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.
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 [16].
The
Lagrangian method is employed to transform the constrained optimization problem into an unconstrained problem, which can be expressed as
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.
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.