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.     

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.     

利用Lagrange乘数法求解两类技巧性初等问题

本文利用Lagrange乘数法求解两类典型的复数、不等式问题.     

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.     

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.     

On the existence and nonexistence of Lagrange multipliers in Banach spaces

The paper deals with the existence of Lagrange multipliers for a general nonlinear programming problem. Some regularity conditions are formulated which are, in a sense, the weakest to assure the existence of multipliers. A number of related conditions are discussed. The connection between the choice of suitable function spaces and the existence of multipliers is analyzed.This work was partly supported by the National Science Foundation, Grant No. GF-37298, to the Institute of Automatic Control, Technical University of Warsaw, Warsaw, Poland, and the Department of Computer and Control Sciences, University of Minnesota, Minneapolis, Minnesota.The author wishes to thank Professor A. P. Wierzbicki for many important remarks concerning the subject of this paper.     

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.     

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.     

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.     

On the mixed finite element method with Lagrange multipliers

In this note we analyze a modified mixed finite element method for second‐order elliptic equations in divergence form. As a model we consider the Poisson problem with mixed boundary conditions in a polygonal domain of R ². The Neumann (essential) condition is imposed here in a weak sense, which yields the introduction of a Lagrange multiplier given by the trace of the solution on the corresponding boundary. This approach allows to handle nonhomogeneous Neumann boundary conditions, theoretically and computationally, in an alternative and usually easier way. Then we utilize the classical Babu?ka‐Brezzi theory to show that the resulting mixed variational formulation is well posed. In addition, we use Raviart‐Thomas spaces to define the associated finite element method and, applying some elliptic regularity results, we prove the stability, unique solvability, and convergence of this discrete scheme, under appropriate assumptions on the mesh sizes. Finally, we provide numerical results illustrating the performance of the algorithm for smooth and singular problems. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 192–210, 2003     

两大部类扩大再生产中的广义拉格朗日乘子

在一个带有非负和不等式约束的优化问题有最优解的情形下,存在着广义拉格朗日乘子即资源的影子价格．本文探索给出马克思两大部类扩大再生产中的影子价格,为经典的马克思扩大再生产理论增添新的重要内容．首先使用“价值系数法”替代单纯形法,简便地求得了扩大再生产优化问题的最优解．然后运用库恩一塔克条件,确立了关于最优解与广义拉格朗日乘子的互补松弛条件的三个不等式组．进而利用这些不等式组和已知的最优解,简便地解出广义拉格朗日乘子,即两大部类扩大再生产中的影子价格．最后引用和借鉴《资本论》中的两个举例,对所获得的影子价格和目标函数最优值做了计算验证．     

Lagrange multiplers for pareto nonsmooth programming problems in banach spaces

We establish the existence of Lagrange multipliers for general Pareto multiobjective mathematical programming problems in Banach spaces. Here the data are general nonsmooth strongly compactly ipschitzian mappings     

Bounded sets of Lagrange multipliers for vector optimization problems in infinite dimension

The aim of this paper is to point out some sufficient constraint qualification conditions ensuring the boundedness of a set of Lagrange multipliers for vectorial optimization problems in infinite dimension. In some (smooth) cases these conditions turn out to be necessary for the existence of multipliers as well.     

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.     

Lagrange multipliers and infinite-dimensional equilibrium problems

We prove the existence of the Lagrange multipliers for a constrained optimization problem, being the constraint set given by the convex set which characterizes the most important equilibrium problems. In order to obtain our result, we’ll make use of the new concept of quasi relative interior.     

Lagrange multipliers theorem and saddle point optimality criteria in mathematical programming

We prove a version of Lagrange multipliers theorem for nonsmooth functionals defined on normed spaces. Applying these results, we extend some results about saddle point optimality criteria in mathematical programming.     

Lagrange Multipliers and mixed formulations for some inequality constrained variational inequalities and some nonsmooth unilateral problems

Abstract

This paper addresses a class of inequality constrained variational inequalities and nonsmooth unilateral variational problems. We present mixed formulations arising from Lagrange multipliers. First we treat in a reflexive Banach space setting the canonical case of a variational inequality that has as essential ingredients a bilinear form and a non-differentiable sublinear, hence convex functional and linear inequality constraints defined by a convex cone. We extend the famous Brezzi splitting theorem that originally covers saddle point problems with equality constraints, only, to these nonsmooth problems and obtain independent Lagrange multipliers in the subdifferential of the convex functional and in the ordering cone of the inequality constraints. For illustration of the theory we provide and investigate an example of a scalar nonsmooth boundary value problem that models frictional unilateral contact problems in linear elastostatics. Finally we discuss how this approach to mixed formulations can be further extended to variational problems with nonlinear operators and equilibrium problems, and moreover, to hemivariational inequalities.     

Constraint qualifications and Lagrange multipliers in nondifferentiable programming problems

In this paper, we present several constraint qualifications, and we show that these conditions guarantee the nonvacuity and the boundedness of the Lagrange multiplier sets for general nondifferentiable programming problems. The relationships with various constraint qualifications are investigated.The author gratefully acknowledges the comments made by the two referees.     

Pseudonormality and a Lagrange Multiplier Theory for Constrained Optimization

We consider optimization problems with equality, inequality, and abstract set constraints, and we explore various characteristics of the constraint set that imply the existence of Lagrange multipliers. We prove a generalized version of the Fritz–John theorem, and we introduce new and general conditions that extend and unify the major constraint qualifications. Among these conditions, two new properties, pseudonormality and quasinormality, emerge as central within the taxonomy of interesting constraint characteristics. In the case where there is no abstract set constraint, these properties provide the connecting link between the classical constraint qualifications and two distinct pathways to the existence of Lagrange multipliers: one involving the notion of quasiregularity and the Farkas lemma, and the other involving the use of exact penalty functions. The second pathway also applies in the general case where there is an abstract set constraint.