Farkas引理的几个等价形式及其推广

本文考虑了Farkas引理,Gordan引理及其拓展形式之间的关系,从理论上证明了其等价性并说明了Farkas引理在各种等价形式中的重要地位,并指出了Gordan引理实际是叮看作是Farkas引理的弱形式,然后研究了Farkas引理及其它形式在锥线性不等式组中的推广.     

一类带DC函数的分式优化的Farkas引理刻画

该文旨在刻画一类带有DC函数(即两个凸函数的差)的约束分式优化问题的Farkas引理.借助Dinkelbach方法,将该分式优化问题转化为DC优化问题.随后借助共轭函数的上图技巧所引入的新的正则性条件,刻画了该DC优化问题与其Fenchel-Lagrange对偶问题之间的对偶关系,从而建立了该分式优化问题的一些新的Fa...     

Zm环上线性组合引理的证明

本文证明了样本空间Ω=Z^nm上m值随机变量Z，与k个互相独立且分布均匀的m值随机变量Y1，…，Yh统计独立的充分必要条件，是Z与Y1，…，Yh的所有非零线性组合λ1Y1 … λhYh统计独立，其中λi∈Zm，i=1，…，m．     

Schwarz引理与Schwarz-Pick引理在单位球B_n上的推广

对单复变中的Schwarz引理与Schwarz-Pick引理在C~n中的超球上进行了推广.考虑C~n中单位球B_n上模小于1的全纯函数f(z),并在f(0)=0的条件下给出函数在原点的任意阶导数的估计.更进一步地,得到了B_n上模小于1的任意全纯函数在任意点的高阶导数的估计.     

一个圆锥曲线引理的补正及其应用

文 [1 ]给出了文 [2 ]的一个统一命题 ,并利用一个引理给出了简证 ,遗憾的是统一命题虽对 ,但是引理不正确 ,为此特作更正 ,并用圆锥曲线的这一几何性质证明几个命题 .定理　设F是圆锥曲线的焦点 ,其相应的准线为L ,过L上一点M作直线交圆锥曲线于P ,Q两点 ,则MF平分∠PFQ ,或其邻补角 .证明　设圆锥曲线的离心率为e,点P ,Q在准线上的射影为R ,S .如图 1中 ,图 (甲 )为e≤ 1 ,图 (乙 )为e >1的情况 .图 1由圆锥曲线的定义得 :｜PF｜｜PR｜=e,｜QF｜｜QS｜=e.由平行线的性质得 :｜PR｜｜QS｜ =｜PM｜｜QM｜.所以 ｜PF｜｜QF｜ =…     

Schwarz引理的推广及其应用

引入Schwarz引理的一个最常见的推广定理,并且作出了详细的证明.同时以引理形式介绍了一个实用的复数性质,并且利用这两个引理,给出了开圆盘内解析函数的的实部,虚部以及模的估计式.     

无限凸规划的约束规范条件和Farkas引理

假设X是局部凸Hausdorff拓扑向量空间,C是X的闭凸子集,T是一下标集(可以是无限集),假设{ft:t∈T}是一X到R ∪{+∞}的真凸下半连续函数簇,而f:x→R∪{+∞}是一真凸下半连续函数.     

罗宾逊序列引理的推广及其应用

A. Robinson＇s sequential lemma is extended to nets in general topological space, and obviously the case of nets in ^＊R is its corollary. As its application, the paper proves a property about topology of uniform convergence.     

关于Morse引理的推广

Morse Lemma是奇点理论中一个极为重要的结论。[1]的作者称其文中的定理1和定理2是Morse Lemma的推广。为此我们愿就[1]中的几个问题与[1]的作者商榷。     

两个引理及其推广的再拓广

文[1]介绍了有关不等式的两个引理及其推广命题1-4．文[2]将两个引理及其推广命题作出了进一步推广．本文将推广的两个引理及其推广的命题再进一步作出拓广．两个引理的再拓广如下：     

A Note on a Matrix Version of the Farkas Lemma

A linear polyomial non-negative on the non-negativity domain of finitely many linear polynomials can be expressed as their non-negative linear combination. Recently, under several additional assumptions, Helton, Klep, and McCullough extended this result to matrix polynomials. The aim of this article is to study which of these additional assumptions are really necessary.     

A Network Algorithm for Realizing Constructivelythe Farkas Lemma Arising from the Quasi-Generalized Network

1.IntroductionLetRIbethespaceofallrealvectorsindexedbyelementsofafiniteseti,icachosenelemelltofi,SandS*apairofcompletelyorthogonalsubspacesofRI.Foragivenpartition(PI,P2,P3,P4)ofI--{ic}(i.e.PIuPZuP3uP4=I--{eo},andPinPj=acfori/j),letF={axled6S,Ax'.>0,Ax.20fore6PI,Ax.50foreEP2,Ax.~0foreEP3},F*={dyIAyES*,ac.>0,ace30foreEPI,ac50foreEP2,ac.~0foreEP4}.Now,theFarkasLemmacanbegenerallydescribedas[1]:oneandonlyoneofthefollowingtwostatemelltsholds:(i)ThereexistsaaxEF.(n)Thereexistsaac…     

A combinatorial proof of the Removal Lemma for Groups

Green [B. Green, A Szemerédi-type regularity lemma in abelian groups, with applications, Geom. Funct. Anal. 15 (2005) 340-376] established a version of the Szemerédi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his Removal Lemma that allows us to extend its statement to all finite groups. We also discuss possible extensions of the Removal Lemma to systems of equations.     

Boundedness and conservativeness of augmented marked graphs

^** Email: cheungks{at}hkbu.edu.hk Augmented marked graphs are often used in modelling shared resourcesystems. A major reason is that they possess a structure especiallydesirable for modelling shared resources. Yet, the propertiesof augmented marked graphs are not extensively studied. Onlysome siphon-based and cycle-based characterisations for deadlockfreeness, liveness and reversibility are reported in the literature.This paper investigates the boundedness and conservativenessof augmented marked graphs. Based on R-transform, a necessaryand sufficient condition for bounded and conservative augmentedmarked graphs is proposed. This is useful in analysing the capacityoverflow for systems modelled as augmented marked graphs.     

Some generalizations of Ahlfors Lemma

One of the most influential versions of the classical Schwarz–Pick Lemma is probably that of Ahlfors. Pulling back a conformal semimetric on a Riemann surface under any holomorphic map from the open unit disk equipped with a Poincaré metric, the curvature of which is assumed to bound from above the curvature of the Riemann surface, he successfully showed that a conformal semimetric to be compared with the Poincaré metric is obtained. In the present paper, we give a comparison theorem between two conformal semimetrics of variable curvature in the same spirit. Our main theorem is a local one by its nature, but global results can be derived therefrom.     

A general Farkas lemma and characterization of optimality for a nonsmooth program involving convex processes

In this paper, a generalization of the Farkas lemma is presented for nonlinear mappings which involve a convex process and a generalized convex function. Using this result, a complete characterization of optimality is obtained for the following nonsmooth programming problem: minimizef(x), subject to – H(x) wheref is a locally Lipschitz function satisfying a generalized convexity hypothesis andH is a closed convex process.This work was partially written while the author was a PhD Student under the supervision of Dr. B. D. Craven, University of Melbourne, whose helpful guidance is much appreciated.     

Approximate Farkas lemmas and stopping rules for iterative infeasible-point algorithms for linear programming

In exact arithmetic, the simplex method applied to a particular linear programming problem instance with real data either shows that it is infeasible, shows that its dual is infeasible, or generates optimal solutions to both problems. Most interior-point methods, on the other hand, do not provide such clear-cut information. If the primal and dual problems have bounded nonempty sets of optimal solutions, they usually generate a sequence of primal or primaldual iterates that approach feasibility and optimality. But if the primal or dual instance is infeasible, most methods give less precise diagnostics. There are methods with finite convergence to an exact solution even with real data. Unfortunately, bounds on the required number of iterations for such methods applied to instances with real data are very hard to calculate and often quite large. Our concern is with obtaining information from inexact solutions after a moderate number of iterations. We provide general tools (extensions of the Farkas lemma) for concluding that a problem or its dual is likely (in a certain well-defined sense) to be infeasible, and apply them to develop stopping rules for a homogeneous self-dual algorithm and for a generic infeasible-interior-point method for linear programming. These rules allow precise conclusions to be drawn about the linear programming problem and its dual: either near-optimal solutions are produced, or we obtain certificates that all optimal solutions, or all feasible solutions to the primal or dual, must have large norm. Our rules thus allow more definitive interpretation of the output of such an algorithm than previous termination criteria. We give bounds on the number of iterations required before these rules apply. Our tools may also be useful for other iterative methods for linear programming. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.