【摘要】：Farkas引理是一個經(jīng)典的結(jié)果,是最優(yōu)化方法中最為基礎(chǔ)的工具之一.Farkas引理最早是由Farkas本人在1902年提出的.我們可以在大多數(shù)最優(yōu)化教程中發(fā)現(xiàn)該引理的證明.如文獻(xiàn)[2]中,早期證明類似于對偶單純形法,但其證明并未考慮到可能出現(xiàn)的循環(huán)現(xiàn)象,因此并不完整.近期的證明通常基于凸集分離定理.該方法有一個簡單和更直觀的幾何解釋.本文給出了多種不同的證明方法,并給出了它的幾種不同的等價(jià)形式.本文的主要目的是以Farkas引理為中心,對其不同證明方法及其各種等價(jià)形式做一個系統(tǒng)的整理.其中,其證明方法大體分成三類,即初等證明、幾何證明和代數(shù)證明.除此之外,本文還給出了Farkas引理的幾種不同的應(yīng)用.它在很多方面都起著不可替代的作用,尤其是在非線性規(guī)劃理論中起著重要作用,如表示最優(yōu)解的Fritz John定理及Kuhn-Tucker定理均可由它導(dǎo)出.文中還給出了Farkas引理的一個簡單的經(jīng)濟(jì)學(xué)解釋實(shí)例,即Farkas引理可以表述為:風(fēng)險(xiǎn)中性概率的存在是無套利條件的結(jié)果.當(dāng)然其應(yīng)用方面遠(yuǎn)不止文中所提到的這些.希望以下的文章能使我們更好的理解和運(yùn)用Farkas引理及其相關(guān)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.
【學(xué)位授予單位】：長江大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O224

【相似文獻(xiàn)】

相關(guān)期刊論文 前5條

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2 劉鐵;黃兆霞;;Farkas定理的一些應(yīng)用[J];安康學(xué)院學(xué)報(bào);2008年05期

3 安潤秋;Farkas引理的證明及其應(yīng)用[J];唐山高等�？茖W(xué)校學(xué)報(bào);2001年04期

4 尚學(xué)海,趙晶;關(guān)于 Farkas 定理的一個注記[J];天津城市建設(shè)學(xué)院學(xué)報(bào);1997年02期

5 王周宏;;Farkas引理的幾個等價(jià)形式及其推廣[J];應(yīng)用數(shù)學(xué)學(xué)報(bào);2008年05期

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本文編號：2364929