14,488 research outputs found
On Quasi-Newton Forward--Backward Splitting: Proximal Calculus and Convergence
We introduce a framework for quasi-Newton forward--backward splitting
algorithms (proximal quasi-Newton methods) with a metric induced by diagonal
rank- symmetric positive definite matrices. This special type of
metric allows for a highly efficient evaluation of the proximal mapping. The
key to this efficiency is a general proximal calculus in the new metric. By
using duality, formulas are derived that relate the proximal mapping in a
rank- modified metric to the original metric. We also describe efficient
implementations of the proximity calculation for a large class of functions;
the implementations exploit the piece-wise linear nature of the dual problem.
Then, we apply these results to acceleration of composite convex minimization
problems, which leads to elegant quasi-Newton methods for which we prove
convergence. The algorithm is tested on several numerical examples and compared
to a comprehensive list of alternatives in the literature. Our quasi-Newton
splitting algorithm with the prescribed metric compares favorably against
state-of-the-art. The algorithm has extensive applications including signal
processing, sparse recovery, machine learning and classification to name a few.Comment: arXiv admin note: text overlap with arXiv:1206.115
A quasi-Newton proximal splitting method
A new result in convex analysis on the calculation of proximity operators in
certain scaled norms is derived. We describe efficient implementations of the
proximity calculation for a useful class of functions; the implementations
exploit the piece-wise linear nature of the dual problem. The second part of
the paper applies the previous result to acceleration of convex minimization
problems, and leads to an elegant quasi-Newton method. The optimization method
compares favorably against state-of-the-art alternatives. The algorithm has
extensive applications including signal processing, sparse recovery and machine
learning and classification
Kinematically optimal hyper-redundant manipulator configurations
“Hyper-redundant” robots have a very large or infinite degree of kinematic redundancy. This paper develops new methods for determining “optimal” hyper-redundant manipulator configurations based on a continuum formulation of kinematics. This formulation uses a backbone curve model to capture the robot's essential macroscopic geometric features. The calculus of variations is used to develop differential equations, whose solution is the optimal backbone curve shape. We show that this approach is computationally efficient on a single processor, and generates solutions in O(1) time for an N degree-of-freedom manipulator when implemented in parallel on O(N) processors. For this reason, it is better suited to hyper-redundant robots than other redundancy resolution methods. Furthermore, this approach is useful for many hyper-redundant mechanical morphologies which are not handled by known methods
Finite Element Simulation of Dense Wire Packings
A finite element program is presented to simulate the process of packing and
coiling elastic wires in two- and three-dimensional confining cavities. The
wire is represented by third order beam elements and embedded into a
corotational formulation to capture the geometric nonlinearity resulting from
large rotations and deformations. The hyperbolic equations of motion are
integrated in time using two different integration methods from the Newmark
family: an implicit iterative Newton-Raphson line search solver, and an
explicit predictor-corrector scheme, both with adaptive time stepping. These
two approaches reveal fundamentally different suitability for the problem of
strongly self-interacting bodies found in densely packed cavities. Generalizing
the spherical confinement symmetry investigated in recent studies, the packing
of a wire in hard ellipsoidal cavities is simulated in the frictionless elastic
limit. Evidence is given that packings in oblate spheroids and scalene
ellipsoids are energetically preferred to spheres.Comment: 17 pages, 7 figures, 1 tabl
Final-State Constrained Optimal Control via a Projection Operator Approach
In this paper we develop a numerical method to solve nonlinear optimal
control problems with final-state constraints. Specifically, we extend the
PRojection Operator based Netwon's method for Trajectory Optimization (PRONTO),
which was proposed by Hauser for unconstrained optimal control problems. While
in the standard method final-state constraints can be only approximately
handled by means of a terminal penalty, in this work we propose a methodology
to meet the constraints exactly. Moreover, our method guarantees recursive
feasibility of the final-state constraint. This is an appealing property
especially in realtime applications in which one would like to be able to stop
the computation even if the desired tolerance has not been reached, but still
satisfy the constraints. Following the same conceptual idea of PRONTO, the
proposed strategy is based on two main steps which (differently from the
standard scheme) preserve the feasibility of the final-state constraints: (i)
solve a quadratic approximation of the nonlinear problem to find a descent
direction, and (ii) get a (feasible) trajectory by means of a feedback law
(which turns out to be a nonlinear projection operator). To find the (feasible)
descent direction we take advantage of final-state constrained Linear Quadratic
optimal control methods, while the second step is performed by suitably
designing a constrained version of the trajectory tracking projection operator.
The effectiveness of the proposed strategy is tested on the optimal state
transfer of an inverted pendulum
Ten Misconceptions from the History of Analysis and Their Debunking
The widespread idea that infinitesimals were "eliminated" by the "great
triumvirate" of Cantor, Dedekind, and Weierstrass is refuted by an
uninterrupted chain of work on infinitesimal-enriched number systems. The
elimination claim is an oversimplification created by triumvirate followers,
who tend to view the history of analysis as a pre-ordained march toward the
radiant future of Weierstrassian epsilontics. In the present text, we document
distortions of the history of analysis stemming from the triumvirate ideology
of ontological minimalism, which identified the continuum with a single number
system. Such anachronistic distortions characterize the received interpretation
of Stevin, Leibniz, d'Alembert, Cauchy, and others.Comment: 46 pages, 4 figures; Foundations of Science (2012). arXiv admin note:
text overlap with arXiv:1108.2885 and arXiv:1110.545
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