IEEE Journal of Selected Topics in Signal Processing, ISSN 1932-4553, 12/2007, Volume 1, Issue 4, pp. 564 - 574

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)...

Noise reduction | nondifferentiable optimization | Projection algorithms | Poisson noise | Convergence | denoising | Douglas-Rachford | Convex optimization | proximal algorithm | Signal processing algorithms | wavelets | Signal processing | Hilbert space | Mathematical model | Signal analysis | Image denoising | frame | Nondifferentiable optimization | Wavelets | Frame | Proximal algorithm | Denoising | FEASIBILITY PROBLEMS | IMAGE-RESTORATION | REPRESENTATIONS | RECONSTRUCTION | THRESHOLDING ALGORITHM | CONSTRAINT | ENGINEERING, ELECTRICAL & ELECTRONIC | INVERSE PROBLEMS | TRANSFORM | PROJECTIONS | OPERATORS | Computation and Language | Computer Science

Noise reduction | nondifferentiable optimization | Projection algorithms | Poisson noise | Convergence | denoising | Douglas-Rachford | Convex optimization | proximal algorithm | Signal processing algorithms | wavelets | Signal processing | Hilbert space | Mathematical model | Signal analysis | Image denoising | frame | Nondifferentiable optimization | Wavelets | Frame | Proximal algorithm | Denoising | FEASIBILITY PROBLEMS | IMAGE-RESTORATION | REPRESENTATIONS | RECONSTRUCTION | THRESHOLDING ALGORITHM | CONSTRAINT | ENGINEERING, ELECTRICAL & ELECTRONIC | INVERSE PROBLEMS | TRANSFORM | PROJECTIONS | OPERATORS | Computation and Language | Computer Science

Journal Article

2011, 1st Edition., CMS Books in Mathematics, ISBN 9781461428695, 624

This book provides a largely self-contained account of the main results of convex analysis and optimization in Hilbert space. A concise exposition of related...

Mathematics | Hilbert space | Visualization | Convex functions | Algorithms | Calculus of Variations and Optimal Control; Optimization | Approximation theory | Monotone operators | Nonlinear functional analysis

Mathematics | Hilbert space | Visualization | Convex functions | Algorithms | Calculus of Variations and Optimal Control; Optimization | Approximation theory | Monotone operators | Nonlinear functional analysis

eBook

Computational Optimization and Applications, ISSN 0926-6003, 04/2012, Volume 51, Issue 3, pp. 1065 - 1088

... inequality constraints Yair Censor Wei Chen Patrick L. Combettes Ran Davidi Gabor T. Herman Received: 22 December 2009 / Published online: 17 March 2011 Springer Science...

Linear inequalities | Sparse matrices | Numerical evaluation | Optimization | Projection methods | Convex feasibility problems | MATHEMATICS, APPLIED | SET | BEHAVIOR | IMAGE-RECONSTRUCTION | RESTORATION | ALGORITHMS | RECOVERY | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | FIXED-POINTS | STRONG-CONVERGENCE | SPARSE SYSTEMS | Computer science | Censorship | Analysis | Methods | Studies | Patents | Effectiveness | Computation | Inequalities | Projection | Feasibility | Dealing

Linear inequalities | Sparse matrices | Numerical evaluation | Optimization | Projection methods | Convex feasibility problems | MATHEMATICS, APPLIED | SET | BEHAVIOR | IMAGE-RECONSTRUCTION | RESTORATION | ALGORITHMS | RECOVERY | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | FIXED-POINTS | STRONG-CONVERGENCE | SPARSE SYSTEMS | Computer science | Censorship | Analysis | Methods | Studies | Patents | Effectiveness | Computation | Inequalities | Projection | Feasibility | Dealing

Journal Article

Mathematical Programming, ISSN 0025-5610, 7/2018, Volume 170, Issue 1, pp. 177 - 206

...Math. Program., Ser. B (2018) 170:177–206 https://doi.org/10.1007/s10107-018-1303-3 FULL LENGTH PAPER Monotone operator theory in convex optimization Patrick L...

65K05 | Self-dual class | Theoretical, Mathematical and Computational Physics | Subdifferential | Proximity operator | Mathematics | Monotone operator | Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | Proximal algorithm | Operator splitting | Proximity-preserving transformation | 49M27 | Combinatorics | Firmly nonexpansive operator | 47H25 | MATHEMATICS, APPLIED | SIGNAL RECOVERY | THRESHOLDING ALGORITHM | DECOMPOSITION | PROXIMAL POINT ALGORITHM | LEAST-SQUARES SOLUTIONS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | NONLINEAR OPERATORS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | ANALYSE FONCTIONNELLE | CONVERGENCE | PARTIAL INVERSES | SPLITTING METHOD | Analysis | Algorithms | Computational geometry | Operators | Proximity | Convexity | Convex analysis | Optimization

65K05 | Self-dual class | Theoretical, Mathematical and Computational Physics | Subdifferential | Proximity operator | Mathematics | Monotone operator | Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | Proximal algorithm | Operator splitting | Proximity-preserving transformation | 49M27 | Combinatorics | Firmly nonexpansive operator | 47H25 | MATHEMATICS, APPLIED | SIGNAL RECOVERY | THRESHOLDING ALGORITHM | DECOMPOSITION | PROXIMAL POINT ALGORITHM | LEAST-SQUARES SOLUTIONS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | NONLINEAR OPERATORS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | ANALYSE FONCTIONNELLE | CONVERGENCE | PARTIAL INVERSES | SPLITTING METHOD | Analysis | Algorithms | Computational geometry | Operators | Proximity | Convexity | Convex analysis | Optimization

Journal Article

Journal of Functional Analysis, ISSN 0022-1236, 2012, Volume 262, Issue 1, pp. 400 - 408

The method of periodic projections consists in iterating projections onto m closed convex subsets of a Hilbert space according to a periodic sweeping strategy....

Alternating projections | Best approximation | Von Neumann algorithm | Limit cycle | MATHEMATICS | ALTERNATING PROJECTION | PRODUCT | SETS | HILBERT-SPACE | Analysis | Methods | Algorithms

Alternating projections | Best approximation | Von Neumann algorithm | Limit cycle | MATHEMATICS | ALTERNATING PROJECTION | PRODUCT | SETS | HILBERT-SPACE | Analysis | Methods | Algorithms

Journal Article

IEEE Transactions on Image Processing, ISSN 1057-7149, 09/2004, Volume 13, Issue 9, pp. 1213 - 1222

Total variation has proven to be a valuable concept in connection with the recovery of images featuring piecewise smooth components. So far, however, it has...

Constraint optimization | Degradation | Additive noise | Deconvolution | Hilbert space | Image restoration | Information filtering | Lagrangian functions | Image denoising | RECOVERY | RECONSTRUCTION | FILTERS | ALGORITHMS | TOTAL VARIATION MINIMIZATION | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | Reproducibility of Results | Information Storage and Retrieval - methods | Image Interpretation, Computer-Assisted - methods | Models, Statistical | Subtraction Technique | Computer Graphics | Algorithms | Numerical Analysis, Computer-Assisted | Computer Simulation | Sensitivity and Specificity | Signal Processing, Computer-Assisted | Image Enhancement - methods | Pattern Recognition, Automated | Image processing | Analysis | Imaging systems | Formulations | Restoration | Images | Blocking | Programming | Joints | Optimization | Numerical Analysis | Mathematics

Constraint optimization | Degradation | Additive noise | Deconvolution | Hilbert space | Image restoration | Information filtering | Lagrangian functions | Image denoising | RECOVERY | RECONSTRUCTION | FILTERS | ALGORITHMS | TOTAL VARIATION MINIMIZATION | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | Reproducibility of Results | Information Storage and Retrieval - methods | Image Interpretation, Computer-Assisted - methods | Models, Statistical | Subtraction Technique | Computer Graphics | Algorithms | Numerical Analysis, Computer-Assisted | Computer Simulation | Sensitivity and Specificity | Signal Processing, Computer-Assisted | Image Enhancement - methods | Pattern Recognition, Automated | Image processing | Analysis | Imaging systems | Formulations | Restoration | Images | Blocking | Programming | Joints | Optimization | Numerical Analysis | Mathematics

Journal Article

SIAM Journal on Optimization, ISSN 1052-6234, 2013, Volume 23, Issue 4, pp. 2420 - 2447

...SIAM J. OPTIM. c Vol. 23, No. 4, pp. 24202447 SYSTEMS OF STRUCTURED MONOTONE INCLUSIONS: DUALITY, ALGORITHMS, AND APPLICATIONS PATRICK L. COMBETTES Abstract...

Convex minimization | Structured minimization problem | Infimal convolution | Monotone inclusion | Parallel algorithm | Coupled system | Operator splitting | Monotone operator | MATHEMATICS, APPLIED | NONEXPANSIVE-MAPPINGS | coupled system | ALTERNATING PROXIMAL ALGORITHMS | convex minimization | parallel algorithm | PDES | MODEL | infimal convolution | monotone operator | IMAGE DECOMPOSITION | VARIATIONAL-INEQUALITIES | RECOVERY | structured minimization problem | POINT | operator splitting | SPLITTING METHOD | OPERATORS | monotone inclusion | Operators | State of the art | Splitting | Algorithms | Asymptotic properties | Inclusions | Linear operators | Optimization | Mathematics | Optimization and Control

Convex minimization | Structured minimization problem | Infimal convolution | Monotone inclusion | Parallel algorithm | Coupled system | Operator splitting | Monotone operator | MATHEMATICS, APPLIED | NONEXPANSIVE-MAPPINGS | coupled system | ALTERNATING PROXIMAL ALGORITHMS | convex minimization | parallel algorithm | PDES | MODEL | infimal convolution | monotone operator | IMAGE DECOMPOSITION | VARIATIONAL-INEQUALITIES | RECOVERY | structured minimization problem | POINT | operator splitting | SPLITTING METHOD | OPERATORS | monotone inclusion | Operators | State of the art | Splitting | Algorithms | Asymptotic properties | Inclusions | Linear operators | Optimization | Mathematics | Optimization and Control

Journal Article

Mathematical Programming, ISSN 0025-5610, 3/2018, Volume 168, Issue 1, pp. 645 - 672

... methods for monotone inclusions Patrick L. Combettes 1 · Jonathan Eckstein 2 Received: 26 November 2015 / Accepted: 16 June 2016 / Published online: 5 July 2016...

Monotone inclusion | Primal-dual algorithm | 65K05 | Theoretical, Mathematical and Computational Physics | Mathematics | Duality | Monotone operator | Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | Splitting algorithm | 49M27 | Block-iterative algorithm | Combinatorics | Asynchronous algorithm | 47H05 | HILBERT-SPACES | MATHEMATICS, APPLIED | KUHN-TUCKER SET | ALGORITHMS | SUMS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SYSTEMS | OPERATORS | PROJECTIVE SPLITTING METHODS | STRONG-CONVERGENCE | Methods | Algorithms | Half spaces | Innovations | Decomposition | Inclusions | Linear operators | Convergence

Monotone inclusion | Primal-dual algorithm | 65K05 | Theoretical, Mathematical and Computational Physics | Mathematics | Duality | Monotone operator | Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | Splitting algorithm | 49M27 | Block-iterative algorithm | Combinatorics | Asynchronous algorithm | 47H05 | HILBERT-SPACES | MATHEMATICS, APPLIED | KUHN-TUCKER SET | ALGORITHMS | SUMS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SYSTEMS | OPERATORS | PROJECTIVE SPLITTING METHODS | STRONG-CONVERGENCE | Methods | Algorithms | Half spaces | Innovations | Decomposition | Inclusions | Linear operators | Convergence

Journal Article

SIAM Journal on Optimization, ISSN 1052-6234, 2015, Volume 25, Issue 2, pp. 1221 - 1248

This work proposes block-coordinate fixed point algorithms with applications to nonlinear analysis and optimization in Hilbert spaces. The asymptotic analysis...

Primal-dual algorithm | Structured convex minimization problem | Monotone operator splitting | Stochastic algorithm | Stochastic quasi-Fejér sequence | Arbitrary sampling | Block-coordinate algorithm | Fixed-point algorithm | structured convex minimization problem | MATHEMATICS, APPLIED | primal-dual algorithm | monotone operator splitting | RACHFORD SPLITTING METHOD | SIGNAL RECOVERY | ALGORITHM | DECOMPOSITION | SUM | PRIMAL-DUAL METHOD | COMPOSITE | arbitrary sampling | block-coordinate algorithm | MONOTONE INCLUSIONS | stochastic algorithm | CONSTRUCTION | CONVERGENCE | stochastic quasi-Fejer sequence | fixed-point algorithm | Signal and Image Processing | Computer Science

Primal-dual algorithm | Structured convex minimization problem | Monotone operator splitting | Stochastic algorithm | Stochastic quasi-Fejér sequence | Arbitrary sampling | Block-coordinate algorithm | Fixed-point algorithm | structured convex minimization problem | MATHEMATICS, APPLIED | primal-dual algorithm | monotone operator splitting | RACHFORD SPLITTING METHOD | SIGNAL RECOVERY | ALGORITHM | DECOMPOSITION | SUM | PRIMAL-DUAL METHOD | COMPOSITE | arbitrary sampling | block-coordinate algorithm | MONOTONE INCLUSIONS | stochastic algorithm | CONSTRUCTION | CONVERGENCE | stochastic quasi-Fejer sequence | fixed-point algorithm | Signal and Image Processing | Computer Science

Journal Article

IEEE Transactions on Signal Processing, ISSN 1053-587X, 07/2003, Volume 51, Issue 7, pp. 1771 - 1782

A block-iterative parallel decomposition method is proposed to solve general quadratic signal recovery problems under convex constraints. The proposed method...

Deconvolution | Signal restoration | Filtering | Computer architecture | Parallel processing | Hilbert space | Quadratic programming | Noise measurement | Signal analysis | Equations | Block-iterative optimization | Signal recovery | Subgradient projection | Convex analysis | CONVEX FEASIBILITY PROBLEMS | NONEXPANSIVE-MAPPINGS | PRODUCT SPACE | IMAGE-RECONSTRUCTION | signal recovery | convex analysis | quadratic programming | LEAST-SQUARES | ENGINEERING, ELECTRICAL & ELECTRONIC | block-iterative optimization | deconvolution | SET-THEORETIC ESTIMATION | HILBERT-SPACE | PROJECTION ALGORITHMS | FIXED-POINTS | subgradient projection | STRONG-CONVERGENCE | Signal processing | Research | Splitting | Computation | Mathematical analysis | Blocking | Mathematical models | Recovery

Deconvolution | Signal restoration | Filtering | Computer architecture | Parallel processing | Hilbert space | Quadratic programming | Noise measurement | Signal analysis | Equations | Block-iterative optimization | Signal recovery | Subgradient projection | Convex analysis | CONVEX FEASIBILITY PROBLEMS | NONEXPANSIVE-MAPPINGS | PRODUCT SPACE | IMAGE-RECONSTRUCTION | signal recovery | convex analysis | quadratic programming | LEAST-SQUARES | ENGINEERING, ELECTRICAL & ELECTRONIC | block-iterative optimization | deconvolution | SET-THEORETIC ESTIMATION | HILBERT-SPACE | PROJECTION ALGORITHMS | FIXED-POINTS | subgradient projection | STRONG-CONVERGENCE | Signal processing | Research | Splitting | Computation | Mathematical analysis | Blocking | Mathematical models | Recovery

Journal Article

Set-Valued and Variational Analysis, ISSN 1877-0533, 6/2012, Volume 20, Issue 2, pp. 307 - 330

..., Lipschitzian, and Parallel-Sum Type Monotone Operators Patrick L. Combettes · Jean-Christophe Pesquet Received: 10 July 2011 / Accepted: 3 August 2011 / Published online...

Monotone inclusion | Nonsmooth convex optimization | Mathematics | Maximal monotone operator | Geometry | 90C25 | Analysis | Splitting algorithm | 49M29 | 49M27 | 49N15 | Parallel sum | 47H05 | Set-valued duality | MATHEMATICS, APPLIED | STABILITY | DECOMPOSITION | PROXIMAL POINT ALGORITHM | VARIATIONAL-INEQUALITIES | RECOVERY | MINIMIZATION | MAPPINGS | Algorithms | Optimization and Control

Monotone inclusion | Nonsmooth convex optimization | Mathematics | Maximal monotone operator | Geometry | 90C25 | Analysis | Splitting algorithm | 49M29 | 49M27 | 49N15 | Parallel sum | 47H05 | Set-valued duality | MATHEMATICS, APPLIED | STABILITY | DECOMPOSITION | PROXIMAL POINT ALGORITHM | VARIATIONAL-INEQUALITIES | RECOVERY | MINIMIZATION | MAPPINGS | Algorithms | Optimization and Control

Journal Article

Multiscale Modeling and Simulation, ISSN 1540-3459, 2005, Volume 4, Issue 4, pp. 1168 - 1200

...MULTISCALE MODEL. SIMUL. c Vol. 4, No. 4, pp. 11681200 SIGNAL RECOVERY BY PROXIMAL FORWARD-BACKWARD SPLITTING PATRICK L. COMBETTES AND VALRIE R. WAJS Abstract...

Proximal Landweber method | Proximity operator | Image decomposition | Iterative soft-thresholding | Signal recovery | Image restoration | Forward-backward algorithm | Multiresolution analysis | Denoising | Inverse problem | inverse problem | forward-backward algorithm | RECONSTRUCTION | ALGORITHM | signal recovery | RESTORATION | PHYSICS, MATHEMATICAL | LEAST-SQUARES | denoising | PROJECTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | image restoration | LINEAR INVERSE PROBLEMS | iterative soft-thresholding | ANALYSE FONCTIONNELLE | proximity operator | proximal Landweber method | multiresolution analysis | TOTAL VARIATION MINIMIZATION | NOISE REMOVAL | image decomposition

Proximal Landweber method | Proximity operator | Image decomposition | Iterative soft-thresholding | Signal recovery | Image restoration | Forward-backward algorithm | Multiresolution analysis | Denoising | Inverse problem | inverse problem | forward-backward algorithm | RECONSTRUCTION | ALGORITHM | signal recovery | RESTORATION | PHYSICS, MATHEMATICAL | LEAST-SQUARES | denoising | PROJECTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | image restoration | LINEAR INVERSE PROBLEMS | iterative soft-thresholding | ANALYSE FONCTIONNELLE | proximity operator | proximal Landweber method | multiresolution analysis | TOTAL VARIATION MINIMIZATION | NOISE REMOVAL | image decomposition

Journal Article

Set-Valued and Variational Analysis, ISSN 1877-0533, 6/2018, Volume 26, Issue 2, pp. 247 - 264

...Set-Valued Var. Anal (2018) 26:247–264 https://doi.org/10.1007/s11228-017-0407-x Perspective Functions: Properties, Constructions, and Examples Patrick L...

Integral functional | Convex optimization | Analysis | Huber function | Probability Theory and Stochastic Processes | Perspective function | Mathematics | Convex function | Berhu function | Statistical divergence | MATHEMATICS, APPLIED | FISHER INFORMATION | CONVEXITY | THIN TORSION RODS | RECONSTRUCTION | DIVERGENCES | ROBUST REGRESSION | DENSITIES | OPTIMIZATION | OPTIMAL TRANSPORT | DERIVATIVE INFORMATION | Information management

Integral functional | Convex optimization | Analysis | Huber function | Probability Theory and Stochastic Processes | Perspective function | Mathematics | Convex function | Berhu function | Statistical divergence | MATHEMATICS, APPLIED | FISHER INFORMATION | CONVEXITY | THIN TORSION RODS | RECONSTRUCTION | DIVERGENCES | ROBUST REGRESSION | DENSITIES | OPTIMIZATION | OPTIMAL TRANSPORT | DERIVATIVE INFORMATION | Information management

Journal Article

SIAM Journal on Optimization, ISSN 1052-6234, 2011, Volume 21, Issue 4, pp. 1230 - 1250

... L. COMBETTES Abstract. The principle underlying this paper is the basic observation that the problem of simultaneously solving a large class of composite...

Monotone inclusion | Minimization algorithm | Convex optimization | Composite operator | Operator splitting | Decomposition | Duality | Fenchel-Rockafellar duality | Monotone operator | Forward-backward-forward algorithm | Mathematics | Optimization and Control

Monotone inclusion | Minimization algorithm | Convex optimization | Composite operator | Operator splitting | Decomposition | Duality | Fenchel-Rockafellar duality | Monotone operator | Forward-backward-forward algorithm | Mathematics | Optimization and Control

Journal Article

Inverse Problems, ISSN 0266-5611, 12/2008, Volume 24, Issue 6, p. 065014

... A proximal decomposition method for solving convex variational inverse problems * Patrick L Combettes 1,3 and Jean-Christophe Pesquet 2 1 UPMC Universit´ e Paris 06...

MATHEMATICS, APPLIED | PHASE | MINIMIZATION | FEASIBILITY | QUADRATIC OPTIMIZATION | IMAGE RECOVERY | THRESHOLDING ALGORITHM | PROJECTIONS | PHYSICS, MATHEMATICAL | NOISE REMOVAL | FIXED-POINTS

MATHEMATICS, APPLIED | PHASE | MINIMIZATION | FEASIBILITY | QUADRATIC OPTIMIZATION | IMAGE RECOVERY | THRESHOLDING ALGORITHM | PROJECTIONS | PHYSICS, MATHEMATICAL | NOISE REMOVAL | FIXED-POINTS

Journal Article

Mathematical Programming, ISSN 0025-5610, 3/2019, Volume 174, Issue 1, pp. 433 - 451

Combettes and Pesquet (SIAM J Optim 25:1221–1248, 2015) investigated the almost sure weak convergence of block-coordinate fixed point algorithms and discussed their applications to nonlinear analysis and optimization...

Theoretical, Mathematical and Computational Physics | Monotone operator splitting | Stochastic algorithm | Mathematics | 90C15 | Block-coordinate algorithm | Mathematical Methods in Physics | Linear convergence | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | Mean-square convergence | 65K10 | Combinatorics | Fixed-point algorithm | 47J25 | 46M10 | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Mathematical optimization | Analysis | Algorithms | Sweeping | Optimization | Nonlinear analysis | Convergence

Theoretical, Mathematical and Computational Physics | Monotone operator splitting | Stochastic algorithm | Mathematics | 90C15 | Block-coordinate algorithm | Mathematical Methods in Physics | Linear convergence | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | Mean-square convergence | 65K10 | Combinatorics | Fixed-point algorithm | 47J25 | 46M10 | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Mathematical optimization | Analysis | Algorithms | Sweeping | Optimization | Nonlinear analysis | Convergence

Journal Article

17.