Applications of a Splitting Algorithm to Decomposition in Convex Programming and Variational Inequalities

Tseng, Paul

doi:10.1137/0329006

Cited by 410 publications

(341 citation statements)

References 37 publications

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“…A number of authors, mainly in the French mathematical community, have extensively studied monotone operator splitting methods, which fall into four principal classes: forward-backward [40,13,56], double-backward [30,40], Peaceman-Rachford [31], and Douglas-Rachford [31]. For a survey, readers may wish to refer to [1 I,Chapter 3].…”

Section: Introductionmentioning

confidence: 99%

On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators

Eckstein¹,

Bertsekas

1992

Mathematical Programming

2,484

2,027

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This paper shows, by means of an operator called a splitting operator, that the Douglas-Rachford splitting method for finding a zero of the sum of two monotone operators is a special case of the proximal point algorithm, Therefore, applications of Douglas-Rachford splitting, such as the alternating direction method of multipliers for convex programming decomposition, are also special cases of the proximal point algorithm. This observation allows the unification and generalization of a variety of convex programming algorithms. By introducing a modified version of the proximal point algorithm, we derive a new, generalized alternating direction method of multipliers for convex programming. Advances of this sort illustrate the power and generality gained by adopting monotone operator theory as a conceptual framework.

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Section: Introductionmentioning

confidence: 99%

On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators

Eckstein¹,

Bertsekas

1992

Mathematical Programming

2,484

2,027

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“…Again it can be shown, see, e.g., (Lions and Mercier 1979;Combettes 2004;Combettes and Wajs 2005), that under the conditions stated in Theorem 2 below the operator T is averaged and convergence follows by Theorem 1. A somewhat different approach to the proof of the following theorem can be found in (Tseng 1991).…”

Section: Operator Splitting Methodsmentioning

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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“…Since the functional (23) is coercive there exists a minimizer. Further, F 2 is proper, convex and closed so that it remains by [24,25] to show that ∇F 1 is Lipschtz continuous with constant < 2/τ which is obviously the case if τ < 2/ ∆ 2 .…”

Section: Algorithmmentioning

confidence: 99%

Convex Hodge Decomposition and Regularization of Image Flows

2008

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The total variation (TV) measure is a key concept in the field of variational image analysis. In this paper, we focus on vector-valued data and derive from the Hodge decomposition of image flows a definition of TV regularization for vector-valued data that extends the standard componentwise definition in a natural way. We show that our approach leads to a convex decomposition of arbitrary vector fields, providing a richer decomposition into piecewise harmonic fields rather than piecewise constant ones, and motion texture. Furthermore, our regularizer provides a measure for motion boundaries of piecewise harmonic image flows in the same way, as the TV measure does for contours of scalar-valued piecewise constant images.

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Applications of a Splitting Algorithm to Decomposition in Convex Programming and Variational Inequalities

Cited by 410 publications

References 37 publications

On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators

On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators

Operator Splittings, Bregman Methods and Frame Shrinkage in Image Processing

Convex Hodge Decomposition and Regularization of Image Flows

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