1,292 research outputs found
First order algorithms in variational image processing
Variational methods in imaging are nowadays developing towards a quite
universal and flexible tool, allowing for highly successful approaches on tasks
like denoising, deblurring, inpainting, segmentation, super-resolution,
disparity, and optical flow estimation. The overall structure of such
approaches is of the form ; where the functional is a data fidelity term also
depending on some input data and measuring the deviation of from such
and is a regularization functional. Moreover is a (often linear)
forward operator modeling the dependence of data on an underlying image, and
is a positive regularization parameter. While is often
smooth and (strictly) convex, the current practice almost exclusively uses
nonsmooth regularization functionals. The majority of successful techniques is
using nonsmooth and convex functionals like the total variation and
generalizations thereof or -norms of coefficients arising from scalar
products with some frame system. The efficient solution of such variational
problems in imaging demands for appropriate algorithms. Taking into account the
specific structure as a sum of two very different terms to be minimized,
splitting algorithms are a quite canonical choice. Consequently this field has
revived the interest in techniques like operator splittings or augmented
Lagrangians. Here we shall provide an overview of methods currently developed
and recent results as well as some computational studies providing a comparison
of different methods and also illustrating their success in applications.Comment: 60 pages, 33 figure
Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems
Optimization methods are at the core of many problems in signal/image
processing, computer vision, and machine learning. For a long time, it has been
recognized that looking at the dual of an optimization problem may drastically
simplify its solution. Deriving efficient strategies which jointly brings into
play the primal and the dual problems is however a more recent idea which has
generated many important new contributions in the last years. These novel
developments are grounded on recent advances in convex analysis, discrete
optimization, parallel processing, and non-smooth optimization with emphasis on
sparsity issues. In this paper, we aim at presenting the principles of
primal-dual approaches, while giving an overview of numerical methods which
have been proposed in different contexts. We show the benefits which can be
drawn from primal-dual algorithms both for solving large-scale convex
optimization problems and discrete ones, and we provide various application
examples to illustrate their usefulness
Forward-backward truncated Newton methods for convex composite optimization
This paper proposes two proximal Newton-CG methods for convex nonsmooth
optimization problems in composite form. The algorithms are based on a a
reformulation of the original nonsmooth problem as the unconstrained
minimization of a continuously differentiable function, namely the
forward-backward envelope (FBE). The first algorithm is based on a standard
line search strategy, whereas the second one combines the global efficiency
estimates of the corresponding first-order methods, while achieving fast
asymptotic convergence rates. Furthermore, they are computationally attractive
since each Newton iteration requires the approximate solution of a linear
system of usually small dimension
A Smooth Primal-Dual Optimization Framework for Nonsmooth Composite Convex Minimization
We propose a new first-order primal-dual optimization framework for a convex
optimization template with broad applications. Our optimization algorithms
feature optimal convergence guarantees under a variety of common structure
assumptions on the problem template. Our analysis relies on a novel combination
of three classic ideas applied to the primal-dual gap function: smoothing,
acceleration, and homotopy. The algorithms due to the new approach achieve the
best known convergence rate results, in particular when the template consists
of only non-smooth functions. We also outline a restart strategy for the
acceleration to significantly enhance the practical performance. We demonstrate
relations with the augmented Lagrangian method and show how to exploit the
strongly convex objectives with rigorous convergence rate guarantees. We
provide numerical evidence with two examples and illustrate that the new
methods can outperform the state-of-the-art, including Chambolle-Pock, and the
alternating direction method-of-multipliers algorithms.Comment: 35 pages, accepted for publication on SIAM J. Optimization. Tech.
Report, Oct. 2015 (last update Sept. 2016
Lagrange optimality system for a class of nonsmooth convex optimization
In this paper, we revisit the augmented Lagrangian method for a class of
nonsmooth convex optimization. We present the Lagrange optimality system of the
augmented Lagrangian associated with the problems, and establish its
connections with the standard optimality condition and the saddle point
condition of the augmented Lagrangian, which provides a powerful tool for
developing numerical algorithms. We apply a linear Newton method to the
Lagrange optimality system to obtain a novel algorithm applicable to a variety
of nonsmooth convex optimization problems arising in practical applications.
Under suitable conditions, we prove the nonsingularity of the Newton system and
the local convergence of the algorithm.Comment: 19 page
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