Journal of optimization theory and applications, ISSN 1573-2878, 2012, Volume 158, Issue 2, pp. 460 - 479

We propose a new first-order splitting algorithm for solving jointly the primal and dual formulations of large-scale convex minimization problems involving the sum of a smooth function...

Monotone inclusion | Mathematics | Theory of Computation | Optimization | Primal–dual algorithm | Douglas–Rachford method | Proximal method | Fenchel–Rockafellar duality | Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | Forward–backward method | Operator splitting | Applications of Mathematics | Engineering, general | Convex and nonsmooth optimization | Douglas-Rachford method | Primal-dual algorithm | Fenchel-Rockafellar duality | Forward-backward method | MATHEMATICS, APPLIED | DECOMPOSITION | SUM | ALGORITHMS | RECOVERY | MONOTONE INCLUSIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MINIMIZATION | CONVERGENCE | Analysis | Methods | Algorithms | Studies | Convex analysis | Formulations | Operators | Splitting | Composite functions | Inversions | Linear operators | Optimization and Control | Engineering Sciences | Signal and Image processing

Monotone inclusion | Mathematics | Theory of Computation | Optimization | Primal–dual algorithm | Douglas–Rachford method | Proximal method | Fenchel–Rockafellar duality | Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | Forward–backward method | Operator splitting | Applications of Mathematics | Engineering, general | Convex and nonsmooth optimization | Douglas-Rachford method | Primal-dual algorithm | Fenchel-Rockafellar duality | Forward-backward method | MATHEMATICS, APPLIED | DECOMPOSITION | SUM | ALGORITHMS | RECOVERY | MONOTONE INCLUSIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MINIMIZATION | CONVERGENCE | Analysis | Methods | Algorithms | Studies | Convex analysis | Formulations | Operators | Splitting | Composite functions | Inversions | Linear operators | Optimization and Control | Engineering Sciences | Signal and Image processing

Journal Article

IEEE transactions on image processing, ISSN 1941-0042, 2010, Volume 19, Issue 7, pp. 1720 - 1730

Multiplicative noise (also known as speckle noise) models are central to the study of coherent imaging systems, such as synthetic aperture radar and sonar, and...

Additive noise | variable splitting | Speckle | multiplicative noise | Synthetic aperture sonar | Lagrangian functions | Constraint optimization | synthetic aperture radar | Ultrasonic imaging | Laser radar | Gaussian noise | Laser noise | Douglas-Rachford splitting | Augmented Lagrangian | Laser modes | speckled images | total variation | Multiplicative noise | Variable splitting | Speckled images | Douglasa | Total variation | Rachford splitting | Synthetic aperture radar | MINIMIZATION | ALGORITHM | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | Lagrange equations | Functions, Gamma | Technology application | Usage | Gaussian processes | Imaging systems | Innovations | Mathematical optimization | Studies | Optimization | Additives | State of the art | Constraints | Noise | Imaging | Images | Gaussian

Additive noise | variable splitting | Speckle | multiplicative noise | Synthetic aperture sonar | Lagrangian functions | Constraint optimization | synthetic aperture radar | Ultrasonic imaging | Laser radar | Gaussian noise | Laser noise | Douglas-Rachford splitting | Augmented Lagrangian | Laser modes | speckled images | total variation | Multiplicative noise | Variable splitting | Speckled images | Douglasa | Total variation | Rachford splitting | Synthetic aperture radar | MINIMIZATION | ALGORITHM | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | Lagrange equations | Functions, Gamma | Technology application | Usage | Gaussian processes | Imaging systems | Innovations | Mathematical optimization | Studies | Optimization | Additives | State of the art | Constraints | Noise | Imaging | Images | Gaussian

Journal Article

Computational optimization and applications, ISSN 1573-2894, 2019, Volume 74, Issue 3, pp. 747 - 778

...Computational Optimization and Applications (2019) 74:747–778 https://doi.org/10.1007/s10589-019-00130-9 Douglas–Rachford splitting and ADMM for pathological...

Strong duality | 90C46 | 90C25 | Operations Research/Decision Theory | Convex and Discrete Geometry | Pathological convex programs | Douglas–Rachford splitting | Mathematics | Operations Research, Management Science | 49N15 | Statistics, general | Optimization | MATHEMATICS, APPLIED | APPROXIMATION | CONVERGENCE RATE ANALYSIS | FACIAL REDUCTION | SUM | ALGORITHMS | ASYMPTOTIC-BEHAVIOR | ALTERNATING DIRECTION METHOD | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford splitting | PROGRAMS | DUALITY | Convexity | Performance degradation | Empirical analysis | Mathematics - Optimization and Control

Strong duality | 90C46 | 90C25 | Operations Research/Decision Theory | Convex and Discrete Geometry | Pathological convex programs | Douglas–Rachford splitting | Mathematics | Operations Research, Management Science | 49N15 | Statistics, general | Optimization | MATHEMATICS, APPLIED | APPROXIMATION | CONVERGENCE RATE ANALYSIS | FACIAL REDUCTION | SUM | ALGORITHMS | ASYMPTOTIC-BEHAVIOR | ALTERNATING DIRECTION METHOD | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford splitting | PROGRAMS | DUALITY | Convexity | Performance degradation | Empirical analysis | Mathematics - Optimization and Control

Journal Article

Journal of optimization theory and applications, ISSN 1573-2878, 2019, Volume 182, Issue 2, pp. 606 - 639

Over the past decades, operator splitting methods have become ubiquitous for non-smooth optimization owing to their simplicity and efficiency...

Forward–Backward | 65K05 | Mathematics | Theory of Computation | Bregman distance | Optimization | Finite identification | Partial smoothness | Calculus of Variations and Optimal Control; Optimization | 49J52 | 90C25 | Operations Research/Decision Theory | Forward–Douglas–Rachford | 65K10 | Applications of Mathematics | Engineering, general | Local linear convergence | MATHEMATICS, APPLIED | SMOOTHNESS | ALGORITHM | Forward-Douglas-Rachford | DESCENT METHODS | Forward-Backward | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | ERROR-BOUNDS | CONVEX MINIMIZATION | Analysis | Methods | Machine learning | Operators (mathematics) | Manifolds | Economic models | Splitting | Divergence | Inverse problems | Image processing | Signal processing | Iterative methods | Convergence | Signal and Image Processing | Information Theory | Functional Analysis | Numerical Analysis | Computer Science | Optimization and Control | Statistics | Statistics Theory | Machine Learning

Forward–Backward | 65K05 | Mathematics | Theory of Computation | Bregman distance | Optimization | Finite identification | Partial smoothness | Calculus of Variations and Optimal Control; Optimization | 49J52 | 90C25 | Operations Research/Decision Theory | Forward–Douglas–Rachford | 65K10 | Applications of Mathematics | Engineering, general | Local linear convergence | MATHEMATICS, APPLIED | SMOOTHNESS | ALGORITHM | Forward-Douglas-Rachford | DESCENT METHODS | Forward-Backward | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | ERROR-BOUNDS | CONVEX MINIMIZATION | Analysis | Methods | Machine learning | Operators (mathematics) | Manifolds | Economic models | Splitting | Divergence | Inverse problems | Image processing | Signal processing | Iterative methods | Convergence | Signal and Image Processing | Information Theory | Functional Analysis | Numerical Analysis | Computer Science | Optimization and Control | Statistics | Statistics Theory | Machine Learning

Journal Article

Mathematical programming, ISSN 1436-4646, 2019, Volume 182, Issue 1-2, pp. 233 - 273

Given the success of Douglas-Rachford splitting (DRS), it is natural to ask whether DRS can be generalized...

COMPUTER SCIENCE, SOFTWARE ENGINEERING | Lower bounds | MATHEMATICS, APPLIED | Splitting methods | MONOTONE INCLUSIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Maximal monotone operators | Douglas-Rachford splitting | ALGORITHM | SUM | OPTIMIZATION | First-order methods | Operators | Splitting | Hoisting

COMPUTER SCIENCE, SOFTWARE ENGINEERING | Lower bounds | MATHEMATICS, APPLIED | Splitting methods | MONOTONE INCLUSIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Maximal monotone operators | Douglas-Rachford splitting | ALGORITHM | SUM | OPTIMIZATION | First-order methods | Operators | Splitting | Hoisting

Journal Article

Mathematical Programming, ISSN 0025-5610, 11/2015, Volume 153, Issue 2, pp. 715 - 722

...–Rachford operator splitting method for finding a root of the sum of two maximal monotone set-valued operators...

Mathematical Methods in Physics | 65N12 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Convergence rate | 65K10 | Mathematics | Combinatorics | Douglas–Rachford operator splitting method | COMPUTER SCIENCE, SOFTWARE ENGINEERING | PROJECTION | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford operator splitting method | MONOTONE | ALGORITHM | Yuan (China) | Methods | Studies | Mathematical programming

Mathematical Methods in Physics | 65N12 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Convergence rate | 65K10 | Mathematics | Combinatorics | Douglas–Rachford operator splitting method | COMPUTER SCIENCE, SOFTWARE ENGINEERING | PROJECTION | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford operator splitting method | MONOTONE | ALGORITHM | Yuan (China) | Methods | Studies | Mathematical programming

Journal Article

Mathematical programming, ISSN 1436-4646, 2018, Volume 177, Issue 1-2, pp. 225 - 253

In this paper, we present a method for identifying infeasible, unbounded, and pathological conic programs based on Douglas–Rachford splitting...

65K15 | 65K05 | Theoretical, Mathematical and Computational Physics | Douglas–Rachford splitting | Mathematics | Unbounded | Mathematical Methods in Physics | Conic programs | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | Pathological | Infeasible | Combinatorics | 47H05 | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford splitting | FACIAL REDUCTION | OPTIMIZATION | SUM | ALGORITHMS | Usage | Algorithms | Splitting | Subroutines | Optimization | Hyperplanes

65K15 | 65K05 | Theoretical, Mathematical and Computational Physics | Douglas–Rachford splitting | Mathematics | Unbounded | Mathematical Methods in Physics | Conic programs | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | Pathological | Infeasible | Combinatorics | 47H05 | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford splitting | FACIAL REDUCTION | OPTIMIZATION | SUM | ALGORITHMS | Usage | Algorithms | Splitting | Subroutines | Optimization | Hyperplanes

Journal Article

Journal of global optimization, ISSN 1573-2916, 2018, Volume 73, Issue 4, pp. 801 - 824

.... Based on the MiKM, we construct some multi-step inertial splitting methods, including the multi-step inertial Douglas...

Nonexpansive operator | Monotone inclusion | Bounded perturbation resilience | Forward–backward splitting method | Mathematics | Optimization | Douglas–Rachford splitting method | Davis–Yin splitting method | Backward–forward splitting method | Operations Research/Decision Theory | Multi-step inertial Krasnosel’skiǐ–Mann algorithm | Computer Science, general | Real Functions | SUPERIORIZATION | MATHEMATICS, APPLIED | Forward-backward splitting method | Backward-forward splitting method | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | GRADIENT METHODS | Multi-step inertial Krasnosel'skii-Mann algorithm | Douglas-Rachford splitting method | Davis-Yin splitting method | Splitting | Hilbert space | Algorithms | Iterative methods | Convergence

Nonexpansive operator | Monotone inclusion | Bounded perturbation resilience | Forward–backward splitting method | Mathematics | Optimization | Douglas–Rachford splitting method | Davis–Yin splitting method | Backward–forward splitting method | Operations Research/Decision Theory | Multi-step inertial Krasnosel’skiǐ–Mann algorithm | Computer Science, general | Real Functions | SUPERIORIZATION | MATHEMATICS, APPLIED | Forward-backward splitting method | Backward-forward splitting method | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | GRADIENT METHODS | Multi-step inertial Krasnosel'skii-Mann algorithm | Douglas-Rachford splitting method | Davis-Yin splitting method | Splitting | Hilbert space | Algorithms | Iterative methods | Convergence

Journal Article

Journal of Mathematical Imaging and Vision, ISSN 0924-9907, 2/2010, Volume 36, Issue 2, pp. 168 - 184

.... We propose to compute the minimizers of our restoration functionals by applying Douglas-Rachford splitting techniques, resp...

Speckle noise | alternating direction method of multipliers | Control , Robotics, Mechatronics | Douglas-Rachford splitting | Computer Science | Image Processing and Computer Vision | Computer Imaging, Vision, Pattern Recognition and Graphics | Artificial Intelligence (incl. Robotics) | Split Bregman algorithm | Poisson noise | Gamma noise | Alternating direction method of multipliers | MATHEMATICS, APPLIED | FILTER | IMAGE-RESTORATION | ALGORITHM | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | COMPUTER SCIENCE, SOFTWARE ENGINEERING | REGULARIZATION | TOTAL VARIATION MINIMIZATION | SCHEMES | Methods | Algorithms

Speckle noise | alternating direction method of multipliers | Control , Robotics, Mechatronics | Douglas-Rachford splitting | Computer Science | Image Processing and Computer Vision | Computer Imaging, Vision, Pattern Recognition and Graphics | Artificial Intelligence (incl. Robotics) | Split Bregman algorithm | Poisson noise | Gamma noise | Alternating direction method of multipliers | MATHEMATICS, APPLIED | FILTER | IMAGE-RESTORATION | ALGORITHM | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | COMPUTER SCIENCE, SOFTWARE ENGINEERING | REGULARIZATION | TOTAL VARIATION MINIMIZATION | SCHEMES | Methods | Algorithms

Journal Article

10.
Full Text
The error structure of the Douglas–Rachford splitting method for stiff linear problems

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 09/2016, Volume 303, pp. 140 - 145

The Lie splitting algorithm is frequently used when splitting stiff ODEs or, more generally, dissipative evolution equations...

Douglas–Rachford splitting | Order reduction | Inhomogeneous evolution equations | Error analysis | Stiff linear problems | Dissipative operators | Douglas-Rachford splitting | Analysis | Methods | Algorithms | Errors | Splitting | Mathematical analysis | Dissipation | Evolution | Remedies | order reduction | Naturvetenskap | Other Mathematics | error analysis | inhomogeneous evolution equations | Annan matematik | Mathematics | Natural Sciences | Matematik | dissipative operators | stiff linear problems

Douglas–Rachford splitting | Order reduction | Inhomogeneous evolution equations | Error analysis | Stiff linear problems | Dissipative operators | Douglas-Rachford splitting | Analysis | Methods | Algorithms | Errors | Splitting | Mathematical analysis | Dissipation | Evolution | Remedies | order reduction | Naturvetenskap | Other Mathematics | error analysis | inhomogeneous evolution equations | Annan matematik | Mathematics | Natural Sciences | Matematik | dissipative operators | stiff linear problems

Journal Article

International Journal of Computer Vision, ISSN 0920-5691, 5/2011, Volume 92, Issue 3, pp. 265 - 280

.... We focus here on operator splittings and Bregman methods based on a unified approach via fixed point iterations and averaged operators...

Augmented Lagrangian method | Pattern Recognition | Douglas-Rachford splitting | Computer Science | Computer Imaging, Vision, Pattern Recognition and Graphics | Image Processing and Computer Vision | Artificial Intelligence (incl. Robotics) | Forward-backward splitting | Alternating split Bregman algorithm | Bregman methods | Image denoising | Douglas-rachford splitting | Alternating split bregman algorithm | Augmented lagrangian method | APPROXIMATION | SUM | ITERATIVE ALGORITHMS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | CONVERGENCE | TOTAL VARIATION MINIMIZATION | SCHEMES | Usage | Image processing equipment industry | Equipment and supplies | Algorithms | Image processing | Methods | Frames | Operators | Splitting | Images | Mathematical models | Shrinkage

Augmented Lagrangian method | Pattern Recognition | Douglas-Rachford splitting | Computer Science | Computer Imaging, Vision, Pattern Recognition and Graphics | Image Processing and Computer Vision | Artificial Intelligence (incl. Robotics) | Forward-backward splitting | Alternating split Bregman algorithm | Bregman methods | Image denoising | Douglas-rachford splitting | Alternating split bregman algorithm | Augmented lagrangian method | APPROXIMATION | SUM | ITERATIVE ALGORITHMS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | CONVERGENCE | TOTAL VARIATION MINIMIZATION | SCHEMES | Usage | Image processing equipment industry | Equipment and supplies | Algorithms | Image processing | Methods | Frames | Operators | Splitting | Images | Mathematical models | Shrinkage

Journal Article

Applied Mathematics & Optimization, ISSN 0095-4616, 12/2019, Volume 80, Issue 3, pp. 665 - 678

...–Rachford splitting algorithm. More precisely, it is obtained by performing an explicit, rather than implicit, discretization with respect to one of the operators involved...

65K15 | Systems Theory, Control | Theoretical, Mathematical and Computational Physics | Mathematics | Monotone operator | Dynamical systems | Mathematical Methods in Physics | 47J20 | Calculus of Variations and Optimal Control; Optimization | 90C25 | Operator splitting | Douglas–Rachford algorithm | 49M29 | Numerical and Computational Physics, Simulation | 47H05 | Mathematics - Optimization and Control

65K15 | Systems Theory, Control | Theoretical, Mathematical and Computational Physics | Mathematics | Monotone operator | Dynamical systems | Mathematical Methods in Physics | 47J20 | Calculus of Variations and Optimal Control; Optimization | 90C25 | Operator splitting | Douglas–Rachford algorithm | 49M29 | Numerical and Computational Physics, Simulation | 47H05 | Mathematics - Optimization and Control

Journal Article

Circuits, Systems, and Signal Processing, ISSN 0278-081X, 10/2017, Volume 36, Issue 10, pp. 4022 - 4049

...–Rachford splitting method is a well-known operator splitting method that has been widely applied for solving a certain class of convex composite problems...

Engineering | Compressive sensing | Signal,Image and Speech Processing | Electronics and Microelectronics, Instrumentation | Circuits and Systems | Sparse signal recovery | Douglas–Rachford splitting method | Electrical Engineering | ITERATION | RECONSTRUCTION | SIGNAL RECOVERY | DECOMPOSITION | PRINCIPLES | ENGINEERING, ELECTRICAL & ELECTRONIC | L-MINIMIZATION | CONVEX | PARALLEL ALGORITHMS | CONVERGENCE | Douglas-Rachford splitting method | POINT ALGORITHM | Methods | Algorithms | Economic models | Splitting | Multipliers | Compressive properties | Run time (computers) | Projection | Signal reconstruction | Iterative methods | Detection | Optimization

Engineering | Compressive sensing | Signal,Image and Speech Processing | Electronics and Microelectronics, Instrumentation | Circuits and Systems | Sparse signal recovery | Douglas–Rachford splitting method | Electrical Engineering | ITERATION | RECONSTRUCTION | SIGNAL RECOVERY | DECOMPOSITION | PRINCIPLES | ENGINEERING, ELECTRICAL & ELECTRONIC | L-MINIMIZATION | CONVEX | PARALLEL ALGORITHMS | CONVERGENCE | Douglas-Rachford splitting method | POINT ALGORITHM | Methods | Algorithms | Economic models | Splitting | Multipliers | Compressive properties | Run time (computers) | Projection | Signal reconstruction | Iterative methods | Detection | Optimization

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 04/2015, Volume 256, pp. 472 - 487

We propose an inertial Douglas–Rachford splitting algorithm for finding the set of zeros of the sum of two maximally monotone operators in Hilbert spaces and investigate its convergence properties...

Douglas–Rachford splitting | Krasnosel’skiı̆–Mann algorithm | Convex optimization | Primal–dual algorithm | Inertial splitting algorithm | Krasnosel'skiѣ-Mann algorithm Primal-dual algorithm Convex optimization | Douglas-Rachford splitting | MATHEMATICS, APPLIED | Primal-dual algorithm | Krasnosel'skii-Mann algorithm | MINIMIZATION | WEAK-CONVERGENCE | PROXIMAL POINT ALGORITHM | OPERATORS | COMPOSITE

Douglas–Rachford splitting | Krasnosel’skiı̆–Mann algorithm | Convex optimization | Primal–dual algorithm | Inertial splitting algorithm | Krasnosel'skiѣ-Mann algorithm Primal-dual algorithm Convex optimization | Douglas-Rachford splitting | MATHEMATICS, APPLIED | Primal-dual algorithm | Krasnosel'skii-Mann algorithm | MINIMIZATION | WEAK-CONVERGENCE | PROXIMAL POINT ALGORITHM | OPERATORS | COMPOSITE

Journal Article

Journal of applied mathematics & computing, ISSN 1865-2085, 2015, Volume 51, Issue 1-2, pp. 569 - 591

In this paper, we study the Douglas–Rachford (DR) splitting method when applied to the standard semidefinite programming, and we introduce a new variant of this method and under weak assumptions prove its global convergence...

Computational Mathematics and Numerical Analysis | Semidefinite programming | 90C06 | Mathematics | Theory of Computation | Douglas–Rachford splitting method | Convergence | Iteration complexity | 90C30 | 90C35 | Mathematics of Computing | 90C22 | Appl.Mathematics/Computational Methods of Engineering | Douglas-Rachford splitting method | Studies | Mathematical analysis | Splitting | Direct reduction | Mathematical models | Robustness | Iterative methods | Symmetry | Mathematical programming

Computational Mathematics and Numerical Analysis | Semidefinite programming | 90C06 | Mathematics | Theory of Computation | Douglas–Rachford splitting method | Convergence | Iteration complexity | 90C30 | 90C35 | Mathematics of Computing | 90C22 | Appl.Mathematics/Computational Methods of Engineering | Douglas-Rachford splitting method | Studies | Mathematical analysis | Splitting | Direct reduction | Mathematical models | Robustness | Iterative methods | Symmetry | Mathematical programming

Journal Article