6,535 research outputs found
An inexact proximal majorization-minimization Algorithm for remote sensing image stripe noise removal
The stripe noise existing in remote sensing images badly degrades the visual
quality and restricts the precision of data analysis. Therefore, many
destriping models have been proposed in recent years. In contrast to these
existing models, in this paper, we propose a nonconvex model with a DC function
(i.e., the difference of convex functions) structure to remove the strip noise.
To solve this model, we make use of the DC structure and apply an inexact
proximal majorization-minimization algorithm with each inner subproblem solved
by the alternating direction method of multipliers. It deserves mentioning that
we design an implementable stopping criterion for the inner subproblem, while
the convergence can still be guaranteed. Numerical experiments demonstrate the
superiority of the proposed model and algorithm.Comment: 19 pages, 3 figure
Alternating Direction Methods for Latent Variable Gaussian Graphical Model Selection
Chandrasekaran, Parrilo and Willsky (2010) proposed a convex optimization
problem to characterize graphical model selection in the presence of unobserved
variables. This convex optimization problem aims to estimate an inverse
covariance matrix that can be decomposed into a sparse matrix minus a low-rank
matrix from sample data. Solving this convex optimization problem is very
challenging, especially for large problems. In this paper, we propose two
alternating direction methods for solving this problem. The first method is to
apply the classical alternating direction method of multipliers to solve the
problem as a consensus problem. The second method is a proximal gradient based
alternating direction method of multipliers. Our methods exploit and take
advantage of the special structure of the problem and thus can solve large
problems very efficiently. Global convergence result is established for the
proposed methods. Numerical results on both synthetic data and gene expression
data show that our methods usually solve problems with one million variables in
one to two minutes, and are usually five to thirty five times faster than a
state-of-the-art Newton-CG proximal point algorithm
On the Convergence of Alternating Direction Lagrangian Methods for Nonconvex Structured Optimization Problems
Nonconvex and structured optimization problems arise in many engineering
applications that demand scalable and distributed solution methods. The study
of the convergence properties of these methods is in general difficult due to
the nonconvexity of the problem. In this paper, two distributed solution
methods that combine the fast convergence properties of augmented
Lagrangian-based methods with the separability properties of alternating
optimization are investigated. The first method is adapted from the classic
quadratic penalty function method and is called the Alternating Direction
Penalty Method (ADPM). Unlike the original quadratic penalty function method,
in which single-step optimizations are adopted, ADPM uses an alternating
optimization, which in turn makes it scalable. The second method is the
well-known Alternating Direction Method of Multipliers (ADMM). It is shown that
ADPM for nonconvex problems asymptotically converges to a primal feasible point
under mild conditions and an additional condition ensuring that it
asymptotically reaches the standard first order necessary conditions for local
optimality are introduced. In the case of the ADMM, novel sufficient conditions
under which the algorithm asymptotically reaches the standard first order
necessary conditions are established. Based on this, complete convergence of
ADMM for a class of low dimensional problems are characterized. Finally, the
results are illustrated by applying ADPM and ADMM to a nonconvex localization
problem in wireless sensor networks.Comment: 13 pages, 6 figure
Distributed Basis Pursuit
We propose a distributed algorithm for solving the optimization problem Basis
Pursuit (BP). BP finds the least L1-norm solution of the underdetermined linear
system Ax = b and is used, for example, in compressed sensing for
reconstruction. Our algorithm solves BP on a distributed platform such as a
sensor network, and is designed to minimize the communication between nodes.
The algorithm only requires the network to be connected, has no notion of a
central processing node, and no node has access to the entire matrix A at any
time. We consider two scenarios in which either the columns or the rows of A
are distributed among the compute nodes. Our algorithm, named D-ADMM, is a
decentralized implementation of the alternating direction method of
multipliers. We show through numerical simulation that our algorithm requires
considerably less communications between the nodes than the state-of-the-art
algorithms.Comment: Preprint of the journal version of the paper; IEEE Transactions on
Signal Processing, Vol. 60, Issue 4, April, 201
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