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32
Projection methods: Swiss army knives for Solving feasibility and best approximation problems with halfspaces
, 2013
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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
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
Projections Onto Convex Sets (POCS) Based Optimization by Lifting
, 1306
"... Two new optimization techniques based on projections onto convex space (POCS) framework for solving convex and some nonconvex optimization problems are presented. The dimension of the minimization problem is lifted by one and sets corresponding to the cost function are defined. If the cost function ..."
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Cited by 4 (2 self)
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Two new optimization techniques based on projections onto convex space (POCS) framework for solving convex and some nonconvex optimization problems are presented. The dimension of the minimization problem is lifted by one and sets corresponding to the cost function are defined. If the cost function is a convex function in RN the corresponding set is a convex set in RN+1. The iterative optimization approach starts with an arbitrary initial estimate in RN+1 and an orthogonal projection is performed onto one of the sets in a sequential manner at each step of the optimization problem. The method provides globally optimal solutions in totalvariation, filtered variation, l1, and entropic cost functions. It is also experimentally observed that cost functions based on lp, p < 1 can be handled by using the supporting hyperplane concept. 1
Problem
, 2011
"... This version contains some further corrections discovered at the galley proofreading stage. We propose a prototypical Split Inverse Problem (SIP) and a new variational problem, called the Split Variational Inequality Problem (SVIP), which is a SIP. It entails
nding a solution of one inverse probl ..."
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This version contains some further corrections discovered at the galley proofreading stage. We propose a prototypical Split Inverse Problem (SIP) and a new variational problem, called the Split Variational Inequality Problem (SVIP), which is a SIP. It entails
nding a solution of one inverse problem (e.g., a Variational Inequality Problem (VIP)), the image of which under a given bounded linear transformation is a solution of another inverse problem such as a VIP. We construct iterative algorithms that solve such problems, under reasonable conditions, in Hilbert space and then discuss special cases, some of which are new even in Euclidean space.
Convex Feasibility Modeling and Projection Methods for Sparse Signal Recovery
, 2011
"... A computationallyefficient method for recovering sparse signals from a series of noisy observations, known as the problem of compressed sensing (CS), is presented. The theory of CS usually leads to a constrained convex minimization problem. In this work, an alternative outlook is proposed. Instead ..."
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A computationallyefficient method for recovering sparse signals from a series of noisy observations, known as the problem of compressed sensing (CS), is presented. The theory of CS usually leads to a constrained convex minimization problem. In this work, an alternative outlook is proposed. Instead of solving the CS problem as an optimization problem, it is suggested to transform the optimization problem into a convex feasibility problem (CFP), and solve it using feasibilityseeking Sequential and Simultaneous Subgradient Projection methods, which are iterative, fast, robust and convergent schemes for solving CFPs. As opposed to some of the commonlyused CS algorithms, such as Bayesian CS and Gradient Projections for sparse reconstruction, which become inefficient as the problem dimension and sparseness degree increase, the proposed methods exhibit robustness with respect to these parameters. Moreover, it is shown that the CFPbased projection methods are superior to some of the stateoftheart methods in recovering the signal’s support. Numerical experiments show that the CFPbased projection methods are viable for solving largescale CS problems with compressible signals. 1
Data Sets of Very Large Linear Feasibility Problems Solved by Projection Methods
, 2011
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unknown title
, 2009
"... From analytic inversion to contemporary IMRT optimization: Radiation therapy planning revisited from a mathematical perspective ..."
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From analytic inversion to contemporary IMRT optimization: Radiation therapy planning revisited from a mathematical perspective
the Common Solutions to Variational Inequalities Problem (CSVIP). This
, 2011
"... We present and study a new variational inequality problem, which we call the Common Solutions to Variational Inequalities Problem (CSVIP). This problem consists of
nding common solutions to a system of unrelated variational inequalities corresponding to setvalued mappings in Hilbert space. We pre ..."
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We present and study a new variational inequality problem, which we call the Common Solutions to Variational Inequalities Problem (CSVIP). This problem consists of
nding common solutions to a system of unrelated variational inequalities corresponding to setvalued mappings in Hilbert space. We present an iterative procedure for solving this problem and establish its strong convergence. Relations with other problems of solving systems of variational inequalities, both old and new, are discussed as well.