5,323 research outputs found
On the Minimization of Convex Functionals of Probability Distributions Under Band Constraints
The problem of minimizing convex functionals of probability distributions is
solved under the assumption that the density of every distribution is bounded
from above and below. A system of sufficient and necessary first-order
optimality conditions as well as a bound on the optimality gap of feasible
candidate solutions are derived. Based on these results, two numerical
algorithms are proposed that iteratively solve the system of optimality
conditions on a grid of discrete points. Both algorithms use a block coordinate
descent strategy and terminate once the optimality gap falls below the desired
tolerance. While the first algorithm is conceptually simpler and more
efficient, it is not guaranteed to converge for objective functions that are
not strictly convex. This shortcoming is overcome in the second algorithm,
which uses an additional outer proximal iteration, and, which is proven to
converge under mild assumptions. Two examples are given to demonstrate the
theoretical usefulness of the optimality conditions as well as the high
efficiency and accuracy of the proposed numerical algorithms.Comment: 13 pages, 5 figures, 2 tables, published in the IEEE Transactions on
Signal Processing. In previous versions, the example in Section VI.B
contained some mistakes and inaccuracies, which have been fixed in this
versio
Guidance, flight mechanics and trajectory optimization. Volume 4 - The calculus of variations and modern applications
Guidance, flight mechanics, and trajectory optimization - calculus of variations and modern application
The Variational Calculus on Time Scales
The discrete, the quantum, and the continuous calculus of variations, have
been recently unified and extended by using the theory of time scales. Such
unification and extension is, however, not unique, and two approaches are
followed in the literature: one dealing with minimization of delta integrals;
the other dealing with minimization of nabla integrals. Here we review a more
general approach to the calculus of variations on time scales that allows to
obtain both delta and nabla results as particular cases.Comment: 15 pages; Published in: Int. J. Simul. Multidisci. Des. Optim. 4
(2010), 11--2
Geodesic boundary value problems with symmetry
This paper shows how left and right actions of Lie groups on a manifold may
be used to complement one another in a variational reformulation of optimal
control problems equivalently as geodesic boundary value problems with
symmetry. We prove an equivalence theorem to this effect and illustrate it with
several examples. In finite-dimensions, we discuss geodesic flows on the Lie
groups SO(3) and SE(3) under the left and right actions of their respective Lie
algebras. In an infinite-dimensional example, we discuss optimal
large-deformation matching of one closed curve to another embedded in the same
plane. In the curve-matching example, the manifold \Emb(S^1, \mathbb{R}^2)
comprises the space of closed curves embedded in the plane
. The diffeomorphic left action \Diff(\mathbb{R}^2) deforms the
curve by a smooth invertible time-dependent transformation of the coordinate
system in which it is embedded, while leaving the parameterisation of the curve
invariant. The diffeomorphic right action \Diff(S^1) corresponds to a smooth
invertible reparameterisation of the domain coordinates of the curve. As
we show, this right action unlocks an important degree of freedom for
geodesically matching the curve shapes using an equivalent fixed boundary value
problem, without being constrained to match corresponding points along the
template and target curves at the endpoint in time.Comment: First version -- comments welcome
Generalized Hamilton-Jacobi equations for nonholonomic dynamics
Employing a suitable nonlinear Lagrange functional, we derive generalized
Hamilton-Jacobi equations for dynamical systems subject to linear velocity
constraints. As long as a solution of the generalized Hamilton-Jacobi equation
exists, the action is actually minimized (not just extremized)
Caratheodory-Equivalence, Noether Theorems, and Tonelli Full-Regularity in the Calculus of Variations and Optimal Control
We study, in a unified way, the following questions related to the properties
of Pontryagin extremals for optimal control problems with unrestricted
controls: i) How the transformations, which define the equivalence of two
problems, transform the extremals? ii) How to obtain quantities which are
conserved along any extremal? iii) How to assure that the set of extremals
include the minimizers predicted by the existence theory? These questions are
connected to: i) the Caratheodory method which establishes a correspondence
between the minimizing curves of equivalent problems; ii) the interplay between
the concept of invariance and the theory of optimality conditions in optimal
control, which are the concern of the theorems of Noether; iii) regularity
conditions for the minimizers and the work pioneered by Tonelli.Comment: 24 pages, Submitted for publication in a Special Issue of the J. of
Mathematical Science
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