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...

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

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...

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

Operations Research Letters, ISSN 0167-6377, 03/2018, Volume 46, Issue 2, pp. 159 - 162

In order to accelerate the Douglas–Rachford method we recently developed the circumcentered-reflection method, which provides the closest iterate to the...

Projection | Reflection | Best approximation problem | Douglas–Rachford method | Douglas-Rachford method | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | FEASIBILITY | DOUGLAS-RACHFORD ALGORITHM | SUBSPACES | ALTERNATING PROJECTIONS | Mathematics - Optimization and Control

Projection | Reflection | Best approximation problem | Douglas–Rachford method | Douglas-Rachford method | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | FEASIBILITY | DOUGLAS-RACHFORD ALGORITHM | SUBSPACES | ALTERNATING PROJECTIONS | Mathematics - Optimization and Control

Journal Article

Operations Research Letters, ISSN 0167-6377, 07/2019, Volume 47, Issue 4, pp. 291 - 293

Douglas–Rachford method is a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Weak convergence in this method to a solution...

Weak convergence | Monotone operators | Douglas–Rachford method | Douglas-Rachford method | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SUM | Methods | Algorithms

Weak convergence | Monotone operators | Douglas–Rachford method | Douglas-Rachford method | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SUM | Methods | Algorithms

Journal Article

Journal of optimization theory and applications, ISSN 1573-2878, 2013, Volume 163, Issue 1, pp. 1 - 30

We discuss recent positive experiences applying convex feasibility algorithms of Douglas–Rachford type to highly combinatorial and far from convex problems.

Mathematics | Theory of Computation | Projections | Combinatorial optimization | Optimization | Nonograms | Calculus of Variations and Optimal Control; Optimization | Sudoku | Operations Research/Decision Theory | Douglas–Rachford | Modelling | Feasibility | Applications of Mathematics | Engineering, general | Satisfiability | Reflections | Douglas-Rachford | MATHEMATICS, APPLIED | ALGORITHM | PROJECTION | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | WEAK-CONVERGENCE | Analysis | Methods | Algorithms | Studies | Combinatorics | Combinatorial analysis

Mathematics | Theory of Computation | Projections | Combinatorial optimization | Optimization | Nonograms | Calculus of Variations and Optimal Control; Optimization | Sudoku | Operations Research/Decision Theory | Douglas–Rachford | Modelling | Feasibility | Applications of Mathematics | Engineering, general | Satisfiability | Reflections | Douglas-Rachford | MATHEMATICS, APPLIED | ALGORITHM | PROJECTION | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | WEAK-CONVERGENCE | Analysis | Methods | Algorithms | Studies | Combinatorics | Combinatorial analysis

Journal Article

Applied and Computational Harmonic Analysis, ISSN 1063-5203, 05/2018, Volume 44, Issue 3, pp. 665 - 699

The Fourier-domain Douglas–Rachford (FDR) algorithm is analyzed for phase retrieval with a single random mask. Since the uniqueness of phase retrieval solution...

Geometric convergence | Douglas–Rachford algorithm | Coded diffraction pattern | Spectral gap | Phase retrieval | MATHEMATICS, APPLIED | MAGNITUDE | Douglas-Rachford algorithm | RECONSTRUCTION | SIGNAL RECOVERY | RAY | spectral gap | geometric convergence | coded diffraction pattern

Geometric convergence | Douglas–Rachford algorithm | Coded diffraction pattern | Spectral gap | Phase retrieval | MATHEMATICS, APPLIED | MAGNITUDE | Douglas-Rachford algorithm | RECONSTRUCTION | SIGNAL RECOVERY | RAY | spectral gap | geometric convergence | coded diffraction pattern

Journal Article

Signal processing, ISSN 0165-1684, 2020, Volume 169, p. 107417

•Proximal algorithms with sound convergence are proposed to solve matrix estimation problems.•New proximity operators for spectral functions within Bregman...

Majorization-minimization | Graphical lasso | Matrix optimization | Bregman divergence | Covariance estimation | Douglas–Rachford method | Douglas-Rachford method | SPARSE | ITERATIVE ALGORITHMS | ENGINEERING, ELECTRICAL & ELECTRONIC | COVARIANCE-MATRIX | MINIMIZATION | MODEL SELECTION | CONVERGENCE | Engineering Sciences | Signal and Image processing

Majorization-minimization | Graphical lasso | Matrix optimization | Bregman divergence | Covariance estimation | Douglas–Rachford method | Douglas-Rachford method | SPARSE | ITERATIVE ALGORITHMS | ENGINEERING, ELECTRICAL & ELECTRONIC | COVARIANCE-MATRIX | MINIMIZATION | MODEL SELECTION | CONVERGENCE | Engineering Sciences | Signal and Image processing

Journal Article

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APPLICATION OF PROJECTION ALGORITHMS TO DIFFERENTIAL EQUATIONS: BOUNDARY VALUE PROBLEMS

The ANZIAM journal, ISSN 1446-1811, 01/2019, Volume 61, Issue 1, pp. 23 - 46

The Douglas–Rachford method has been employed successfully to solve many kinds of nonconvex feasibility problems. In particular, recent research has shown...

MATHEMATICS, APPLIED | Douglas-Rachford method | hypersurface | DOUGLAS-RACHFORD ALGORITHM | Newton's method | boundary value problem | LINEAR CONVERGENCE

MATHEMATICS, APPLIED | Douglas-Rachford method | hypersurface | DOUGLAS-RACHFORD ALGORITHM | Newton's method | boundary value problem | LINEAR CONVERGENCE

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 01/2016, Volume 85, Issue 297, pp. 209 - 238

We provide a simple analysis of the Douglas-Rachford splitting algorithm in the context of \ell ^1-regularization and over-relaxation including the dual split...

Asymptotic linear convergence rate | Douglas-Rachford | Peaceman-Rachford | Basis pursuit | Generalized Douglas-Rachford | Relaxation parameter | MATHEMATICS, APPLIED | generalized Douglas-Rachford | relaxation parameter | SPLIT BREGMAN METHOD | ALGORITHM | asymptotic linear convergence rate | SUM

Asymptotic linear convergence rate | Douglas-Rachford | Peaceman-Rachford | Basis pursuit | Generalized Douglas-Rachford | Relaxation parameter | MATHEMATICS, APPLIED | generalized Douglas-Rachford | relaxation parameter | SPLIT BREGMAN METHOD | ALGORITHM | asymptotic linear convergence rate | SUM

Journal Article

The ANZIAM journal, ISSN 1446-1811, 04/2014, Volume 55, Issue 4, pp. 299 - 326

In this paper, we give general recommendations for successful application of the Douglas–Rachford reflection method to convex and nonconvex real matrix...

protein reconstruction | Hadamard matrices | feasibility problems | matrix completion | reflections | Phrases Douglas-Rachford projections | MATHEMATICS, APPLIED | APPROXIMATION | Douglas-Rachford projections | CONVERGENCE | ALTERNATING PROJECTION ALGORITHM | Feasibility | Reflection | Guidelines | Mathematics - Optimization and Control

protein reconstruction | Hadamard matrices | feasibility problems | matrix completion | reflections | Phrases Douglas-Rachford projections | MATHEMATICS, APPLIED | APPROXIMATION | Douglas-Rachford projections | CONVERGENCE | ALTERNATING PROJECTION ALGORITHM | Feasibility | Reflection | Guidelines | Mathematics - Optimization and Control

Journal Article

Journal of mathematical analysis and applications, ISSN 0022-247X, 2015, Volume 421, Issue 1, pp. 1 - 20

We introduce regularity notions for averaged nonexpansive operators. Combined with regularity notions of their fixed point sets, we obtain linear and strong...

Douglas–Rachford algorithm | Averaged nonexpansive mapping | Projection | Nonexpansive operator | Convex feasibility problem | Bounded linear regularity | Douglas-Rachford algorithm | MATHEMATICS | MATHEMATICS, APPLIED | REGULARITY | PROJECTIONS | Analysis | Algorithms

Douglas–Rachford algorithm | Averaged nonexpansive mapping | Projection | Nonexpansive operator | Convex feasibility problem | Bounded linear regularity | Douglas-Rachford algorithm | MATHEMATICS | MATHEMATICS, APPLIED | REGULARITY | PROJECTIONS | Analysis | Algorithms

Journal Article

SIAM Journal on Optimization, ISSN 1052-6234, 2015, Volume 25, Issue 3, pp. 1760 - 1786

Operator splitting schemes are a class of powerful algorithms that solve complicated monotone inclusion and convex optimization problems that are built from...

Forward-backward splitting | Forward-Douglas-Rachford splitting | Primal-dual algorithm | Fixed-point algorithm | Douglas-Rachford splitting | Generalized forward-backward splitting | MATHEMATICS, APPLIED | MONOTONE INCLUSIONS | generalized forward-backward splitting | primal-dual algorithm | forward-backward splitting | forward-Douglas-Rachford splitting | SUM | ALGORITHMS | fixed-point algorithm

Forward-backward splitting | Forward-Douglas-Rachford splitting | Primal-dual algorithm | Fixed-point algorithm | Douglas-Rachford splitting | Generalized forward-backward splitting | MATHEMATICS, APPLIED | MONOTONE INCLUSIONS | generalized forward-backward splitting | primal-dual algorithm | forward-backward splitting | forward-Douglas-Rachford splitting | SUM | ALGORITHMS | fixed-point algorithm

Journal Article

Journal of Approximation Theory, ISSN 0021-9045, 09/2014, Volume 185, pp. 63 - 79

The Douglas–Rachford splitting algorithm is a classical optimization method that has found many applications. When specialized to two normal cone operators, it...

Projection operator | Linear convergence | Method of alternating projections | Firmly nonexpansive | Normal cone operator | Subspaces | Douglas–Rachford splitting method | Friedrichs angle | Secondary | Primary | Douglas-Rachford splitting method | MATHEMATICS | Analysis | Algorithms

Projection operator | Linear convergence | Method of alternating projections | Firmly nonexpansive | Normal cone operator | Subspaces | Douglas–Rachford splitting method | Friedrichs angle | Secondary | Primary | Douglas-Rachford splitting method | MATHEMATICS | Analysis | Algorithms

Journal Article

Mathematical programming, ISSN 1436-4646, 2016, Volume 164, Issue 1-2, pp. 263 - 284

The Douglas-Rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators. The behaviour of the...

Maximally monotone operator | Weak convergence | Sum problem | Douglas–Rachford algorithm | Inconsistent case | Paramonotone operator | Nonexpansive mapping | Attouch–Théra duality | FEASIBILITY PROBLEMS | MATHEMATICS, APPLIED | MAXIMAL MONOTONE-OPERATORS | Attouch-Thera duality | SUM | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford algorithm | CONSTRUCTION | CONVERGENCE | OPTIMIZATION | SUBSPACES | LINEAR-OPERATORS | HILBERT-SPACE | PROJECTIVE SPLITTING METHODS | Algorithms | Operators | Splitting | Approximation | Mathematical analysis | Clusters | Feasibility | Behavior | Convergence

Maximally monotone operator | Weak convergence | Sum problem | Douglas–Rachford algorithm | Inconsistent case | Paramonotone operator | Nonexpansive mapping | Attouch–Théra duality | FEASIBILITY PROBLEMS | MATHEMATICS, APPLIED | MAXIMAL MONOTONE-OPERATORS | Attouch-Thera duality | SUM | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford algorithm | CONSTRUCTION | CONVERGENCE | OPTIMIZATION | SUBSPACES | LINEAR-OPERATORS | HILBERT-SPACE | PROJECTIVE SPLITTING METHODS | Algorithms | Operators | Splitting | Approximation | Mathematical analysis | Clusters | Feasibility | Behavior | Convergence

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. When an...

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

SIAM journal on optimization, ISSN 1095-7189, 2013, Volume 23, Issue 4, pp. 2541 - 2565

In this paper we propose two different primal-dual splitting algorithms for solving inclusions involving mixtures of composite and parallel-sum type monotone...

Fenchel duality | Monotone inclusion | Convex optimization | Douglas-Rachford splitting | MATHEMATICS, APPLIED | SPLITTING ALGORITHM | convex optimization | GENERALIZED HERON PROBLEM | monotone inclusion

Fenchel duality | Monotone inclusion | Convex optimization | Douglas-Rachford splitting | MATHEMATICS, APPLIED | SPLITTING ALGORITHM | convex optimization | GENERALIZED HERON PROBLEM | monotone inclusion

Journal Article

Operations Research Letters, ISSN 0167-6377, 11/2018, Volume 46, Issue 6, pp. 585 - 587

Aragón Artacho and Campoy recently proposed a new method for computing the projection onto the intersection of two closed convex sets in Hilbert space;...

Maximally monotone operator | Douglas–Rachford algorithm | Resolvent average | Convex function | Proximal average | Aragón Artacho–Campoy algorithm | Aragon Artacho-Campoy algorithm | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford algorithm | RESOLVENT | AVERAGE | SUM | Electrical engineering | Analysis | Algorithms

Maximally monotone operator | Douglas–Rachford algorithm | Resolvent average | Convex function | Proximal average | Aragón Artacho–Campoy algorithm | Aragon Artacho-Campoy algorithm | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford algorithm | RESOLVENT | AVERAGE | SUM | Electrical engineering | Analysis | Algorithms

Journal Article

Optimization letters, ISSN 1862-4480, 2018, Volume 13, Issue 4, pp. 717 - 740

We shed light on the structure of the three-operator version of the forward-Douglas-Rachford splitting algorithm for finding a zero of a sum of maximally...

Douglas–Rachford splitting | Nonsmooth convex optimization | Proximal splitting | Monotone operator splitting | Forward–backward splitting | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford splitting | Forward-backward splitting | Algorithms

Douglas–Rachford splitting | Nonsmooth convex optimization | Proximal splitting | Monotone operator splitting | Forward–backward splitting | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford splitting | Forward-backward splitting | Algorithms

Journal Article

Journal of Global Optimization, ISSN 0925-5001, 1/2019, Volume 73, Issue 1, pp. 83 - 112

The Douglas–Rachford projection algorithm is an iterative method used to find a point in the intersection of closed constraint sets. The algorithm has been...

Global convergence | Stability | Newton’s method | Projection | 37B25 | Mathematics | 90C26 | Optimization | Graph of a function | Linear convergence | 47H10 | Method of alternating projections | Operations Research/Decision Theory | Zero of a function | Douglas–Rachford algorithm | Nonconvex set | Computer Science, general | Feasibility problem | Real Functions | Lyapunov function | CONVEX FEASIBILITY PROBLEMS | MATHEMATICS, APPLIED | FINITE CONVERGENCE | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford algorithm | Newton's method | ALTERNATING PROJECTIONS | Algorithms | Convexity | Convergence

Global convergence | Stability | Newton’s method | Projection | 37B25 | Mathematics | 90C26 | Optimization | Graph of a function | Linear convergence | 47H10 | Method of alternating projections | Operations Research/Decision Theory | Zero of a function | Douglas–Rachford algorithm | Nonconvex set | Computer Science, general | Feasibility problem | Real Functions | Lyapunov function | CONVEX FEASIBILITY PROBLEMS | MATHEMATICS, APPLIED | FINITE CONVERGENCE | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Douglas-Rachford algorithm | Newton's method | ALTERNATING PROJECTIONS | Algorithms | Convexity | Convergence

Journal Article