Results 1  10
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89
SIGNAL RECOVERY BY PROXIMAL FORWARDBACKWARD SPLITTING
 MULTISCALE MODEL. SIMUL. TO APPEAR
"... We show that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. This formulation makes it possible to derive existence, uniqueness, characterization, and stability results in a unifi ..."
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Cited by 509 (24 self)
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We show that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. This formulation makes it possible to derive existence, uniqueness, characterization, and stability results in a unified and standardized fashion for a large class of apparently disparate problems. Recent results on monotone operator splitting methods are applied to establish the convergence of a forwardbackward algorithm to solve the generic problem. In turn, we recover, extend, and provide a simplified analysis for a variety of existing iterative methods. Applications to geometry/texture image decomposition schemes are also discussed. A novelty of our framework is to use extensively the notion of a proximity operator, which was introduced by Moreau in the 1960s.
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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Cited by 30 (18 self)
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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).
Extrapolation algorithm for affineconvex feasibility problems
, 2005
"... The convex feasibility problem under consideration is to find a common point of a countable family of closed affine subspaces and convex sets in a Hilbert space. To solve such problems, we propose a general parallel blockiterative algorithmic framework in which the affine subspaces are exploited to ..."
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Cited by 21 (5 self)
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The convex feasibility problem under consideration is to find a common point of a countable family of closed affine subspaces and convex sets in a Hilbert space. To solve such problems, we propose a general parallel blockiterative algorithmic framework in which the affine subspaces are exploited to introduce extrapolated overrelaxations. This framework encompasses a wide range of projection, subgradient projection, proximal, and fixed point methods encountered in various branches of applied mathematics. The asymptotic behavior of the method is investigated and numerical experiments are provided to illustrate the benefits of the extrapolations.
Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems,”
 Journal of Inequalities and Applications,
, 2008
"... We introduce an iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of a variational inclusion with setvalued maximal monotone mapping and inverse strongly monotone mappings, the set of solutions of an equilibrium problem, and the set of fixed ..."
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Cited by 10 (1 self)
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We introduce an iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of a variational inclusion with setvalued maximal monotone mapping and inverse strongly monotone mappings, the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces. The results in this paper unify, extend, and improve some wellknown results in the literature.
A General Iterative Method for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces
, 2007
"... We introduce a general iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Our results improve and extend the corresponding ones announced by S. ..."
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Cited by 9 (1 self)
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We introduce a general iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Our results improve and extend the corresponding ones announced by S. Takahashi and W. Takahashi in 2007, Marino and Xu in 2006, Combettes and Hirstoaga in 2005, and many others.
A Hybrid Iterative Scheme for a Maximal Monotone Operator and Two Countable Families of Relatively QuasiNonexpansive
 Mappings for Generalized Mixed Equilibrium and Variational Inequality Problems, Abstract and Applied Analysis, Article ID 123027
"... We introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasinonexpansive mappings, the set of the variational inequality for an αinversestrongly monotone operator, the set of solutions of the generalized m ..."
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Cited by 5 (1 self)
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We introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasinonexpansive mappings, the set of the variational inequality for an αinversestrongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2 uniformly convex and uniformly smooth Banach space. The results presented in this paper improve and extend some recent results.
A Hybrid Iterative Scheme for Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems in Banach Spaces
, 2009
"... The purpose of this paper is to introduce a new hybrid projection algorithm for finding a common element of the set of solutions of the equilibrium problem and the set of the variational inequality for an inversestronglymonotone operator and the set of fixed points of relatively quasinonexpansive ..."
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Cited by 5 (3 self)
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The purpose of this paper is to introduce a new hybrid projection algorithm for finding a common element of the set of solutions of the equilibrium problem and the set of the variational inequality for an inversestronglymonotone operator and the set of fixed points of relatively quasinonexpansive mappings in a Banach space. Then we show a strong convergence theorem. Using this result, we obtain some applications in a Banach space.
A new iterative method for finding common solutions of a system of equilibrium problems, fixedpoint problems, and variational inequalities,” Abstract and Applied Analysis,
 Article ID 428293,
, 2010
"... We introduce a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed point sets of an infinite family of nonexpansive mappings, and the solution set of a variational inequali ..."
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Cited by 2 (0 self)
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We introduce a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed point sets of an infinite family of nonexpansive mappings, and the solution set of a variational inequality for a relaxed cocoercive mapping in a Hilbert space. We prove strong convergence theorem. The results in this paper unify and generalize some wellknown results in the literature.
Composite algorithms for minimization over the solu tions of equilibrium problems and fixed point problems
 Abstract and Applied Analysis, Volume 2010 (2010), Article ID 763506
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Convergence theorems based on the shrinking projection method for hemirelatively nonexpansive mappings, variational inequalities and equilibrium problem
 Banach J. Math. Anal
"... chi ve of S ID ..."
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