利用Lagrange乘数法证明加权平均不等式

在高等数学中,证明不等式的常用方法是利用函数的单调性及函数的极值或最值．文献[1]用多元函数极值性质证明了算术-几何平均不等式,本文用Lagrange乘数法证明在应用上很重要的一个不等式—加权平均不等式．不等式称为加权平均不等式其中等号当且仅当时成立．行证明即可．构造Lagrange函数对诸X;求偏导并令其为零,则有解得,将其代中就得到山（下转第37页）为唯一驻点．因为是诸的连续函数,由文献[3]知,处取得最小值所以等号当且仅当时成立．利用Lagrange乘数法证明加权平均不等式@张俊祖$西安公路交通大学[1]薛红,条件极值在证明不…     

利用拉格朗日乘数法证明两个不等式

文中利用利用拉格朗日乘数法证明了两个不等式,其中一个不等式验证了发表文献中一个类比猜想不成立,另一个不等式验证了发表文献中一个类比猜想成立.     

鞍点定理在Lagrange乘数法上的应用

将鞍点的概念运用在Lagrange乘数法上,给出了多元函数的条件极值问题存在的一个充要条件.     

拉格朗日乘数法求解极值点的讨论

文中对拉格朗日乘数法极值点的求法进行了补充说明.     

巧用Lagrange乘数法求一类多元对称函数的条件最值

巧用Lagrange乘数法,将一类多元对称函数的条件最值转化为一元函数的无条件最值,避免了具体求复杂而困难的驻点方程组的解,使问题化难为易.     

Lagrange乘数法与规划问题

本文运用Ｌａｇｒａｎｇｅ乘数法论证了线性规划问题中的对偶定理与“悖论”现象，并分析了为什么不用此方法解线性规划问题。     

用Lagrange乘数研究二次曲线的几何性质

文中把二次曲线的几何性质的研究转化成条件极值问题,但又不关心问题的解,而是利用Lagrange乘数来研究二次曲线的几何性质,找到了用Lagrange 乘数判别二次曲线形状的方法,给出了用Lagrange乘数计算二次曲线的对称轴和轴长的公式.     

拉格朗日乘数法的几何理解

结合目标函数等值线族与条件曲线的图像,从几何上认识拉格朗日乘数法,理解拉格朗日乘数法在解决条件极值问题过程中的必要非充分性.     

关于拉格朗日乘数法的一点注记

建立了多元函数在任意有限多个约束条件下的极值点和拉格朗日函数极值点之间的一一对应关系,从而找到拉格朗日函数的极值点也就找到了多元函数在这些约束条件下的极值点.从另一角度给出了拉格朗日乘数法的证明.     

拉格朗日乘数法在条件不等式证明中的应用

拉格朗日乘数法是高等数学中求多元函数条件极值的重要方法,应用广泛,思想深刻．该方法程序性强,非常容易掌握,但由于涉及到求多元函数的偏微分,因此并不适合中学生直接学习。那么,能否将该法加以改进,使普通中学生也能轻松掌握呢？回答是肯定的．下面笔者将通过例题说明如何用改进后的拉格朗日乘数法证明条件不等式．     

The Mortar Element Method with Lagrange Multipliers for Stokes Problem

In this paper,we propose a mortar element method with Lagrange multiplier for incompressible Stokes problem,i.e.,the matching constraints of velocity on mortar edges are expressed in terms of Lagrange multipliers.We also present P_1 noncon- forming element attached to the subdomains.By proving inf-sup condition,we derive optimal error estimates for velocity and pressure.Moreover,we obtain satisfactory approximation for normal derivatives of the velocity across the interfaces.     

On the Genesis of the Lagrange Multipliers

The genesis of the Lagrange multipliers is analyzed in this work. Particularly, the author shows that this mathematical approach was introduced by Lagrange in the framework of statics in order to determine the general equations of equilibrium for problems with constraints. Indeed, the multipliers allowed Lagrange to treat the questions of maxima and minima in differential calculus and in calculus of variations in the same way as problems of statics: if the equilibrium of a point or a system of points is required, there is an analogy between statics and differential calculus; if the equilibrium of a rigid body is required, there is an analogy between statics and calculus of variations.     

Lagrange Multipliers and saddle points in multiobjective programming

In this paper, we present several conditions for the existence of a Lagrange multiplier or a weak saddle point in multiobjective optimization. Relations between a Lagrange multiplier and a weak saddle point are established. A sufficient condition is also given for the equivalence of the Benson proper efficiency and the Borwein proper efficiency.This research was supported by NSFC under Grant No. 78900011 and by BMADIS. The authors are grateful to two referees for supplying valuable comments and pointing out detailed corrections to the draft paper. The authors also wish to thank Dr. P. L. Yu for valuable comments and suggestions.The revised version of this paper was completed while the second author visited the Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands.     

Lagrange Multipliers for Multiobjective Programs with a General Preference

We consider a nonsmooth multiobjective optimization problems related to a new general preference between infinite dimensional Banach spaces. This preference contains preferences given by generalized Pareto as well as those given by an utility function. We use the concepts of compactly epi-Lipschitzian sets and strongly compactly Lipschitzian mappings to derive Lagrange multipliers of Karush–Kuhn–Tucker type and Fritz-John type in terms of the Ioffe-approximate subdifferentials. Dedicated to Boris Mordukhovich in honour of his 60th birthday.     

Separation of Sets,Lagrange Multipliers,and Totally Regular Extremum Problems

We introduce the concept of total regularity for the separation of sets, and we give a characterization of it. Also, we prove the equivalence between total regularity and boundedness of the generalized multipliers associated to the separation, and we compute the value of the bound. Then, we give a theorem concerning the uniqueness of such multipliers. Afterward, the previous results are applied to the study of the impossibility of generalized systems; particular attention is devoted to systems arising from the optimality conditions of constrained extremum problems.     

On generalized Nash equilibrium in infinite dimension: the Lagrange multipliers approach

In this note we study a class of generalized Nash equilibrium problems and characterize the solutions which have the property that all players share the same Lagrange multipliers. Nash equilibria of this kind were introduced by Rosen in 1965, in finite-dimensional spaces. In order to obtain the same property in infinite dimension, we use very recent developments of a new duality theory. In view of its usefulness in the study of time-dependent or stochastic equilibrium problems, an application in Lebesgue spaces is given.     

Lagrange multipliers for functions derivable along directions in a linear subspace

We prove a Lagrange multipliers theorem for a class of functions that are derivable along directions in a linear subspace of a Banach space where they are defined. Our result is available for topological linear vector spaces and is stronger than the classical one even for two-dimensional spaces, because we only require the differentiablity of functions at critical points. Applying these results we generalize the Lax-Milgram theorem. Some applications in variational inequalities and quasilinear elliptic equations are given.

     

On Normality of Lagrange Multipliers for State Constrained Optimal Control Problems

An optimal control problem for nonlinear ODEs, subject to mixed control-state and pure state constraints is considered. Sufficient conditions are formulated, under which unique normal Lagrange multipliers exist and are given by regular functions. These conditions include pointwise linear independence of gradients of f -active constraints and controllability of the linearized state equation. Under some additional assumptions, further regularity of the multipliers is shown.