2,507 research outputs found
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
A duality-based approach for distributed min-max optimization with application to demand side management
In this paper we consider a distributed optimization scenario in which a set
of processors aims at minimizing the maximum of a collection of "separable
convex functions" subject to local constraints. This set-up is motivated by
peak-demand minimization problems in smart grids. Here, the goal is to minimize
the peak value over a finite horizon with: (i) the demand at each time instant
being the sum of contributions from different devices, and (ii) the local
states at different time instants being coupled through local dynamics. The
min-max structure and the double coupling (through the devices and over the
time horizon) makes this problem challenging in a distributed set-up (e.g.,
well-known distributed dual decomposition approaches cannot be applied). We
propose a distributed algorithm based on the combination of duality methods and
properties from min-max optimization. Specifically, we derive a series of
equivalent problems by introducing ad-hoc slack variables and by going back and
forth from primal and dual formulations. On the resulting problem we apply a
dual subgradient method, which turns out to be a distributed algorithm. We
prove the correctness of the proposed algorithm and show its effectiveness via
numerical computations.Comment: arXiv admin note: substantial text overlap with arXiv:1611.0916
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
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