Abstract The method of characteristics is extended to set-valued Hamilton-Jacobi equations. This problem arises from a calculus variations’ with multicriteria Lagrangian function: through an embedding into framework, equation derived, where the Hamiltonian function Fenchel conjugate function. In this paper described and some results are given for conjugate.

We apply variational methods to Converse KAM theory. These are useful for symplectomorphisms in the annulus that satisfy weaker hypotheses than those usually required. For instance, we do not need the existence of a global Lagrangian generating function. We obtain the variational principles from the primitive function of our symplectomorphism. They are introduced not only for the orbits of a sy...

Many integrable hierarchies of differential equations allow a variational description, called Lagrangian multiform or pluri-Lagrangian structure. The fundamental object in this theory is not Lagrange function but $d$-form that integrated over arbitrary $d$-dimensional submanifolds. All such action integrals must be stationary for field to solution the problem. In paper we present procedure obta...

This paper is about distributed derivative-based algorithms for solving optimization problems with a separable (potentially nonconvex) objective function and coupled affine constraints. A parallelizable method is proposed that combines ideas from the fields of sequential quadratic programming and augmented Lagrangian algorithms. The method negotiates shared dual variables that may be interprete...

A large class of optimization problems can be modeled as minimization of an objective function subject to constraints given in a form of set inclusions. We discuss in this paper augmented Lagrangian duality for such optimization problems. We formulate the augmented Lagrangian dual problems and study conditions ensuring existence of the corresponding augmented Lagrange multipliers. We also discu...

II Symmetries in Classical Particle Mechanics 4 1 Hamiltonian and Lagrangian Formulations of Classical Particle Mechanics 5 1.1 Basic De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.3 Generalized...

We study the Born-Infeld equation from a Lagrangian point of view emphasizing the duality symmetry present in such systems. We obtain the Hamiltonian formulation directly from the Lagrangian. We also show that this system admits a dispersionless nonstandard Lax representation which directly leads to all the conserved charges (including the ones not previously obtained). We also present the gene...

Newton's method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized Jacobian instead of the derivative. This extension includes the B-derivative version of Newton's method as a special case. Convergence theorems are proved under the condition of semismoothness. It is shown that the gradient function of the augmented Lagrangian for C2-no...

The exact unclosed equation for the phase-space density function ~or corresponding Lagrangian pdf! in turbulent flows is obtained using conditional techniques. The equation has direct implications for stochastic Lagrangian models based on the assumption of similarity with a Markov process. The problem of random particle sources is examined and the appropriate correcting term is suggested. © 199...