成熟,將向量優(yōu)化問題轉(zhuǎn)化為標量優(yōu)化問題來求解的標量化方法,被證明是一種重要和有效的方法.線性標量化方法簡單易行,但同時因其對問題的凸性要求必不可少而使其應用受到了較大的限制.因此,為了處理實際中更多的非凸問題,不受到凸性限制的非線性標量化方法逐漸成為了研究的熱點.這其中最為關鍵與核心的是非線性標量化函數(shù)的選取.本文圍繞向量優(yōu)化理論中的非線性標量化函數(shù)的性質(zhì)分析及應用而展開,具體的工作分為以下的六個部分：第一,我們首先討論了最大嚴格單調(diào)函數(shù)這一非線性標量化函數(shù)的若干基本性質(zhì)并且給出該函數(shù)的對偶形式.然后提出了一般實拓撲向量空間中錐形鄰域的概念和一類新的向量值映射錐半連續(xù)性的定義.此外,通過使用Gerstewitz泛函和最大嚴格單調(diào)函數(shù)這兩個非線性標量化函數(shù),我們得到了對向量值映射的錐半連續(xù)性完整統(tǒng)一的刻畫.第二,利用兩個非線性標量化函數(shù),我們構(gòu)造出了一種半范數(shù)并且在一種等價關系下導出了一個相關的賦范線性空間.然后基于通常的嚴有效性和超有效性,文中提出了錐嚴有效性和錐超有效性的概念并分析了新舊概念之間的關系.最后,我們得到了錐嚴有效性的若干標量化刻畫,其中涉及到了相應標量化問題的適定性.第三,將賦范線性空間中的增廣對偶錐的概念推廣到了一般的局部凸空間中,在兩種情形下分別給出了廣義增廣對偶錐的定義.然后討論了它們的主要性質(zhì),并在合適的假設下建立了廣義增廣對偶錐非平凡的存在性條件.此外,在更一般的Hausdorff拓撲向量空間中,關于Gerstewitz泛函和最大嚴格單調(diào)函數(shù)的廣義增廣對偶錐的概念被提出.同時還給出了它們的一些性質(zhì)及保證其非平凡性的存在性定理.第四,本文利用基泛函和增廣對偶錐的概念,首次指出了范數(shù)、Gerstewitz泛函和面向距離函數(shù)等三種非線性標量化函數(shù)均具有某種和基泛函相同的特性.然后,在序錐存在有界基的假設下,通過借助增廣對偶錐的結(jié)構(gòu),建立了這三種次線性函數(shù)在序錐上的等價性.然而我們證明兩種超線性函數(shù)同范數(shù)之間卻并沒有類似的等價關系.更一般地,這三種次線性函數(shù)在負序錐外的等價性在本文中也被得到.第五,通過分別使用一種嚴格下水平集和最大嚴格單調(diào)函數(shù),文中建立了對向量值映射的恰當錐擬凸性的水平集和標量化刻畫.進一步,基于一般實拓撲向量空間中的兩種常見集合偏序關系,我們先后給出了對集值映射的恰當錐擬凸性的相應刻畫.我們使用的方法包括兩種不同形式的水平集和最大嚴格單調(diào)函數(shù).第六,在上述涉及到的一種常見的集合偏序關系下,我們提出了集值映射的標量錐擬凸概念,討論了它與各種錐凸性的關系.同時建立了集值映射的各種錐凸性通過實值單調(diào)增加凸函數(shù)表示的標量化復合法則.最后給出了利用Gerstewitz泛函表示的對集值映射的錐擬凸性的標量化刻畫.
[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.
【學位授予單位】：內(nèi)蒙古大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O224

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