On Projection Algorithms for Solving Convex Feasibility Problems

Bauschke, Heinz H.; Borwein, Jonathan M.

doi:10.1137/s0036144593251710

Cited by 1,434 publications

(1,238 citation statements)

References 81 publications

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“…In contrast to the property of being contractive, the nonexpansivity of an operator does not guarantee the convergence of the corresponding Picard iterations. Therefore, we use the stronger notion of an averaged operator, cf., e.g., (Bauschke and Borwein 1996;Byrne 2004;Combettes 2004). By definition, T : H → H is averaged if for a nonexpansive operator R and some α ∈ (0, 1) we can write T as…”

Section: Picard Iterations For the Solution Of Variational Problemsmentioning

confidence: 99%

Operator Splittings, Bregman Methods and Frame Shrinkage in Image Processing

2010

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We examine the underlying structure of popular algorithms for variational methods used in image processing. We focus here on operator splittings and Bregman methods based on a unified approach via fixed point iterations and averaged operators. In particular, the recently proposed alternating split Bregman method can be interpreted from different points of view -as a Bregman, as an augmented Lagrangian and as a Douglas-Rachford splitting algorithm which is a classical operator splitting method. We also study similarities between this method and the forward-backward splitting method when applied to two frequently used models for image denoising which employ a Besov-norm and a total variation regularization term, respectively. In the first setting, we show that for a discretization based on Parseval frames the gradient descent reprojection and the alternating split Bregman algorithm are equivalent and turn out to be a frame shrinkage method. For the total variation regularizer, we also present a numerical comparison with multistep methods.Keywords Douglas-Rachford splitting · forward-backward splitting · Bregman methods · augmented Lagrangian method · alternating split Bregman algorithm · image denoising

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Section: Picard Iterations For the Solution Of Variational Problemsmentioning

confidence: 99%

Operator Splittings, Bregman Methods and Frame Shrinkage in Image Processing

2010

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“…The most famous one is the successive projection method for polyhedral sets known as the Agmon-MotzkinSchoenberg algorithm [Agm54,MoS54]. Further examples can be found in the excellent surveys in [BeT89,BaB96,CeZ97]. Such methods can also in some cases be interpreted as subgradient methods for the minimization of a non-differentiable convex function over a closed convex set (e.g., [Gof78]), several methods for which also use projections onto level sets of convex functions or surrogate linearized subgradient inequalities (as in "poor man's bundle methods"); see, for example, the level methods in [LNN95,Kiw96a,Kiw96b], references found therein, and [Bra93, pp.…”

Section: Euclidean Projectionmentioning

confidence: 99%

A survey on the continuous nonlinear resource allocation problem

Patriksson

2008

European Journal of Operational Research

184

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Our problem of interest consists of minimizing a separable, convex and differentiable function over a convex set, defined by bounds on the variables and an explicit constraint described by a separable convex function. Applications are abundant, and vary from equilibrium problems in the engineering and economic sciences, through resource allocation and balancing problems in manufacturing, statistics, military operations research and production and financial economics, to subproblems in algorithms for a variety of more complex optimization models. This paper surveys the history and applications of the problem, as well as algorithmic approaches to its solution. The most common techniques are based on finding the optimal value of the Lagrange multiplier for the explicit constraint, most often through the use of a type of line search procedure. We analyze the most relevant references, especially regarding their originality and numerical findings, summarizing with remarks on possible extensions and future research.

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“…It is, therefore, a sequential block-projection algorithm, an instance of a widely used projection technique (PT) for solving the convex feasibility problem (Bauschke & Borwein, 1996;García-Palomares, 1994), and its convergence properties are well understood (Bauschke & Borwein, 1996;Flåm & Zowe, 1990;García-Palomares, 1998.…”

Section: Psvm Iterationmentioning

confidence: 99%

“…Projection techniques have become a successful approach for solving convex systems (See for instance Bauschke & Borwein, 1996;Butnariu, Censor, & Reich, 2001;Censor & Zenios, 1997 and references therein). The main objective of this paper is to demonstrate that projection techniques yield an efficient approach to SVM classification.…”

Section: Introductionmentioning

confidence: 99%

Projection Support Vector Machine Generators

2004

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Abstract. Large-scale Support Vector Machine (SVM) classification is a very active research line in data mining. In recent years, several efficient SVM generation algorithms based on quadratic problems have been proposed, including: Successive OverRelaxation (SOR), Active Support Vector Machines (ASVM) and Lagrangian Support Vector Machines (LSVM). These algorithms have been used to solve classification problems with millions of points. ASVM is perhaps the fastest among them. This paper compares a new projection-based SVM algorithm with ASVM on a selection of real and synthetic data sets. The new algorithm seems competitive in terms of speed and testing accuracy.

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On Projection Algorithms for Solving Convex Feasibility Problems

Cited by 1,434 publications

References 81 publications

Operator Splittings, Bregman Methods and Frame Shrinkage in Image Processing

Operator Splittings, Bregman Methods and Frame Shrinkage in Image Processing

A survey on the continuous nonlinear resource allocation problem

Projection Support Vector Machine Generators

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