Results 1  10
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16
A monotone+skew splitting model for composite monotone inclusions in duality
, 2011
"... The principle underlying this paper is the basic observation that the problem of simultaneously solving a large class of composite monotone inclusions and their duals can be reduced to that of finding a zero of the sum of a maximally monotone operator and a linear skewadjoint operator. An algorith ..."
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Cited by 33 (5 self)
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The principle underlying this paper is the basic observation that the problem of simultaneously solving a large class of composite monotone inclusions and their duals can be reduced to that of finding a zero of the sum of a maximally monotone operator and a linear skewadjoint operator. An algorithmic framework is developed for solving this generic problem in a Hilbert space setting. New primaldual splitting algorithms are derived from this framework for inclusions involving composite monotone operators, and convergence results are established. These algorithms draw their simplicity and efficacy from the fact that they operate in a fully decomposed fashion in the sense that the monotone operators and the linear transformations involved are activated separately at each iteration. Comparisons with existing methods are made and applications to composite variational problems are demonstrated.
Accelerated and inexact forwardbackward algorithms
 Optimization Online, EPrint
"... Abstract. We propose a convergence analysis of accelerated forwardbackward splitting methods for composite function minimization, when the proximity operator is not available in closed form, and can only be computed up to a certain precision. We prove that the 1/k 2 convergence rate for the functio ..."
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Cited by 18 (10 self)
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Abstract. We propose a convergence analysis of accelerated forwardbackward splitting methods for composite function minimization, when the proximity operator is not available in closed form, and can only be computed up to a certain precision. We prove that the 1/k 2 convergence rate for the function values can be achieved if the admissible errors are of a certain type and satisfy a sufficiently fast decay condition. Our analysis is based on the machinery of estimate sequences first introduced by Nesterov for the study of accelerated gradient descent algorithms. Furthermore, we give a global complexity analysis, taking into account the cost of computing admissible approximations of the proximal point. An experimental analysis is also presented.
Variable Metric ForwardBackward Splitting with Applications to Monotone Inclusions in Duality
"... We propose a variable metric forwardbackward splitting algorithm and prove its convergence in real Hilbert spaces. We then use this framework to derive primaldual splitting algorithms for solving various classes of monotone inclusions in duality. Some of these algorithms are new even when speciali ..."
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Cited by 15 (7 self)
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We propose a variable metric forwardbackward splitting algorithm and prove its convergence in real Hilbert spaces. We then use this framework to derive primaldual splitting algorithms for solving various classes of monotone inclusions in duality. Some of these algorithms are new even when specialized to the fixed metric case. Various applications are discussed.
Proximity for Sums of Composite Functions
"... We propose an algorithm for computing the proximity operator of a sum of composite convex functions in Hilbert spaces and investigate its asymptotic behavior. Applications to best approximation and image recovery are described. ..."
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Cited by 8 (3 self)
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We propose an algorithm for computing the proximity operator of a sum of composite convex functions in Hilbert spaces and investigate its asymptotic behavior. Applications to best approximation and image recovery are described.
Duality and Convex Programming
, 2010
"... We survey some key concepts in convex duality theory and their application to the analysis and numerical solution of problem archetypes in imaging. ..."
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Cited by 6 (4 self)
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We survey some key concepts in convex duality theory and their application to the analysis and numerical solution of problem archetypes in imaging.
A forwardbackward view of some primaldual optimization methods in image recovery
 IN PROC. INT. CONF. IMAGE PROCESS
, 2014
"... A wide array of image recovery problems can be abstracted into the problem of minimizing a sum of composite convex functions in a Hilbert space. To solve such problems, primaldual proximal approaches have been developed which provide efficient solutions to largescale optimization problems. The obj ..."
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Cited by 6 (4 self)
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A wide array of image recovery problems can be abstracted into the problem of minimizing a sum of composite convex functions in a Hilbert space. To solve such problems, primaldual proximal approaches have been developed which provide efficient solutions to largescale optimization problems. The objective of this paper is to show that a number of existing algorithms can be derived from a general form of the forwardbackward algorithm applied in a suitable product space. Our approach also allows us to develop useful extensions of existing algorithms by introducing a variable metric. An illustration to image restoration is provided.
Playing with duality: An overview of recent primaldual approaches for . . .
, 2014
"... Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify its solution. Deriving efficient strategies jointly bringing i ..."
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Cited by 5 (1 self)
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Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify its solution. Deriving efficient strategies jointly bringing into play the primal and the dual problems is however a more recent idea which has generated many important new contributions in the last years. These novel developments are grounded on recent advances in convex analysis, discrete optimization, parallel processing, and nonsmooth optimization with emphasis on sparsity issues. In this paper, we aim at presenting the principles of primaldual approaches, while giving an overview of numerical methods which have been proposed in different contexts. We show the benefits which can be drawn from primaldual algorithms both for solving largescale convex optimization problems and discrete ones, and we provide various application examples to illustrate their usefulness.