On a characterization of convergence for the Hestenes method of multipliers

The connection between the convergence of the Hestenes method of multipliers and the existence of augmented Lagrange multipliers for the constrained minimum problem (P): minimizef(x), subject tog(x)=0, is investigated under very general assumptions onX,f, andg.In the first part, we use the existence of augmented Lagrange multipliers as a sufficient condition for the convergence of the algorithm. In the second part, we prove that this is also a necessary condition for the convergence of the method and the boundedness of the sequence of the multiplier estimates.Further, we give very simple examples to show that the existence of augmented Lagrange multipliers is independent of smoothness condition onf andg. Finally, an application to the linear-convex problem is given.     

Error analysis of variable degree mixed methods for elliptic problems via hybridization

A new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree Raviart-Thomas method for second order elliptic problems. The approach gives error estimates for the Lagrange multipliers without using error estimates for the other variables. Error estimates for the primal and flux variables then follow from those for the Lagrange multipliers. In contrast, traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the Lagrange multipliers. The new approach not only gives new error estimates for the new variable degree Raviart-Thomas method, but also new error estimates for the classical uniform degree method with less stringent regularity requirements than previously known estimates. The error analysis is achieved by using a variational characterization of the Lagrange multipliers wherein the other unknowns do not appear. This approach can be applied to other hybridized mixed methods as well.

     

New formulation of iterative substructuring methods without Lagrange multipliers: Neumann–Neumann and FETI

This article is devoted to introduce a new approach to iterative substructuring methods that, without recourse to Lagrange multipliers, yields positive definite preconditioned formulations of the Neumann–Neumann and FETI types. To my knowledge, this is the first time that such formulations have been made without resource to Lagrange multipliers. A numerical advantage that is concomitant to such multipliers‐free formulations is the reduction of the degrees of freedom associated with the Lagrange multipliers. Other attractive features are their generality, directness, and simplicity. The general framework of the new approach is rather simple and stems directly from the discretization procedures that are applied; in it, the differential operators act on discontinuous piecewise‐defined functions. Then, the Lagrange multipliers are not required because in such an environment the functions‐discontinuities are not an anomaly that need to be corrected. The resulting algorithms and equations‐systems are also derived with considerable detail. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Optimal control of parabolic problems with state constraints: a penalization method for optimality conditions

In this paper state constrained optimal control problems governed by parabolic evolution equations are studied. Our purpose is to obtain a (first-order) decoupled optimality system (that ensures the Lagrange multipliers existence). In a first step we are led to Slater-like assumptions and we are then allowed to extend the application field of the decoupled system we obtain. With a weaker assumption the existence of Lagrange multipliers (that are measures) for nonqualified problems may be established.     

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.     

改进的拉氏乘子法和多变量广义变分原理

拉氏乘子法是构造广义变分原理的重要途径 ,在识别拉氏乘子时 ,拉氏乘子是独立变分的 ,而识别后 ,它却是其他变量的函数 ,这是产生临界变分的原因 .本文对拉氏乘子法作了改进 ,提出了一种新的理论——凑合反推法 ,应用该方法可以方便地构造多变量的广义变分原理 ,并且不会出现临界变分现象     

对合变换和薄板弯曲问题的多变量变分原理

本文利用拉氏乘子法把薄板弯曲问题的最小位能原理和最小余能原理的变分约束条件解除.从而导出了常见的广义变分原理.为了降低泛函中变量导数的阶次.我们用对合变换引进新的正则变量.于是,我们可以进一步利用拉氏乘子法,把这些对合变换当作变分约束而予以消除,从而导出了各种多变量的薄板弯曲广义变分原理.事实证明,使用上述拉氏乘子法,并不能消除一切变分约束;为此,我们进一步引用高阶拉氏乘子法消除这些剩下来的约束条件,从而导得了薄板弯曲问题的更一般的广义变分原理.     

Méthode d'éléments finis avec hybridisation frontière pour les problèmes de contact avec frottementFinite element method with Lagrange multipliers for contact problems with friction

In this Note, we propose a finite element method with Lagrange multipliers in order to approximate contact problems with friction. The discretized normal and tangential constraints at the candidate contact interface are expressed by using continuous piecewise linear Lagrange multipliers in the saddle-point formulation. An optimal error estimate is established and several numerical studies corresponding to this choice of the discretized normal and tangential constraints are achieved. To cite this article: L. Baillet, T. Sassi, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 917–922.     

Non-linearity of multibody dynamic equations with respect to Lagrange multipliers: application to railway dynamics

In this paper, the dynamics of multibody systems with closed kinematical chains of bodies is considered. The main focus is set on non-linearity of the multibody equations with respect to the Lagrange multipliers. When closed chains are considered, loop cutting procedure is a solution to express the constraint equations associated with the loops. Dynamic equations of the multibody tree-like structure are thus completed with the constraint forces via the Lagrange multipliers. In the considered case of railway vehicles, constraints arise from the contact between the rigid wheels and the rails. Corresponding contact forces applied to the wheels appears via the Lagrange multipliers λ and the tangent creep forces as well. Resulting differential-algebraic equations can be transformed into an ODE system and then time-integrated using the coordinate partitioning method [3], when the system is linear with respect to λ. This paper presents an algorithm allowing us to solve this system in case of nonlinearities with respect to λ, which is typical of wheel/rail contact force models. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of He's variational iteration method to nonlinear Jaulent–Miodek equations and comparing it with ADM

Instead of finding a small parameter for solving nonlinear problems through perturbation method, a new analytical method called He's variational iteration method (VIM) is introduced to be applied to solve nonlinear Jaulent–Miodek, coupled KdV and coupled MKdV equations in this article. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers can be identified optimally via the variational theory. The results are compared with exact solutions.     

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.     

On the boundedness of Lagrange multipliers of the Kuhn-Tucker theorem applied to the problem of Tikhonov function minimization

In the proof of the convergence of sequences of approximations derived by the regularized method of linearization, the Kuhn-Tucker theorem with bounded sequences of Lagrange multipliers is applied to sequences of Tikhonov functions. This paper demonstrates that in the case of three existing forms of constrains: (i) functional inequalities strict at some point, (ii) linear functional inequalities, and (iii) a linear operator equality, there exist bounded sequences of Lagrange multipliers of the Kuhn-Thucker theorem applied to the sequences of Tikhonov functions.     

On differentiability with respect to parameter of solutions to convex optimal control problems subject to state space constraints

A family of convex optimal control problems that depend on a real parameterh is considered. The optimal control problems are subject to state space constraints.It is shown that under some regularity conditions on data the solutions of these problems as well as the associated Lagrange multipliers are directionally-differentiable functions of the parameter.The respective right-derivatives are given as the solution and respective Lagrange multipliers for an auxiliary quadratic optimal control problem subject to linear state space constraints.If a condition of strict complementarity type holds, then directional derivatives become continuous ones.     

On the attraction of Newton’s method to critical lagrange multipliers

The attraction of dual trajectories of Newton’s method for the Lagrange system to critical Lagrange multipliers is analyzed. This stable effect, which has been confirmed by numerical practice, leads to the Newton-Lagrange method losing its superlinear convergence when applied to problems with irregular constraints. At the same time, available theoretical results are of “negative” character; i.e., they show that convergence to a noncritical multiplier is not possible or unlikely. In the case of a purely quadratic problem with a single constraint, a “positive” result is proved for the first time demonstrating that the critical multipliers are attractors for the dual trajectories. Additionally, the influence exerted by the attraction to critical multipliers on the convergence rate of direct and dual trajectories is characterized.     

The Singular Function Boundary Integral Method for singular Laplacian problems over circular sections

The Singular Function Boundary Integral Method (SFBIM) for solving two-dimensional elliptic problems with boundary singularities is revisited. In this method the solution is approximated by the leading terms of the asymptotic expansion of the local solution, which are also used to weight the governing partial differential equation. The singular coefficients, i.e., the coefficients of the local asymptotic expansion, are thus primary unknowns. By means of the divergence theorem, the discretized equations are reduced to boundary integrals and integration is needed only far from the singularity. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers, the discrete values of which are additional unknowns. In the case of two-dimensional Laplacian problems, the SFBIM converges exponentially with respect to the numbers of singular functions and Lagrange multipliers. In the present work the method is applied to Laplacian test problems over circular sectors, the analytical solution of which is known. The convergence of the method is studied for various values of the order p of the polynomial approximation of the Lagrange multipliers (i.e., constant, linear, quadratic, and cubic), and the exact approximation errors are calculated. These are compared to the theoretical results provided in the literature and their agreement is demonstrated.     

一个求解条件极值问题的极值点的新方法

利用齐次线性方程组理论,建立了一个求解条件极值问题的极值点的新方法.该方法的优点是:能有效地避免在运用Lagrange乘数法求解条件极值时,因引进了参数而给解方程组带来的困扰.也可以说,对于有些问题我们仅从已知条件入手,不必引进参数就可以直接求得极值点.     

The infinite dimensional Lagrange multiplier rule for convex optimization problems

In this paper an infinite dimensional generalized Lagrange multipliers rule for convex optimization problems is presented and necessary and sufficient optimality conditions are given in order to guarantee the strong duality. Furthermore, an application is presented, in particular the existence of Lagrange multipliers associated to the bi-obstacle problem is obtained.     

两个泛函优化定理的证明

在泛函优化理论中,Lagrange乘子定理、对偶定理占有重要地位.建立了带有等式和不等式约束的泛函优化问题,并给出了广义Lagrange乘子定理、广义Lagrange对偶定理的证明.     

Two-level primal-dual decomposition technique for large-scale nonconvex optimization problems with constraints

A two-level decomposition method for nonconvex separable optimization problems with additional local constraints of general inequality type is presented and thoroughly analyzed in the paper. The method is of primal-dual type, based on an augmentation of the Lagrange function. Previous methods of this type were in fact three-level, with adjustment of the Lagrange multipliers at one of the levels. This level is eliminated in the present approach by replacing the multipliers by a formula depending only on primal variables and Kuhn-Tucker multipliers for the local constraints. The primal variables and the Kuhn-Tucker multipliers are together the higher-level variables, which are updated simultaneously. Algorithms for this updating are proposed in the paper, together with their convergence analysis, which gives also indications on how to choose penalty coefficients of the augmented Lagrangian. Finally, numerical examples are presented.     

The Euler-Lagrange method in Pontryagin’s formulation

The classical variational problem with nonholonomic constraints is solvable by the Euler-Lagrange method in Pontryagin’s formulation; however, in this case Lagrange multipliers are merely measurable functions. In this paper, we put forward a modified Euler-Lagrange method, in which the original problem involves a Lagrangian dependent only on the independent components of the velocity vector. Under this approach, the Lagrange multipliers make up an absolutely continuous vector function. Our method is applied to the problem of horizontal geodesics for a nonholonomic distribution on a manifold. These equations are established as having two types of connections: connection on the distribution and connection on the manifold; this was not accounted for by other researchers.