Farkas引理及其应用
发布时间：2018-11-29 11:57
【摘要】：Farkas引理是一个经典的结果,是最优化方法中最为基础的工具之一.Farkas引理最早是由Farkas本人在1902年提出的.我们可以在大多数最优化教程中发现该引理的证明.如文献[2]中,早期证明类似于对偶单纯形法,但其证明并未考虑到可能出现的循环现象,因此并不完整.近期的证明通常基于凸集分离定理.该方法有一个简单和更直观的几何解释.本文给出了多种不同的证明方法,并给出了它的几种不同的等价形式.本文的主要目的是以Farkas引理为中心,对其不同证明方法及其各种等价形式做一个系统的整理.其中,其证明方法大体分成三类,即初等证明、几何证明和代数证明.除此之外,本文还给出了Farkas引理的几种不同的应用.它在很多方面都起着不可替代的作用,尤其是在非线性规划理论中起着重要作用,如表示最优解的Fritz John定理及Kuhn-Tucker定理均可由它导出.文中还给出了Farkas引理的一个简单的经济学解释实例,即Farkas引理可以表述为:风险中性概率的存在是无套利条件的结果.当然其应用方面远不止文中所提到的这些.希望以下的文章能使我们更好的理解和运用Farkas引理及其相关定理。
[Abstract]:Farkas Lemma is a classical result and one of the most basic tools in optimization methods. Farkas Lemma was first proposed by Farkas himself in 1902. We can find proof of this Lemma in most optimization tutorials. For example, in reference [2], the early proof is similar to the dual simplex method, but its proof does not take into account the possible cyclic phenomenon, so it is not complete. Recent proofs are usually based on the separation theorem of convex sets. The method has a simple and more intuitive geometric explanation. In this paper, several different proof methods are given, and several different equivalent forms are given. The main purpose of this paper is to make a systematic arrangement of the different proof methods and their equivalent forms with Farkas Lemma as the center. Among them, there are three kinds of proof methods: elementary proof, geometric proof and algebraic proof. In addition, several different applications of Farkas Lemma are given. It plays an irreplaceable role in many aspects, especially in the theory of nonlinear programming. For example, the Fritz John theorem and Kuhn-Tucker theorem, which represent the optimal solution, can be derived from it. A simple economic example of Farkas Lemma is given in this paper. That is, Farkas Lemma can be expressed as: the existence of risk-neutral probability is the result of no arbitrage condition. Of course, its application is far more than those mentioned in the paper. It is hoped that the following articles will help us to better understand and apply Farkas Lemma and its related theorems.
【学位授予单位】：长江大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O224
本文编号：2364928
[Abstract]:Farkas Lemma is a classical result and one of the most basic tools in optimization methods. Farkas Lemma was first proposed by Farkas himself in 1902. We can find proof of this Lemma in most optimization tutorials. For example, in reference [2], the early proof is similar to the dual simplex method, but its proof does not take into account the possible cyclic phenomenon, so it is not complete. Recent proofs are usually based on the separation theorem of convex sets. The method has a simple and more intuitive geometric explanation. In this paper, several different proof methods are given, and several different equivalent forms are given. The main purpose of this paper is to make a systematic arrangement of the different proof methods and their equivalent forms with Farkas Lemma as the center. Among them, there are three kinds of proof methods: elementary proof, geometric proof and algebraic proof. In addition, several different applications of Farkas Lemma are given. It plays an irreplaceable role in many aspects, especially in the theory of nonlinear programming. For example, the Fritz John theorem and Kuhn-Tucker theorem, which represent the optimal solution, can be derived from it. A simple economic example of Farkas Lemma is given in this paper. That is, Farkas Lemma can be expressed as: the existence of risk-neutral probability is the result of no arbitrage condition. Of course, its application is far more than those mentioned in the paper. It is hoped that the following articles will help us to better understand and apply Farkas Lemma and its related theorems.
【学位授予单位】：长江大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O224
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