21,909 research outputs found
A Unified Approach to Variational Derivatives of Modified Gravitational Actions
Our main aim in this paper is to promote the coframe variational method as a
unified approach to derive field equations for any given gravitational action
containing the algebraic functions of the scalars constructed from the Riemann
curvature tensor and its contractions. We are able to derive a master equation
which expresses the variational derivatives of the generalized gravitational
actions in terms of the variational derivatives of its constituent curvature
scalars. Using the Lagrange multiplier method relative to an orthonormal
coframe, we investigate the variational procedures for modified gravitational
Lagrangian densities in spacetime dimensions . We study
well-known gravitational actions such as those involving the Gauss-Bonnet and
Ricci-squared, Kretchmann scalar, Weyl-squared terms and their algebraic
generalizations similar to generic theories and the algebraic
generalization of sixth order gravitational Lagrangians. We put forth a new
model involving the gravitational Chern-Simons term and also give three
dimensional New massive gravity equations in a new form in terms of the Cotton
2-form
A Primal-Dual Augmented Lagrangian
Nonlinearly constrained optimization problems can be solved by minimizing a sequence of simpler unconstrained or linearly constrained subproblems. In this paper, we discuss the formulation of subproblems in which the objective is a primal-dual generalization of the Hestenes-Powell augmented Lagrangian function. This generalization has the crucial feature that it is minimized with respect to both the primal and the dual variables simultaneously. A benefit of this approach is that the quality of the dual variables is monitored explicitly during the solution of the subproblem. Moreover, each subproblem may be regularized by imposing explicit bounds on the dual variables. Two primal-dual variants of conventional primal methods are proposed: a primal-dual bound constrained Lagrangian (pdBCL) method and a primal-dual 1 linearly constrained Lagrangian (pd1-LCL) method
A Distributed Asynchronous Method of Multipliers for Constrained Nonconvex Optimization
This paper presents a fully asynchronous and distributed approach for
tackling optimization problems in which both the objective function and the
constraints may be nonconvex. In the considered network setting each node is
active upon triggering of a local timer and has access only to a portion of the
objective function and to a subset of the constraints. In the proposed
technique, based on the method of multipliers, each node performs, when it
wakes up, either a descent step on a local augmented Lagrangian or an ascent
step on the local multiplier vector. Nodes realize when to switch from the
descent step to the ascent one through an asynchronous distributed logic-AND,
which detects when all the nodes have reached a predefined tolerance in the
minimization of the augmented Lagrangian. It is shown that the resulting
distributed algorithm is equivalent to a block coordinate descent for the
minimization of the global augmented Lagrangian. This allows one to extend the
properties of the centralized method of multipliers to the considered
distributed framework. Two application examples are presented to validate the
proposed approach: a distributed source localization problem and the parameter
estimation of a neural network.Comment: arXiv admin note: substantial text overlap with arXiv:1803.0648
The inverse problem of the calculus of variations and the stabilization of controlled Lagrangian systems
We apply methods of the so-called `inverse problem of the calculus of
variations' to the stabilization of an equilibrium of a class of
two-dimensional controlled mechanical systems. The class is general enough to
include, among others, the inverted pendulum on a cart and the inertia wheel
pendulum. By making use of a condition that follows from Douglas'
classification, we derive feedback controls for which the control system is
variational. We then use the energy of a suitable controlled Lagrangian to
provide a stability criterion for the equilibrium
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