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18
On the Effectiveness of Projection Methods for Convex Feasibility Problems with Linear Inequality Constraints
"... The effectiveness of projection methods for solving systems of linear inequalities is investigated. It is shown that they often have a computational advantage over alternatives that have been proposed for solving the same problem and that this makes them successful in many realworld applications. ..."
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The effectiveness of projection methods for solving systems of linear inequalities is investigated. It is shown that they often have a computational advantage over alternatives that have been proposed for solving the same problem and that this makes them successful in many realworld applications. This is supported by experimental evidence provided in this paper on problems of various sizes (up to tens of thousands of unknowns satisfying up to hundreds of thousands of constraints) and by a discussion of the demonstrated efficacy of projection methods in numerous scientific publications and commercial patents (dealing with problems that can have over a billion unknowns and a similar number of constraints).
Convergence and perturbation resilience of dynamic stringaveraging projection methods
 Computational Optimization and Applications
, 2013
"... We consider the convex feasibility problem (CFP) in Hilbert space and concentrate on the study of stringaveraging 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 ..."
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Cited by 8 (5 self)
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We consider the convex feasibility problem (CFP) in Hilbert space and concentrate on the study of stringaveraging 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 iterationindexdependent variable strings and weights and term such methods dynamic stringaveraging 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. 1
A Heuristic SuperiorizationLike Approach to Bioluminescence Tomography
"... Abstract — Bioluminescence tomography (BLT) is a powerful molecular imaging technology designed for the localization and quantification of bioluminescent sources in vivo. With the forward process modeled by the diffusion approximation equation, BLT is the inverse problem to reconstruct the distribu ..."
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Abstract — Bioluminescence tomography (BLT) is a powerful molecular imaging technology designed for the localization and quantification of bioluminescent sources in vivo. With the forward process modeled by the diffusion approximation equation, BLT is the inverse problem to reconstruct the distribution of internal bioluminescent sources subject to Cauchy data. Due to the nonuniqueness of BLT in general, adequate prior information such as nonnegativity and source support constraints must be utilized to obtain a physically favorable BLT solution. Iterative algorithms such as the wellknown Expectation Maximization (EM) algorithm and the Landweber algorithm which are suitable for incorporating these knowledgebased constraints are widely used in practice. In the current work, we investigate the application of a superiorizationlike approach to BLT. A superiorizationlike version of prototypical iterative algorithms for BLT in a general framework, denoted by SBLT, is presented. For the EM algorithm as the underlying iterative algorithms for BLT and SBLT, superiorized by the total variation (TV) merit function, preliminary simulation results for a heterogeneous phantom are reported to demonstrate the viability of the approach and evaluate the performance of the proposed algorithm. It is found that total variation superiorization of BLT can significantly improve the visualization effect of the reconstruction with the sources set as a particular case of radial basis functions.
StringAveraging Projected Subgradient Methods for Constrained Minimization
, 2013
"... 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 ..."
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Cited by 4 (1 self)
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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 feasibilityseeking iterative projection method. For this purpose we use the recently developed family of Dynamic StringAveraging Projection (DSAP) methods wherein iterationindexdependent 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 Nedíc, of Nurminski, and of Ram et al. 1
SIGNAL AND IMAGE PROCESSING ALGORITHMS USING INTERVAL CONVEX PROGRAMMING AND SPARSITY
, 2012
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Strict Fejér Monotonicity by Superiorization of FeasibilitySeeking Projection Methods
, 2014
"... We consider the superiorization methodology, which can be thought of as lying between feasibilityseeking 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 t ..."
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We consider the superiorization methodology, which can be thought of as lying between feasibilityseeking 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 the objective function value) to one returned by a feasibilityseeking only algorithm. Our main result reveals new information about the mathematical behavior of the superiorization methodology. We deal with a constrained minimization problem with a feasible region, which is the intersection of finitely many closed convex constraint sets, and use the dynamic stringaveraging projection method, with variable strings and variable weights, as a feasibilityseeking algorithm. We show that any sequence, generated by the superiorized version of a dynamic stringaveraging projection algorithm, not only converges to a feasible point but, additionally, either its limit point solves the constrained minimization problem or the sequence is strictly Fejér monotone with respect to a subset of the solution set of the original problem.
Weak and Strong Superiorization: Between FeasibilitySeeking and Minimization
"... We review the superiorization methodology, which can be thought of, in some cases, as lying between feasibilityseeking 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 (wit ..."
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We review the superiorization methodology, which can be thought of, in some cases, as lying between feasibilityseeking 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 feasibilityseeking 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. 1
1StringAveraging ExpectationMaximization for Maximum Likelihood Estimation in Emission Tomography
"... We study the maximum likelihood model in emission tomography and propose a new family of algorithms for its solution, called StringAveraging ExpectationMaximization (SAEM). In the StringAveraging algorithmic regime, the index set of all underlying equations is split into subsets, called “strings, ..."
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We study the maximum likelihood model in emission tomography and propose a new family of algorithms for its solution, called StringAveraging ExpectationMaximization (SAEM). In the StringAveraging algorithmic regime, the index set of all underlying equations is split into subsets, called “strings, ” and the algorithm separately proceeds along each string, possibly in parallel. Then, the endpoints of all strings are averaged to form the next iterate. SAEM algorithms with 2several strings presents better practical merits than the classical RowAction MaximumLikelihood Algorithm (RAMLA). We present numerical experiments showing the effectiveness of the algorithmic scheme in realistic situations. Performance is evaluated from the computational cost and
Contemporary Mathematics Projection Methods: Swiss Army Knives for Solving Feasibility and Best Approximation Problems with Halfspaces
"... Abstract. We model a problem motivated by road design as a feasibility problem. Projections onto the constraint sets are obtained, and projection methods for solving the feasibility problem are studied. We present results of numerical experiments which demonstrate the efficacy of projection methods ..."
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Abstract. We model a problem motivated by road design as a feasibility problem. Projections onto the constraint sets are obtained, and projection methods for solving the feasibility problem are studied. We present results of numerical experiments which demonstrate the efficacy of projection methods even for challenging nonconvex problems. 1. Introduction and
Proton co...
"... variation superiorization schemes in proton computed tomography image reconstruction ..."
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variation superiorization schemes in proton computed tomography image reconstruction