41,923 research outputs found
String-Averaging Projected Subgradient Methods for Constrained Minimization
We consider constrained minimization problems and propose to replace the
projection onto the entire feasible region, required in the Projected
Subgradient Method (PSM), by projections onto the individual sets whose
intersection forms the entire feasible region. Specifically, we propose to
perform such projections onto the individual sets in an algorithmic regime of a
feasibility-seeking iterative projection method. For this purpose we use the
recently developed family of Dynamic String-Averaging Projection (DSAP) methods
wherein iteration-index-dependent variable strings and variable weights are
permitted. This gives rise to an algorithmic scheme that generalizes, from the
algorithmic structural point of view, earlier work of Helou Neto and De Pierro,
of Nedi\'c, of Nurminski, and of Ram et al.Comment: Optimization Methods and Software, accepted for publicatio
An Augmented Lagrangian Approach to the Constrained Optimization Formulation of Imaging Inverse Problems
We propose a new fast algorithm for solving one of the standard approaches to
ill-posed linear inverse problems (IPLIP), where a (possibly non-smooth)
regularizer is minimized under the constraint that the solution explains the
observations sufficiently well. Although the regularizer and constraint are
usually convex, several particular features of these problems (huge
dimensionality, non-smoothness) preclude the use of off-the-shelf optimization
tools and have stimulated a considerable amount of research. In this paper, we
propose a new efficient algorithm to handle one class of constrained problems
(often known as basis pursuit denoising) tailored to image recovery
applications. The proposed algorithm, which belongs to the family of augmented
Lagrangian methods, can be used to deal with a variety of imaging IPLIP,
including deconvolution and reconstruction from compressive observations (such
as MRI), using either total-variation or wavelet-based (or, more generally,
frame-based) regularization. The proposed algorithm is an instance of the
so-called "alternating direction method of multipliers", for which convergence
sufficient conditions are known; we show that these conditions are satisfied by
the proposed algorithm. Experiments on a set of image restoration and
reconstruction benchmark problems show that the proposed algorithm is a strong
contender for the state-of-the-art.Comment: 13 pages, 8 figure, 8 tables. Submitted to the IEEE Transactions on
Image Processin
Superiorization and Perturbation Resilience of Algorithms: A Continuously Updated Bibliography
This document presents a, (mostly) chronologically ordered, bibliography of
scientific publications on the superiorization methodology and perturbation
resilience of algorithms which is compiled and continuously updated by us at:
http://math.haifa.ac.il/yair/bib-superiorization-censor.html. Since the
beginings of this topic we try to trace the work that has been published about
it since its inception. To the best of our knowledge this bibliography
represents all available publications on this topic to date, and while the URL
is continuously updated we will revise this document and bring it up to date on
arXiv approximately once a year. Abstracts of the cited works, and some links
and downloadable files of preprints or reprints are available on the above
mentioned Internet page. If you know of a related scientific work in any form
that should be included here kindly write to me on: [email protected] with
full bibliographic details, a DOI if available, and a PDF copy of the work if
possible. The Internet page was initiated on March 7, 2015, and has been last
updated on March 12, 2020.Comment: Original report: June 13, 2015 contained 41 items. First revision:
March 9, 2017 contained 64 items. Second revision: March 8, 2018 contained 76
items. Third revision: March 11, 2019 contains 90 items. Fourth revision:
March 16, 2020 contains 112 item
Sparse Solution of Underdetermined Linear Equations via Adaptively Iterative Thresholding
Finding the sparset solution of an underdetermined system of linear equations
has attracted considerable attention in recent years. Among a large
number of algorithms, iterative thresholding algorithms are recognized as one
of the most efficient and important classes of algorithms. This is mainly due
to their low computational complexities, especially for large scale
applications. The aim of this paper is to provide guarantees on the global
convergence of a wide class of iterative thresholding algorithms. Since the
thresholds of the considered algorithms are set adaptively at each iteration,
we call them adaptively iterative thresholding (AIT) algorithms. As the main
result, we show that as long as satisfies a certain coherence property, AIT
algorithms can find the correct support set within finite iterations, and then
converge to the original sparse solution exponentially fast once the correct
support set has been identified. Meanwhile, we also demonstrate that AIT
algorithms are robust to the algorithmic parameters. In addition, it should be
pointed out that most of the existing iterative thresholding algorithms such as
hard, soft, half and smoothly clipped absolute deviation (SCAD) algorithms are
included in the class of AIT algorithms studied in this paper.Comment: 33 pages, 1 figur
Bounded perturbation resilience of projected scaled gradient methods
We investigate projected scaled gradient (PSG) methods for convex
minimization problems. These methods perform a descent step along a diagonally
scaled gradient direction followed by a feasibility regaining step via
orthogonal projection onto the constraint set. This constitutes a generalized
algorithmic structure that encompasses as special cases the gradient projection
method, the projected Newton method, the projected Landweber-type methods and
the generalized Expectation-Maximization (EM)-type methods. We prove the
convergence of the PSG methods in the presence of bounded perturbations. This
resilience to bounded perturbations is relevant to the ability to apply the
recently developed superiorization methodology to PSG methods, in particular to
the EM algorithm.Comment: Computational Optimization and Applications, accepted for publicatio
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