Farkas Lemma And Its Application

Farkas lemma is one of the most classical basic tools in the optimization method.Farkas' s lemma was first published by Farkas in 1902.We can find the proof of Farkas lemma in most of the optimization courses.In literature [2],the early proof of this view is similar to the dual simplex method.However,it is not complete since it does not take into account the possible circulation.The recent proof is usually based on the separation theorem of convex sets.This method has a simple and more intuitive geometric interpretation.In this paper,many different methods based on its basic theory and the different understanding and extension of the theorem.The main purpose of this paper focus on the Farkas lemma to make a systematic collation of its different proof methods and its various equivalent forms at the center of Farkas' s lemma.In this paper,the proof method of Farkas is divided into three classes,namely elementary proof,geometric proof and algebra proof.In addition,several different applications of Farkas lemma are given.The Farkas lemma plays an irreplaceable role in many aspects,especially in nonlinear programming theory,such as John theorem and Kuhn theorem can be derived.A simple example of economics lemma can be expressed as: the existence of risk neutral probability is the result of there is no conditions.Hope that the following article can make us better understand and use the farkas lemma and its related theorems.