581 research outputs found
Convergence and Perturbation Resilience of Dynamic String-Averaging Projection Methods
We consider the convex feasibility problem (CFP) in Hilbert space and
concentrate on the study of string-averaging projection (SAP) methods for the
CFP, analyzing their convergence and their perturbation resilience. In the
past, SAP methods were formulated with a single predetermined set of strings
and a single predetermined set of weights. Here we extend the scope of the
family of SAP methods to allow iteration-index-dependent variable strings and
weights and term such methods dynamic string-averaging projection (DSAP)
methods. The bounded perturbation resilience of DSAP methods is relevant and
important for their possible use in the framework of the recently developed
superiorization heuristic methodology for constrained minimization problems.Comment: Computational Optimization and Applications, accepted for publicatio
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
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
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
Weak and Strong Superiorization: Between Feasibility-Seeking and Minimization
We review the superiorization methodology, which can be thought of, in some
cases, as lying between feasibility-seeking and constrained minimization. It is
not quite trying to solve the full fledged constrained minimization problem;
rather, the task is to find a feasible point which is superior (with respect to
an objective function value) to one returned by a feasibility-seeking only
algorithm. We distinguish between two research directions in the
superiorization methodology that nourish from the same general principle: Weak
superiorization and strong superiorization and clarify their nature.Comment: Revised version. Presented at the Tenth Workshop on Mathematical
Modelling of Environmental and Life Sciences Problems, October 16-19, 2014,
Constantza, Romania. http://www.ima.ro/workshop/tenth_workshop
New Douglas-Rachford algorithmic structures and their convergence analyses
In this paper we study new algorithmic structures with Douglas- Rachford (DR)
operators to solve convex feasibility problems. We propose to embed the basic
two-set-DR algorithmic operator into the String-Averaging Projections (SAP) and
into the Block-Iterative Pro- jection (BIP) algorithmic structures, thereby
creating new DR algo- rithmic schemes that include the recently proposed cyclic
Douglas- Rachford algorithm and the averaged DR algorithm as special cases. We
further propose and investigate a new multiple-set-DR algorithmic operator.
Convergence of all these algorithmic schemes is studied by using properties of
strongly quasi-nonexpansive operators and firmly nonexpansive operators.Comment: SIAM Journal on Optimization, accepted for publicatio
Accelerating two projection methods via perturbations with application to Intensity-Modulated Radiation Therapy
Constrained convex optimization problems arise naturally in many real-world
applications. One strategy to solve them in an approximate way is to translate
them into a sequence of convex feasibility problems via the recently developed
level set scheme and then solve each feasibility problem using projection
methods. However, if the problem is ill-conditioned, projection methods often
show zigzagging behavior and therefore converge slowly.
To address this issue, we exploit the bounded perturbation resilience of the
projection methods and introduce two new perturbations which avoid zigzagging
behavior. The first perturbation is in the spirit of -step methods and uses
gradient information from previous iterates. The second uses the approach of
surrogate constraint methods combined with relaxed, averaged projections.
We apply two different projection methods in the unperturbed version, as well
as the two perturbed versions, to linear feasibility problems along with
nonlinear optimization problems arising from intensity-modulated radiation
therapy (IMRT) treatment planning. We demonstrate that for all the considered
problems the perturbations can significantly accelerate the convergence of the
projection methods and hence the overall procedure of the level set scheme. For
the IMRT optimization problems the perturbed projection methods found an
approximate solution up to 4 times faster than the unperturbed methods while at
the same time achieving objective function values which were 0.5 to 5.1% lower.Comment: Accepted for publication in Applied Mathematics & Optimizatio
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