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41
On Projection Algorithms for Solving Convex Feasibility Problems
, 1996
"... Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences (most notably, computerized tomography), algorithms for solving convex feasibility problems continue to receive great attention. To unify, generalize, and review some of the ..."
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Cited by 330 (44 self)
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Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences (most notably, computerized tomography), algorithms for solving convex feasibility problems continue to receive great attention. To unify, generalize, and review some of these algorithms, a very broad and flexible framework is investigated . Several crucial new concepts which allow a systematic discussion of questions on behaviour in general Hilbert spaces and on the quality of convergence are brought out. Numerous examples are given. 1991 M.R. Subject Classification. Primary 47H09, 49M45, 6502, 65J05, 90C25; Secondary 26B25, 41A65, 46C99, 46N10, 47N10, 52A05, 52A41, 65F10, 65K05, 90C90, 92C55. Key words and phrases. Angle between two subspaces, averaged mapping, Cimmino's method, computerized tomography, convex feasibility problem, convex function, convex inequalities, convex programming, convex set, Fej'er monotone sequence, firmly nonexpansive mapping, H...
Solving monotone inclusions via compositions of nonexpansive averaged operators
 Optimization
, 2004
"... A unified fixed point theoretic framework is proposed to investigate the asymptotic behavior of algorithms for finding solutions to monotone inclusion problems. The basic iterative scheme under consideration involves nonstationary compositions of perturbed averaged nonexpansive operators. The analys ..."
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Cited by 145 (31 self)
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A unified fixed point theoretic framework is proposed to investigate the asymptotic behavior of algorithms for finding solutions to monotone inclusion problems. The basic iterative scheme under consideration involves nonstationary compositions of perturbed averaged nonexpansive operators. The analysis covers proximal methods for common zero problems as well as various splitting methods for finding a zero of the sum of monotone operators.
A douglasRachford splitting approach to nonsmooth convex variational signal recovery
 IEEE Journal of Selected Topics in Signal Processing
, 2007
"... Abstract — Under consideration is the large body of signal recovery problems that can be formulated as the problem of minimizing the sum of two (not necessarily smooth) lower semicontinuous convex functions in a real Hilbert space. This generic problem is analyzed and a decomposition method is propo ..."
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Cited by 89 (23 self)
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Abstract — Under consideration is the large body of signal recovery problems that can be formulated as the problem of minimizing the sum of two (not necessarily smooth) lower semicontinuous convex functions in a real Hilbert space. This generic problem is analyzed and a decomposition method is proposed to solve it. The convergence of the method, which is based on the DouglasRachford algorithm for monotone operatorsplitting, is obtained under general conditions. Applications to nonGaussian image denoising in a tight frame are also demonstrated. Index Terms — Convex optimization, denoising, DouglasRachford, frame, nondifferentiable optimization, Poisson noise,
A WEAKTOSTRONGCONVERGENCE PRINCIPLE FOR FEJÉRMONOTONE METHODS IN HILBERT SPACES
, 2001
"... We consider a wide class of iterative methods arising in numerical mathematics and optimization that are known to converge only weakly. Exploiting an idea originally proposed by Haugazeau, we present a simple modification of these methods that makes them strongly convergent without additional assump ..."
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Cited by 81 (13 self)
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We consider a wide class of iterative methods arising in numerical mathematics and optimization that are known to converge only weakly. Exploiting an idea originally proposed by Haugazeau, we present a simple modification of these methods that makes them strongly convergent without additional assumptions. Several applications are discussed.
A distributed algorithm for managing multitarget identities in wireless adhoc sensor networks
 In IPSN ’03: Information Processing in Sensor Networks
, 2003
"... Abstract. This paper presents a scalable distributed algorithm for computing and maintaining multitarget identity information. The algorithm builds on a novel representational framework, IdentityMass Flow, to overcome the problem of exponential computational complexity in managing multitarget ide ..."
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Cited by 69 (12 self)
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Abstract. This paper presents a scalable distributed algorithm for computing and maintaining multitarget identity information. The algorithm builds on a novel representational framework, IdentityMass Flow, to overcome the problem of exponential computational complexity in managing multitarget identity explicitly. The algorithm uses local information to efficiently update the global multitarget identity information represented as a doubly stochastic matrix, and can be efficiently mapped to nodes in a wireless ad hoc sensor network. The paper describes a distributed implementation of the algorithm in sensor networks. Simulation results have validated the IdentityMass Flow framework and demonstrated the feasibility of the algorithm. 1
Finding Best Approximation Pairs Relative to Two Closed Convex Sets in Hilbert Spaces
 J. APPROX. THEORY
, 2003
"... We consider the problem of nding a best approximation pair, i.e., two points which achieve the minimum distance between two closed convex sets in a Hilbert space. When the sets intersect, the method under consideration, termed ASR for averaged successive reections, is a special instance of an al ..."
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Cited by 60 (23 self)
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We consider the problem of nding a best approximation pair, i.e., two points which achieve the minimum distance between two closed convex sets in a Hilbert space. When the sets intersect, the method under consideration, termed ASR for averaged successive reections, is a special instance of an algorithm due to Lions and Mercier for nding a zero of the sum of two maximal monotone operators. We investigate systematically the asymptotic behavior of ASR in the general case when the sets do not necessarily intersect and show that the method produces best approximation pairs provided they exist. Finitely many sets are handled in a product space, in which case the ASR method is shown to coincide with a special case of Spingarn's method of partial inverses.
QuasiFejérian Analysis of Some Optimization Algorithms
"... A quasiFejér sequence is a sequence which satisfies the standard Fejér monotonicity property to within an additional error term. This notion is studied in detail in a Hilbert space setting and shown to provide a powerful framework to analyze the convergence of a wide range of optimization algorithm ..."
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Cited by 56 (14 self)
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A quasiFejér sequence is a sequence which satisfies the standard Fejér monotonicity property to within an additional error term. This notion is studied in detail in a Hilbert space setting and shown to provide a powerful framework to analyze the convergence of a wide range of optimization algorithms in a systematic fashion. A number of convergence theorems covering and extending existing results are thus established. Special emphasis is placed on the design and the analysis of parallel algorithms.
Projection and proximal point methods: convergence results and counterexamples
, 2003
"... Recently, Hundal has constructed a hyperplane H, a cone K, and a starting point y0 in `2 such that the sequence of alternating projections (PKPH)ny0 n∈N converges weakly to some point in H ∩K, but not in norm. We show how this construction results in a counterexample to norm convergence for iterates ..."
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Cited by 49 (19 self)
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Recently, Hundal has constructed a hyperplane H, a cone K, and a starting point y0 in `2 such that the sequence of alternating projections (PKPH)ny0 n∈N converges weakly to some point in H ∩K, but not in norm. We show how this construction results in a counterexample to norm convergence for iterates of averaged projections; hence, we give an affirmative answer to a question raised by Reich two decades ago. Furthermore, new counterexamples to norm convergence for iterates of firmly nonexpansive maps (a ̀ la Genel and Lindenstrauss) and for the proximal point algorithm (a ̀ la Güler) are provided. We also present a counterexample, along with some weak and norm convergence results, for the new framework of stringaveraging projection methods introduced by Censor, Elfving, and Herman. Extensions to Banach spaces and the situation for the Hilbert ball are discussed as well.
A parallel splitting method for coupled monotone inclusions
"... A parallel splitting method is proposed for solving systems of coupled monotone inclusions in Hilbert spaces, and its convergence is established under the assumption that solutions exist. Unlike existing alternating algorithms, which are limited to two variables and linear coupling, our parallel met ..."
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Cited by 35 (11 self)
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A parallel splitting method is proposed for solving systems of coupled monotone inclusions in Hilbert spaces, and its convergence is established under the assumption that solutions exist. Unlike existing alternating algorithms, which are limited to two variables and linear coupling, our parallel method can handle an arbitrary number of variables as well as nonlinear coupling schemes. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of evolution inclusions, variational problems, best approximation, and network flows.