# Lagrangian method

## Lagrangian method

[lə′grän·jē·ən ‚meth·əd]
(fluid mechanics)
A method of studying fluid motion and the mechanics of deformable bodies in which one considers volume elements which are carried along with the fluid or body, and across whose boundaries material does not flow; in contrast to Euler method. Also known as Lagrangian description.
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1) Finding property of optimal solution by Lagrangian method and KKT condition: Lagrangian function is usually constructed by adding all the constraints to the objective function to transform the constrained problem to unconstrained one.
Unlike k-t RPCA (a method that uses the low-rank plus sparse decomposition prior to reconstruction of dynamic MRI from part of the k-space measurements), the authors propose inexact augmented Lagrangian method (IALM) to solve the optimization of RPCA and to accelerate the dynamic MRI reconstruction from highly undersampled k-space data, which has a generalized formulation capability of separating dynamic MR data into low-rank and sparse component.
In this article, we would like to derive the well-known BPS equations of monopole and dyon in the SU(2) Yang-Mills-Higgs model and their Born-Infeld type extensions, which we shall call them Nakamula-Shiraishi models, using a procedure called BPS Lagrangian method developed in .
The Lagrangian method is employed to transform the constrained optimization problem into an unconstrained problem, which can be expressed as
Governing Equations in Lagrangian Method. The three fundamental conservation equations are the conservation of mass, the conservation of momentum, and the conservation of energy.
In the particle method, which is a complete Lagrangian method free from a computational grid, interface-neighboring particles are given interfacial boundary condition.
The method presented in this paper is a variant of the augmented Lagrangian method (denoted by AL).
For contact/impact dynamics of discretized elastic body, two typical methods are presented to model contact: penalty method and Lagrangian method. A brief overview of these two formulations is presented as follows.
Several rapid numerical algorithms can solve the numerical difficulties, which are, for example, alternating direction method of multipliers (ADMM) , augmented Lagrangian method (ALM) , splitting Bregman algorithm (SBA) , splitting Barzilai-Borwein (SBB) , and Bregman operator splitting (BOS) .
Brazilian mathematicians explain the augmented Lagrangian method for solving constrained optimization problems for engineers, chemists, physicists, economists, and others who use constrained optimization for solving real-life problems.

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