Prospects for expanding current applications of the Lagrangian method
to more complex cloud systems are reviewed.
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.